2. Career development process is complex and rapidly evolving and new theories are continually developing presenting challenges to traditional understandings. Discuss why an understanding of career development processes is critical to management, employee and organizational success.

2. Career development process is complex and rapidly evolving and new theories are continually developing presenting challenges to traditional understandings. Discuss why an understanding of career development processes is critical to management, employee and organizational success.

Studies are at the present extrapolative huge employment income in … Read More...
4. Name a big idea (major concept) in your subject area and write a one paragraph rationale for why students should learn it.

4. Name a big idea (major concept) in your subject area and write a one paragraph rationale for why students should learn it.

Computer education improves students’ investigation skill by encouraging them to … Read More...
Que 1: in women who suffer from migraine …………………. are classified menstrual migraines, which tend to be more severe and longer lasting . a) 5% – 10% b) 45% – 55% c) 20% – 50% d) 65%-75% Que 2: why are the women on average, slightly shorter than men a) They have fat then man which contributes to stature b) Their long bones are sealed and stop growing earlier than men c) Their brains are somewhat smaller than man’s brain d) Their brains are somewhat larger than man, brain que 3: menopause frequently occurs between ………………..age of year a) 25-30 b) 45-55 c) 35-40 d) 65-75 que 4: this hormone causes enlargement of the larynx and an increase in the length and thickness of the vocal cords. A) estrogen 2) cholesterol 3) progesterone 4) testosterone Que 5: the reproductive cycle includes which of the following interconnected sets of events a) Ovarian cycle b) Urinary cycle c) Placental cycle d) Female prostate cycle Que 6: high level of circulating progesterone have been associated with : a) Excessive milk production b) Ovarian cancer c) Inability to breast feed a new born child d) Pregnancy Que 7: although variations exist, ovulation typically occurs on the …………day before mensuaration a) 1st b) 14th c) 7th d) 28th Que 8: LH stimulates interstitial cells a) To decrease GnRH b) To produce FSH c) To produce testosterones d) To produce sperm Que 9:what region of the uterus is shed during menstration? a) Stratum basalis of the myometrium b) Stratum basalis of the endometrium c) Stratum functionalis of the endometrium d) Perimetrium Que 10: the phenomenon is which women living in close proximity tend to menstruate at approximately the time is called. a) Precocious puberty b) Menstrual synchrony’ c) Delayed puberty d) Ovarian synchrony Que 11.studies have shown that healthy menstruating women a) Should not participate in sports b) Often feel ill or weak when exercising c) Are able to safety engage in athletic activities d) Can contaminate others and should not engage in contacts sports. Que 12: which term below describes a chemical that resembles steroid hormones and posses threat to maintain homeostasis. a) Androgens b) Prostaglandins c) Endocrine disruptors d) All of the above Que 13: one of the primary function of ……….is preparing and sustaining the uterus of pregnancy a) Testosterone b) Progesterone c) Estradiol d) inhibin Que 14: typically ovulation occurs a) at the end of the uterine phase b) at the start of follicular phase c) during an increase of LH in the ovarian cycle d) at the middle of the luteal phase

Que 1: in women who suffer from migraine …………………. are classified menstrual migraines, which tend to be more severe and longer lasting . a) 5% – 10% b) 45% – 55% c) 20% – 50% d) 65%-75% Que 2: why are the women on average, slightly shorter than men a) They have fat then man which contributes to stature b) Their long bones are sealed and stop growing earlier than men c) Their brains are somewhat smaller than man’s brain d) Their brains are somewhat larger than man, brain que 3: menopause frequently occurs between ………………..age of year a) 25-30 b) 45-55 c) 35-40 d) 65-75 que 4: this hormone causes enlargement of the larynx and an increase in the length and thickness of the vocal cords. A) estrogen 2) cholesterol 3) progesterone 4) testosterone Que 5: the reproductive cycle includes which of the following interconnected sets of events a) Ovarian cycle b) Urinary cycle c) Placental cycle d) Female prostate cycle Que 6: high level of circulating progesterone have been associated with : a) Excessive milk production b) Ovarian cancer c) Inability to breast feed a new born child d) Pregnancy Que 7: although variations exist, ovulation typically occurs on the …………day before mensuaration a) 1st b) 14th c) 7th d) 28th Que 8: LH stimulates interstitial cells a) To decrease GnRH b) To produce FSH c) To produce testosterones d) To produce sperm Que 9:what region of the uterus is shed during menstration? a) Stratum basalis of the myometrium b) Stratum basalis of the endometrium c) Stratum functionalis of the endometrium d) Perimetrium Que 10: the phenomenon is which women living in close proximity tend to menstruate at approximately the time is called. a) Precocious puberty b) Menstrual synchrony’ c) Delayed puberty d) Ovarian synchrony Que 11.studies have shown that healthy menstruating women a) Should not participate in sports b) Often feel ill or weak when exercising c) Are able to safety engage in athletic activities d) Can contaminate others and should not engage in contacts sports. Que 12: which term below describes a chemical that resembles steroid hormones and posses threat to maintain homeostasis. a) Androgens b) Prostaglandins c) Endocrine disruptors d) All of the above Que 13: one of the primary function of ……….is preparing and sustaining the uterus of pregnancy a) Testosterone b) Progesterone c) Estradiol d) inhibin Que 14: typically ovulation occurs a) at the end of the uterine phase b) at the start of follicular phase c) during an increase of LH in the ovarian cycle d) at the middle of the luteal phase

Read this article and answer this question in 2 pages : Answers should be from the below article only. What is the difference between “standards-based” and “standards-embedded” curriculum? what are the curricular implications of this difference? Article: In 2007, at the dawn of 21st century in education, it is impossible to talk about teaching, curriculum, schools, or education without discussing standards . standards-based v. standards-embedded curriculum We are in an age of accountability where our success as educators is determined by individual and group mastery of specific standards dem- onstrated by standardized test per- formance. Even before No Child Left Behind (NCLB), standards and measures were used to determine if schools and students were success- ful (McClure, 2005). But, NCLB has increased the pace, intensity, and high stakes of this trend. Gifted and talented students and their teach- ers are significantly impacted by these local or state proficiency stan- dards and grade-level assessments (VanTassel-Baska & Stambaugh, 2006). This article explores how to use these standards in the develop- ment of high-quality curriculum for gifted students. NCLB, High-Stakes State Testing, and Standards- Based Instruction There are a few potentially positive outcomes of this evolution to public accountability. All stakeholders have had to ask themselves, “Are students learning? If so, what are they learning and how do we know?” In cases where we have been allowed to thoughtfully evaluate curriculum and instruction, we have also asked, “What’s worth learning?” “When’s the best time to learn it?” and “Who needs to learn it?” Even though state achievement tests are only a single measure, citizens are now offered a yardstick, albeit a nar- row one, for comparing communities, schools, and in some cases, teachers. Some testing reports allow teachers to identify for parents what their chil- dren can do and what they can not do. Testing also has focused attention on the not-so-new observations that pov- erty, discrimination and prejudices, and language proficiency impacts learning. With enough ceiling (e.g., above-grade-level assessments), even gifted students’ actual achievement and readiness levels can be identi- fied and provide a starting point for appropriately differentiated instruc- tion (Tomlinson, 2001). Unfortunately, as a veteran teacher for more than three decades and as a teacher-educator, my recent observa- tions of and conversations with class- room and gifted teachers have usually revealed negative outcomes. For gifted children, their actual achievement level is often unrecognized by teachers because both the tests and the reporting of the results rarely reach above the student’s grade-level placement. Assessments also focus on a huge number of state stan- dards for a given school year that cre- ate “overload” (Tomlinson & McTighe, 2006) and have a devastating impact on the development and implementation of rich and relevant curriculum and instruction. In too many scenarios, I see teachers teach- ing directly to the test. And, in the worst cases, some teachers actually teach The Test. In those cases, The Test itself becomes the curriculum. Consistently I hear, “Oh, I used to teach a great unit on ________ but I can’t do it any- more because I have to teach the standards.” Or, “I have to teach my favorite units in April and May after testing.” If the outcomes can’t be boiled down to simple “I can . . .” state- ments that can be posted on a school’s walls, then teachers seem to omit poten- tially meaningful learning opportunities from the school year. In many cases, real education and learning are being trivial- ized. We seem to have lost sight of the more significant purpose of teaching and learning: individual growth and develop- ment. We also have surrendered much of the joy of learning, as the incidentals, the tangents, the “bird walks” are cut short or elimi- nated because teachers hear the con- stant ticking clock of the countdown to the state test and feel the pressure of the way-too-many standards that have to be covered in a mere 180 school days. The accountability movement has pushed us away from seeing the whole child: “Students are not machines, as the standards movement suggests; they are volatile, complicated, and paradoxical” (Cookson, 2001, p. 42). How does this impact gifted chil- dren? In many heterogeneous class- rooms, teachers have retreated to traditional subject delineations and traditional instruction in an effort to ensure direct standards-based instruc- tion even though “no solid basis exists in the research literature for the ways we currently develop, place, and align educational standards in school cur- ricula” (Zenger & Zenger, 2002, p. 212). Grade-level standards are often particularly inappropriate for the gifted and talented whose pace of learning, achievement levels, and depth of knowledge are significantly beyond their chronological peers. A broad-based, thematically rich, and challenging curriculum is the heart of education for the gifted. Virgil Ward, one of the earliest voices for a differen- tial education for the gifted, said, “It is insufficient to consider the curriculum for the gifted in terms of traditional subjects and instructional processes” (Ward, 1980, p. 5). VanTassel-Baska Standards-Based v. Standards-Embedded Curriculum gifted child today 45 Standards-Based v. Standards-Embedded Curriculum and Stambaugh (2006) described three dimensions of successful curriculum for gifted students: content mastery, pro- cess and product, and epistemological concept, “understanding and appre- ciating systems of knowledge rather than individual elements of those systems” (p. 9). Overemphasis on testing and grade-level standards limits all three and therefore limits learning for gifted students. Hirsch (2001) concluded that “broad gen- eral knowledge is the best entrée to deep knowledge” (p. 23) and that it is highly correlated with general ability to learn. He continued, “the best way to learn a subject is to learn its gen- eral principles and to study an ample number of diverse examples that illustrate those principles” (Hirsch, 2001, p. 23). Principle-based learn- ing applies to both gifted and general education children. In order to meet the needs of gifted and general education students, cur- riculum should be differentiated in ways that are relevant and engaging. Curriculum content, processes, and products should provide challenge, depth, and complexity, offering multiple opportunities for problem solving, creativity, and exploration. In specific content areas, the cur- riculum should reflect the elegance and sophistication unique to the discipline. Even with this expanded view of curriculum in mind, we still must find ways to address the current reality of state standards and assess- ments. Standards-Embedded Curriculum How can educators address this chal- lenge? As in most things, a change of perspective can be helpful. Standards- based curriculum as described above should be replaced with standards- embedded curriculum. Standards- embedded curriculum begins with broad questions and topics, either discipline specific or interdisciplinary. Once teachers have given thoughtful consideration to relevant, engaging, and important content and the con- nections that support meaning-making (Jensen, 1998), they next select stan- dards that are relevant to this content and to summative assessments. This process is supported by the backward planning advocated in Understanding by Design by Wiggins and McTighe (2005) and its predecessors, as well as current thinkers in other fields, such as Covey (Tomlinson & McTighe, 2006). It is a critical component of differenti- ating instruction for advanced learners (Tomlinson, 2001) and a significant factor in the Core Parallel in the Parallel Curriculum Model (Tomlinson et al., 2002). Teachers choose from standards in multiple disciplines at both above and below grade level depending on the needs of the students and the classroom or program structure. Preassessment data and the results of prior instruc- tion also inform this process of embed- ding appropriate standards. For gifted students, this formative assessment will result in “more advanced curricula available at younger ages, ensuring that all levels of the standards are traversed in the process” (VanTassel-Baska & Little, 2003, p. 3). Once the essential questions, key content, and relevant standards are selected and sequenced, they are embedded into a coherent unit design and instructional decisions (grouping, pacing, instructional methodology) can be made. For gifted students, this includes the identification of appropri- ate resources, often including advanced texts, mentors, and independent research, as appropriate to the child’s developmental level and interest. Applying Standards- Embedded Curriculum What does this look like in practice? In reading the possible class- room applications below, consider these three Ohio Academic Content Standards for third grade: 1. Math: “Read thermometers in both Fahrenheit and Celsius scales” (“Academic Content Standards: K–12 Mathematics,” n.d., p. 71). 2. Social Studies: “Compare some of the cultural practices and products of various groups of people who have lived in the local community including artistic expression, religion, language, and food. Compare the cultural practices and products of the local community with those of other communities in Ohio, the United States, and countries of the world” (Academic Content Standards: K–12 Social Studies, n.d., p. 122). 3. Life Science: “Observe and explore how fossils provide evidence about animals that lived long ago and the nature of the environment at that time” (Academic Content Standards: K–12 Science, n.d., p. 57). When students are fortunate to have a teacher who is dedicated to helping all of them make good use of their time, the gifted may have a preassessment opportunity where they can demonstrate their familiarity with the content and potential mastery of a standard at their grade level. Students who pass may get to read by them- selves for the brief period while the rest of the class works on the single outcome. Sometimes more experienced teachers will create opportunities for gifted and advanced students Standards-Based v. Standards-Embedded Curriculum to work on a standard in the same domain or strand at the next higher grade level (i.e., accelerate through the standards). For example, a stu- dent might be able to work on a Life Science standard for fourth grade that progresses to other communities such as ecosystems. These above-grade-level standards can provide rich material for differentiation, advanced problem solving, and more in-depth curriculum integration. In another classroom scenario, a teacher may focus on the math stan- dard above, identifying the standard number on his lesson plan. He creates or collects paper thermometers, some showing measurement in Celsius and some in Fahrenheit. He also has some real thermometers. He demonstrates thermometer use with boiling water and with freezing water and reads the different temperatures. Students complete a worksheet that has them read thermometers in Celsius and Fahrenheit. The more advanced students may learn how to convert between the two scales. Students then practice with several questions on the topic that are similar in structure and content to those that have been on past proficiency tests. They are coached in how to answer them so that the stan- dard, instruction, formative assess- ment, and summative assessment are all aligned. Then, each student writes a statement that says, “I can read a thermometer using either Celsius or Fahrenheit scales.” Both of these examples describe a standards-based environment, where the starting point is the standard. Direct instruction to that standard is followed by an observable student behavior that demonstrates specific mastery of that single standard. The standard becomes both the start- ing point and the ending point of the curriculum. Education, rather than opening up a student’s mind, becomes a series of closed links in a chain. Whereas the above lessons may be differentiated to some extent, they have no context; they may relate only to the next standard on the list, such as, “Telling time to the nearest minute and finding elapsed time using a cal- endar or a clock.” How would a “standards-embed- ded” model of curriculum design be different? It would begin with the development of an essential ques- tion such as, “Who or what lived here before me? How were they different from me? How were they the same? How do we know?” These questions might be more relevant to our con- temporary highly mobile students. It would involve place and time. Using this intriguing line of inquiry, students might work on the social studies stan- dard as part of the study of their home- town, their school, or even their house or apartment. Because where people live and what they do is influenced by the weather, students could look into weather patterns of their area and learn how to measure temperature using a Fahrenheit scale so they could see if it is similar now to what it was a century ago. Skipping ahead to consideration of the social studies standard, students could then choose another country, preferably one that uses Celsius, and do the same investigation of fossils, communities, and the like. Students could complete a weather comparison, looking at the temperature in Celsius as people in other parts of the world, such as those in Canada, do. Thus, learning is contextualized and connected, dem- onstrating both depth and complexity. This approach takes a lot more work and time. It is a sophisticated integrated view of curriculum devel- opment and involves in-depth knowl- edge of the content areas, as well as an understanding of the scope and sequence of the standards in each dis- cipline. Teachers who develop vital single-discipline units, as well as inter- disciplinary teaching units, begin with a central topic surrounded by subtopics and connections to other areas. Then they connect important terms, facts, or concepts to the subtopics. Next, the skilled teacher/curriculum devel- oper embeds relevant, multileveled standards and objectives appropriate to a given student or group of stu- dents into the unit. Finally, teachers select the instructional strategies and develop student assessments. These assessments include, but are not lim- ited to, the types of questions asked on standardized and state assessments. Comparing Standards- Based and Standards- Embedded Curriculum Design Following is an articulation of the differences between standards-based and standards-embedded curriculum design. (See Figure 1.) 1. The starting point. Standards- based curriculum begins with the grade-level standard and the underlying assumption that every student needs to master that stan- dard at that moment in time. In standards-embedded curriculum, the multifaceted essential ques- tion and students’ needs are the starting points. 2. Preassessment. In standards- based curriculum and teaching, if a preassessment is provided, it cov- ers a single standard or two. In a standards-embedded curriculum, preassessment includes a broader range of grade-level and advanced standards, as well as students’ knowledge of surrounding content such as background experiences with the subject, relevant skills (such as reading and writing), and continued on page ?? even learning style or interests. gifted child today 47 Standards-Based v. Standards-Embedded Curriculum Standards Based Standards Embedded Starting Points The grade-level standard. Whole class’ general skill level Essential questions and content relevant to individual students and groups. Preassessment Targeted to a single grade-level standard. Short-cycle assessments. Background knowledge. Multiple grade-level standards from multiple areas connected by the theme of the unit. Includes annual learning style and interest inventories. Acceleration/ Enrichment To next grade-level standard in the same strand. To above-grade-level standards, as well as into broader thematically connected content. Language Arts Divided into individual skills. Reading and writing skills often separated from real-world relevant contexts. The language arts are embedded in all units and themes and connected to differentiated processes and products across all content areas. Instruction Lesson planning begins with the standard as the objective. Sequential direct instruction progresses through the standards in each content area separately. Strategies are selected to introduce, practice, and demonstrate mastery of all grade-level standards in all content areas in one school year. Lesson planning begins with essential questions, topics, and significant themes. Integrated instruction is designed around connections among content areas and embeds all relevant standards. Assessment Format modeled after the state test. Variety of assessments including questions similar to the state test format. Teacher Role Monitor of standards mastery. Time manager. Facilitator of instructional design and student engagement with learning, as well as assessor of achievement. Student Self- Esteem “I can . . .” statements. Star Charts. Passing “the test.” Completed projects/products. Making personal connections to learning and the theme/topic. Figure 1. Standards based v. standards-embedded instruction and gifted students. and the potential political outcry of “stepping on the toes” of the next grade’s teacher. Few classroom teachers have been provided with the in-depth professional develop- ment and understanding of curric- ulum compacting that would allow them to implement this effectively. In standards-embedded curricu- lum, enrichment and extensions of learning are more possible and more interesting because ideas, top- ics, and questions lend themselves more easily to depth and complex- ity than isolated skills. 4. Language arts. In standards- based classrooms, the language arts have been redivided into sepa- rate skills, with reading separated from writing, and writing sepa- rated from grammar. To many concrete thinkers, whole-language approaches seem antithetical to teaching “to the standards.” In a standards-embedded classroom, integrated language arts skills (reading, writing, listening, speak- ing, presenting, and even pho- nics) are embedded into the study of every unit. Especially for the gifted, the communication and language arts are essential, regard- less of domain-specific talents (Ward, 1980) and should be com- ponents of all curriculum because they are the underpinnings of scholarship in all areas. 5. Instruction. A standards-based classroom lends itself to direct instruction and sequential pro- gression from one standard to the next. A standards-embedded class- room requires a variety of more open-ended instructional strate- gies and materials that extend and diversify learning rather than focus it narrowly. Creativity and differ- entiation in instruction and stu- dent performance are supported more effectively in a standards- embedded approach. 6. Assessment. A standards-based classroom uses targeted assess- ments focused on the structure and content of questions on the externally imposed standardized test (i.e., proficiency tests). A stan- dards-embedded classroom lends itself to greater use of authentic assessment and differentiated 3. Acceleration/Enrichment. In a standards-based curriculum, the narrow definition of the learning outcome (a test item) often makes acceleration or curriculum compact- ing the only path for differentiating instruction for gifted, talented, and/ or advanced learners. This rarely happens, however, because of lack of materials, knowledge, o

Read this article and answer this question in 2 pages : Answers should be from the below article only. What is the difference between “standards-based” and “standards-embedded” curriculum? what are the curricular implications of this difference? Article: In 2007, at the dawn of 21st century in education, it is impossible to talk about teaching, curriculum, schools, or education without discussing standards . standards-based v. standards-embedded curriculum We are in an age of accountability where our success as educators is determined by individual and group mastery of specific standards dem- onstrated by standardized test per- formance. Even before No Child Left Behind (NCLB), standards and measures were used to determine if schools and students were success- ful (McClure, 2005). But, NCLB has increased the pace, intensity, and high stakes of this trend. Gifted and talented students and their teach- ers are significantly impacted by these local or state proficiency stan- dards and grade-level assessments (VanTassel-Baska & Stambaugh, 2006). This article explores how to use these standards in the develop- ment of high-quality curriculum for gifted students. NCLB, High-Stakes State Testing, and Standards- Based Instruction There are a few potentially positive outcomes of this evolution to public accountability. All stakeholders have had to ask themselves, “Are students learning? If so, what are they learning and how do we know?” In cases where we have been allowed to thoughtfully evaluate curriculum and instruction, we have also asked, “What’s worth learning?” “When’s the best time to learn it?” and “Who needs to learn it?” Even though state achievement tests are only a single measure, citizens are now offered a yardstick, albeit a nar- row one, for comparing communities, schools, and in some cases, teachers. Some testing reports allow teachers to identify for parents what their chil- dren can do and what they can not do. Testing also has focused attention on the not-so-new observations that pov- erty, discrimination and prejudices, and language proficiency impacts learning. With enough ceiling (e.g., above-grade-level assessments), even gifted students’ actual achievement and readiness levels can be identi- fied and provide a starting point for appropriately differentiated instruc- tion (Tomlinson, 2001). Unfortunately, as a veteran teacher for more than three decades and as a teacher-educator, my recent observa- tions of and conversations with class- room and gifted teachers have usually revealed negative outcomes. For gifted children, their actual achievement level is often unrecognized by teachers because both the tests and the reporting of the results rarely reach above the student’s grade-level placement. Assessments also focus on a huge number of state stan- dards for a given school year that cre- ate “overload” (Tomlinson & McTighe, 2006) and have a devastating impact on the development and implementation of rich and relevant curriculum and instruction. In too many scenarios, I see teachers teach- ing directly to the test. And, in the worst cases, some teachers actually teach The Test. In those cases, The Test itself becomes the curriculum. Consistently I hear, “Oh, I used to teach a great unit on ________ but I can’t do it any- more because I have to teach the standards.” Or, “I have to teach my favorite units in April and May after testing.” If the outcomes can’t be boiled down to simple “I can . . .” state- ments that can be posted on a school’s walls, then teachers seem to omit poten- tially meaningful learning opportunities from the school year. In many cases, real education and learning are being trivial- ized. We seem to have lost sight of the more significant purpose of teaching and learning: individual growth and develop- ment. We also have surrendered much of the joy of learning, as the incidentals, the tangents, the “bird walks” are cut short or elimi- nated because teachers hear the con- stant ticking clock of the countdown to the state test and feel the pressure of the way-too-many standards that have to be covered in a mere 180 school days. The accountability movement has pushed us away from seeing the whole child: “Students are not machines, as the standards movement suggests; they are volatile, complicated, and paradoxical” (Cookson, 2001, p. 42). How does this impact gifted chil- dren? In many heterogeneous class- rooms, teachers have retreated to traditional subject delineations and traditional instruction in an effort to ensure direct standards-based instruc- tion even though “no solid basis exists in the research literature for the ways we currently develop, place, and align educational standards in school cur- ricula” (Zenger & Zenger, 2002, p. 212). Grade-level standards are often particularly inappropriate for the gifted and talented whose pace of learning, achievement levels, and depth of knowledge are significantly beyond their chronological peers. A broad-based, thematically rich, and challenging curriculum is the heart of education for the gifted. Virgil Ward, one of the earliest voices for a differen- tial education for the gifted, said, “It is insufficient to consider the curriculum for the gifted in terms of traditional subjects and instructional processes” (Ward, 1980, p. 5). VanTassel-Baska Standards-Based v. Standards-Embedded Curriculum gifted child today 45 Standards-Based v. Standards-Embedded Curriculum and Stambaugh (2006) described three dimensions of successful curriculum for gifted students: content mastery, pro- cess and product, and epistemological concept, “understanding and appre- ciating systems of knowledge rather than individual elements of those systems” (p. 9). Overemphasis on testing and grade-level standards limits all three and therefore limits learning for gifted students. Hirsch (2001) concluded that “broad gen- eral knowledge is the best entrée to deep knowledge” (p. 23) and that it is highly correlated with general ability to learn. He continued, “the best way to learn a subject is to learn its gen- eral principles and to study an ample number of diverse examples that illustrate those principles” (Hirsch, 2001, p. 23). Principle-based learn- ing applies to both gifted and general education children. In order to meet the needs of gifted and general education students, cur- riculum should be differentiated in ways that are relevant and engaging. Curriculum content, processes, and products should provide challenge, depth, and complexity, offering multiple opportunities for problem solving, creativity, and exploration. In specific content areas, the cur- riculum should reflect the elegance and sophistication unique to the discipline. Even with this expanded view of curriculum in mind, we still must find ways to address the current reality of state standards and assess- ments. Standards-Embedded Curriculum How can educators address this chal- lenge? As in most things, a change of perspective can be helpful. Standards- based curriculum as described above should be replaced with standards- embedded curriculum. Standards- embedded curriculum begins with broad questions and topics, either discipline specific or interdisciplinary. Once teachers have given thoughtful consideration to relevant, engaging, and important content and the con- nections that support meaning-making (Jensen, 1998), they next select stan- dards that are relevant to this content and to summative assessments. This process is supported by the backward planning advocated in Understanding by Design by Wiggins and McTighe (2005) and its predecessors, as well as current thinkers in other fields, such as Covey (Tomlinson & McTighe, 2006). It is a critical component of differenti- ating instruction for advanced learners (Tomlinson, 2001) and a significant factor in the Core Parallel in the Parallel Curriculum Model (Tomlinson et al., 2002). Teachers choose from standards in multiple disciplines at both above and below grade level depending on the needs of the students and the classroom or program structure. Preassessment data and the results of prior instruc- tion also inform this process of embed- ding appropriate standards. For gifted students, this formative assessment will result in “more advanced curricula available at younger ages, ensuring that all levels of the standards are traversed in the process” (VanTassel-Baska & Little, 2003, p. 3). Once the essential questions, key content, and relevant standards are selected and sequenced, they are embedded into a coherent unit design and instructional decisions (grouping, pacing, instructional methodology) can be made. For gifted students, this includes the identification of appropri- ate resources, often including advanced texts, mentors, and independent research, as appropriate to the child’s developmental level and interest. Applying Standards- Embedded Curriculum What does this look like in practice? In reading the possible class- room applications below, consider these three Ohio Academic Content Standards for third grade: 1. Math: “Read thermometers in both Fahrenheit and Celsius scales” (“Academic Content Standards: K–12 Mathematics,” n.d., p. 71). 2. Social Studies: “Compare some of the cultural practices and products of various groups of people who have lived in the local community including artistic expression, religion, language, and food. Compare the cultural practices and products of the local community with those of other communities in Ohio, the United States, and countries of the world” (Academic Content Standards: K–12 Social Studies, n.d., p. 122). 3. Life Science: “Observe and explore how fossils provide evidence about animals that lived long ago and the nature of the environment at that time” (Academic Content Standards: K–12 Science, n.d., p. 57). When students are fortunate to have a teacher who is dedicated to helping all of them make good use of their time, the gifted may have a preassessment opportunity where they can demonstrate their familiarity with the content and potential mastery of a standard at their grade level. Students who pass may get to read by them- selves for the brief period while the rest of the class works on the single outcome. Sometimes more experienced teachers will create opportunities for gifted and advanced students Standards-Based v. Standards-Embedded Curriculum to work on a standard in the same domain or strand at the next higher grade level (i.e., accelerate through the standards). For example, a stu- dent might be able to work on a Life Science standard for fourth grade that progresses to other communities such as ecosystems. These above-grade-level standards can provide rich material for differentiation, advanced problem solving, and more in-depth curriculum integration. In another classroom scenario, a teacher may focus on the math stan- dard above, identifying the standard number on his lesson plan. He creates or collects paper thermometers, some showing measurement in Celsius and some in Fahrenheit. He also has some real thermometers. He demonstrates thermometer use with boiling water and with freezing water and reads the different temperatures. Students complete a worksheet that has them read thermometers in Celsius and Fahrenheit. The more advanced students may learn how to convert between the two scales. Students then practice with several questions on the topic that are similar in structure and content to those that have been on past proficiency tests. They are coached in how to answer them so that the stan- dard, instruction, formative assess- ment, and summative assessment are all aligned. Then, each student writes a statement that says, “I can read a thermometer using either Celsius or Fahrenheit scales.” Both of these examples describe a standards-based environment, where the starting point is the standard. Direct instruction to that standard is followed by an observable student behavior that demonstrates specific mastery of that single standard. The standard becomes both the start- ing point and the ending point of the curriculum. Education, rather than opening up a student’s mind, becomes a series of closed links in a chain. Whereas the above lessons may be differentiated to some extent, they have no context; they may relate only to the next standard on the list, such as, “Telling time to the nearest minute and finding elapsed time using a cal- endar or a clock.” How would a “standards-embed- ded” model of curriculum design be different? It would begin with the development of an essential ques- tion such as, “Who or what lived here before me? How were they different from me? How were they the same? How do we know?” These questions might be more relevant to our con- temporary highly mobile students. It would involve place and time. Using this intriguing line of inquiry, students might work on the social studies stan- dard as part of the study of their home- town, their school, or even their house or apartment. Because where people live and what they do is influenced by the weather, students could look into weather patterns of their area and learn how to measure temperature using a Fahrenheit scale so they could see if it is similar now to what it was a century ago. Skipping ahead to consideration of the social studies standard, students could then choose another country, preferably one that uses Celsius, and do the same investigation of fossils, communities, and the like. Students could complete a weather comparison, looking at the temperature in Celsius as people in other parts of the world, such as those in Canada, do. Thus, learning is contextualized and connected, dem- onstrating both depth and complexity. This approach takes a lot more work and time. It is a sophisticated integrated view of curriculum devel- opment and involves in-depth knowl- edge of the content areas, as well as an understanding of the scope and sequence of the standards in each dis- cipline. Teachers who develop vital single-discipline units, as well as inter- disciplinary teaching units, begin with a central topic surrounded by subtopics and connections to other areas. Then they connect important terms, facts, or concepts to the subtopics. Next, the skilled teacher/curriculum devel- oper embeds relevant, multileveled standards and objectives appropriate to a given student or group of stu- dents into the unit. Finally, teachers select the instructional strategies and develop student assessments. These assessments include, but are not lim- ited to, the types of questions asked on standardized and state assessments. Comparing Standards- Based and Standards- Embedded Curriculum Design Following is an articulation of the differences between standards-based and standards-embedded curriculum design. (See Figure 1.) 1. The starting point. Standards- based curriculum begins with the grade-level standard and the underlying assumption that every student needs to master that stan- dard at that moment in time. In standards-embedded curriculum, the multifaceted essential ques- tion and students’ needs are the starting points. 2. Preassessment. In standards- based curriculum and teaching, if a preassessment is provided, it cov- ers a single standard or two. In a standards-embedded curriculum, preassessment includes a broader range of grade-level and advanced standards, as well as students’ knowledge of surrounding content such as background experiences with the subject, relevant skills (such as reading and writing), and continued on page ?? even learning style or interests. gifted child today 47 Standards-Based v. Standards-Embedded Curriculum Standards Based Standards Embedded Starting Points The grade-level standard. Whole class’ general skill level Essential questions and content relevant to individual students and groups. Preassessment Targeted to a single grade-level standard. Short-cycle assessments. Background knowledge. Multiple grade-level standards from multiple areas connected by the theme of the unit. Includes annual learning style and interest inventories. Acceleration/ Enrichment To next grade-level standard in the same strand. To above-grade-level standards, as well as into broader thematically connected content. Language Arts Divided into individual skills. Reading and writing skills often separated from real-world relevant contexts. The language arts are embedded in all units and themes and connected to differentiated processes and products across all content areas. Instruction Lesson planning begins with the standard as the objective. Sequential direct instruction progresses through the standards in each content area separately. Strategies are selected to introduce, practice, and demonstrate mastery of all grade-level standards in all content areas in one school year. Lesson planning begins with essential questions, topics, and significant themes. Integrated instruction is designed around connections among content areas and embeds all relevant standards. Assessment Format modeled after the state test. Variety of assessments including questions similar to the state test format. Teacher Role Monitor of standards mastery. Time manager. Facilitator of instructional design and student engagement with learning, as well as assessor of achievement. Student Self- Esteem “I can . . .” statements. Star Charts. Passing “the test.” Completed projects/products. Making personal connections to learning and the theme/topic. Figure 1. Standards based v. standards-embedded instruction and gifted students. and the potential political outcry of “stepping on the toes” of the next grade’s teacher. Few classroom teachers have been provided with the in-depth professional develop- ment and understanding of curric- ulum compacting that would allow them to implement this effectively. In standards-embedded curricu- lum, enrichment and extensions of learning are more possible and more interesting because ideas, top- ics, and questions lend themselves more easily to depth and complex- ity than isolated skills. 4. Language arts. In standards- based classrooms, the language arts have been redivided into sepa- rate skills, with reading separated from writing, and writing sepa- rated from grammar. To many concrete thinkers, whole-language approaches seem antithetical to teaching “to the standards.” In a standards-embedded classroom, integrated language arts skills (reading, writing, listening, speak- ing, presenting, and even pho- nics) are embedded into the study of every unit. Especially for the gifted, the communication and language arts are essential, regard- less of domain-specific talents (Ward, 1980) and should be com- ponents of all curriculum because they are the underpinnings of scholarship in all areas. 5. Instruction. A standards-based classroom lends itself to direct instruction and sequential pro- gression from one standard to the next. A standards-embedded class- room requires a variety of more open-ended instructional strate- gies and materials that extend and diversify learning rather than focus it narrowly. Creativity and differ- entiation in instruction and stu- dent performance are supported more effectively in a standards- embedded approach. 6. Assessment. A standards-based classroom uses targeted assess- ments focused on the structure and content of questions on the externally imposed standardized test (i.e., proficiency tests). A stan- dards-embedded classroom lends itself to greater use of authentic assessment and differentiated 3. Acceleration/Enrichment. In a standards-based curriculum, the narrow definition of the learning outcome (a test item) often makes acceleration or curriculum compact- ing the only path for differentiating instruction for gifted, talented, and/ or advanced learners. This rarely happens, however, because of lack of materials, knowledge, o

Standard based Curriculum In standard based curriculum, the initial point … Read More...
1. Frieda Birnbaum, at age 60, became the oldest U.S. women to give birth to twins after traveling to __________ for a special in-vitro fertilization treatment for older women. A.California. B.France. C.South Africa. D.New York. 2. A life course perspective: A.examines the entire course of human life from childhood to old age. B.examines the first twelve years of human life. C.examines life from age 18 to old age. D.focuses on the later years in life. 3. __________ is the specific study of aging and the elderly. A.Seniorology. B.Scientology. C.Genealogy. D.Gerontology. 4. Population aging in the U.S. is also referred to as: A.Angelology. B.“the graying of America.” C.“the whiting of America.” D.Agrology. 5. Median age refers to: A.the average age of a population. B.the most frequently occurring age in a population. C.the age where half the population is older and the other half is younger. D.the age calculated one half between the average and mode. 6. Place the appropriate median age in the U.S. with the appropriate year listed as follows: __________- 1820; __________ – 1900; __________ – 2000; __________ – 2030. A.17 years; 35 years; 23 years; 42 years. B.23 years; 17 years; 35 years; 42 years. C.35 years; 17 years; 23 years; 42 years. D.17 years; 23 years; 35 years; 42 years. 7. __________ refers to the study of the changes and trends in the population. A.Agrology. B.Demography. C.Gerontology. D.Urbanization. 8. All but which of the following have been identified as reasons why our population is aging? A.increase in birth rates. B.improvements in medical and technological advances. C.influence of birth cohorts. D.all the above are correct. 9. According to the Centers for Disease Control, the life expectancy for a child born in 2010 was: A.59.8 years. B.68.2 years. C.78.7 years. D.85.5 years. 10. It is estimated that approximately __________ people become eligible for Social Security every __________. A.1,000; day. B.1,000; month. C.10,000; month. D.10,000; day. 11. The persistent social ideals of women as homemakers and men as breadmakers is/are problematic from which of the following sociological perspectives? A.Conflict. B.Functional. C.Feminist. D.both a and c are correct. 12. From the Interactionist perspective, we create and maintain our definition of a family through: A.endogamy. B.egalitarianism. C.power struggles. D.social interaction. 13. According to the author, political and religious forces uphold and encourage a(n) __________ family form as the standard for what all families should be like. A.blended. B.egalitarian. C.matriarchal. D.patriarchal. 14. No-fault divorces were introduced in the: A.1960s. B.1970s. C.1980s. D.1990s. 15. It was noted that divorce rates have increased for all but which one of the following reasons? A.the transition from nuclear to extended family forms. B.the stigma attached to divorce has decreased. C.increasing geographic and occupational mobility of families. D.increasing economic independence of women. 16. The divorce rate in the late 1970s and early 1980s was approximately __________ per 1,000 individuals. A.2.3. B.6.7. C.1.4. D.5.3. 17. The U.S. marital rate _____ over the last ten years most recently recorded at _____ per 1,000 individuals for 2009. A.declined; 6.8. B.declined; 4.3. C.increased; 2.4. D.increased; 9.7. 18. In 2010, __________ children were more likely than children from other ethnic/racial groups to live in their grandparent’s household. A.Hispanic. B.black. C.white. D. Asian. 19. According to research, getting married at a young age __________ chances of divorce, while living with high levels of poverty __________ chances of divorce. A.increases; decreases. B.decreases; decreases. C.increases; increases. D.decreases; increases. 20. In the United States, nearly ___ of surveyed women reported that they had been raped or phyiscally assaulted by a current or former intimate partner. A.15%. B.25%. C.35%. D.40%.

1. Frieda Birnbaum, at age 60, became the oldest U.S. women to give birth to twins after traveling to __________ for a special in-vitro fertilization treatment for older women. A.California. B.France. C.South Africa. D.New York. 2. A life course perspective: A.examines the entire course of human life from childhood to old age. B.examines the first twelve years of human life. C.examines life from age 18 to old age. D.focuses on the later years in life. 3. __________ is the specific study of aging and the elderly. A.Seniorology. B.Scientology. C.Genealogy. D.Gerontology. 4. Population aging in the U.S. is also referred to as: A.Angelology. B.“the graying of America.” C.“the whiting of America.” D.Agrology. 5. Median age refers to: A.the average age of a population. B.the most frequently occurring age in a population. C.the age where half the population is older and the other half is younger. D.the age calculated one half between the average and mode. 6. Place the appropriate median age in the U.S. with the appropriate year listed as follows: __________- 1820; __________ – 1900; __________ – 2000; __________ – 2030. A.17 years; 35 years; 23 years; 42 years. B.23 years; 17 years; 35 years; 42 years. C.35 years; 17 years; 23 years; 42 years. D.17 years; 23 years; 35 years; 42 years. 7. __________ refers to the study of the changes and trends in the population. A.Agrology. B.Demography. C.Gerontology. D.Urbanization. 8. All but which of the following have been identified as reasons why our population is aging? A.increase in birth rates. B.improvements in medical and technological advances. C.influence of birth cohorts. D.all the above are correct. 9. According to the Centers for Disease Control, the life expectancy for a child born in 2010 was: A.59.8 years. B.68.2 years. C.78.7 years. D.85.5 years. 10. It is estimated that approximately __________ people become eligible for Social Security every __________. A.1,000; day. B.1,000; month. C.10,000; month. D.10,000; day. 11. The persistent social ideals of women as homemakers and men as breadmakers is/are problematic from which of the following sociological perspectives? A.Conflict. B.Functional. C.Feminist. D.both a and c are correct. 12. From the Interactionist perspective, we create and maintain our definition of a family through: A.endogamy. B.egalitarianism. C.power struggles. D.social interaction. 13. According to the author, political and religious forces uphold and encourage a(n) __________ family form as the standard for what all families should be like. A.blended. B.egalitarian. C.matriarchal. D.patriarchal. 14. No-fault divorces were introduced in the: A.1960s. B.1970s. C.1980s. D.1990s. 15. It was noted that divorce rates have increased for all but which one of the following reasons? A.the transition from nuclear to extended family forms. B.the stigma attached to divorce has decreased. C.increasing geographic and occupational mobility of families. D.increasing economic independence of women. 16. The divorce rate in the late 1970s and early 1980s was approximately __________ per 1,000 individuals. A.2.3. B.6.7. C.1.4. D.5.3. 17. The U.S. marital rate _____ over the last ten years most recently recorded at _____ per 1,000 individuals for 2009. A.declined; 6.8. B.declined; 4.3. C.increased; 2.4. D.increased; 9.7. 18. In 2010, __________ children were more likely than children from other ethnic/racial groups to live in their grandparent’s household. A.Hispanic. B.black. C.white. D. Asian. 19. According to research, getting married at a young age __________ chances of divorce, while living with high levels of poverty __________ chances of divorce. A.increases; decreases. B.decreases; decreases. C.increases; increases. D.decreases; increases. 20. In the United States, nearly ___ of surveyed women reported that they had been raped or phyiscally assaulted by a current or former intimate partner. A.15%. B.25%. C.35%. D.40%.

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Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: xbus(t) a t x(t) = x(0) + v(0)t + (1/2)at2 vbus(0) = 0 xbus = 1 a 2 t2 tcatch tcatch Typesetting math: 15% Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? xman(tcatch) > xbus(tcatch) xman(tcatch) = xbus(tcatch) xman(tcatch) < xbus(tcatch) c = a  tcatch xman(t) xbus(t) xman(tcatch) = xbus(tcatch) −b+ct = a catch 1 2 t2 catch 1 a −c +b = 0 2 t2 catch tcatch c tcatch cmin c a b 1 a − c + b = 0 2 t2 catch tcatch tcatch Typesetting math: 15% Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint 1. What is the general quadratic equation? The general quadratic equation is , where , \texttip{B}{B}, and \texttip{C}{C} are constants. Depending on the value of the discriminant, \Delta = c^2-2ab, the equation may have Ax2 + Bx + C = 0 x = −B±B2−4AC 2A  = cc − 2ab tcatch c cmin cmin = (2ab) −−−−  c > cmin Ax2 + Bx + C = 0 A Typesetting math: 15% two real valued solutions 1. if \Delta > 0, 2. one real valued solution if \Delta = 0, or 3. two complex valued solutions if \Delta < 0. In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A No; there is no chance he is going to get aboard. Yes; he will get a second chance Typesetting math: 15% Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: C and E E and F A and F C and D B and D Typesetting math: 15% Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: C and D A and F E and F A and B E and D A and F A and E D and B C and D E and F Typesetting math: 15% To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector \texttip{\vec{A}}{A_vec} is decomposed into component vectors \texttip{\vec{A}_{\mit x}}{A_vec_x} and \texttip{\vec{A}_{\mit y}}{A_vec_y} parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector \texttip{\vec{A}}{A_vec}, denoted \texttip{A_{\mit x}}{A_x} and \texttip{A_{\mit y}}{A_y}. TACTICS BOX 3.1 Determining the components of a vector The absolute value |A_x| of the x component \texttip{A_{\mit x}}{A_x} is the magnitude of the component vector \texttip{\vec{A}_{\1. mit x}}{A_vec_x}. The sign of \texttip{A_{\mit x}}{A_x} is positive if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the positive x direction; it is negative if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the negative x direction. 2. 3. The y component \texttip{A_{\mit y}}{A_y} is determined similarly. Part A What is the magnitude of the component vector \texttip{\vec{A}_{\mit x}}{A_vec_x} shown in the figure? Express your answer in meters to one significant figure. ANSWER: Correct |A_x| = 5 \rm m Typesetting math: 15% Part B What is the sign of the y component \texttip{A_{\mit y}}{A_y} of vector \texttip{\vec{A}}{A_vec} shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, \texttip{B_{\mit x}}{B_x} and \texttip{B_{\mit y}}{B_y}, of vector \texttip{\vec{B}}{B_vec} shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. positive negative Typesetting math: 15% ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, \texttip{v_{\mit x}}{1. v_x}, is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by \texttip{g}{g}, is equal to 9.80 \rm{m/s^2} near the surface of the earth. Hence, the y component of its velocity, \texttip{v_{\mit y}}{v_y}, changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time \texttip{t}{t} the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t_0 = 0\;\rm{s} corresponds to the moment just after the ball is launched from position x_0 = 0\;\rm{m} and y_0 = 0\;\rm{m}. Its launch velocity, also called the initial velocity, is \texttip{\vec{v}_{\rm 0}}{v_vec_0}. Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time \texttip{t_{\rm 1}}{t_1} with velocity \texttip{\vec{v}_{\rm 1}}{v_1_vec}. Its position at this moment is denoted by (x_1, y_1) or (x_1, y_{\max}) since it is at its maximum \texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = -2,-5 \rm m, \rm m Typesetting math: 15% The other point, at time \texttip{t_{\rm 2}}{t_2} with velocity \texttip{\vec{v}_{\rm 2}}{v_2_vec}, corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is (x_2, y_2), also known as (x_{\max}, y_2) since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence y_2 = y_0 = 0\;\rm{m}. Part A How do the speeds \texttip{v_{\rm 0}}{v_0}, \texttip{v_{\rm 1}}{v_1}, and \texttip{v_{\rm 2}}{v_2} (at times \texttip{t_{\rm 0}}{t_0}, \texttip{t_{\rm 1}}{t_1}, and \texttip{t_{\rm 2}}{t_2}) compare? ANSWER: Correct Here \texttip{v_{\rm 0}}{v_0} equals \texttip{v_{\rm 2}}{v_2} by symmetry and both exceed \texttip{v_{\rm 1}}{v_1}. This is because \texttip{v_{\rm 0}}{v_0} and \texttip{v_{\rm 2}}{v_2} include vertical speed as well as the constant horizontal speed. Consider a diagram of the ball at time \texttip{t_{\rm 0}}{t_0}. Recall that \texttip{t_{\rm 0}}{t_0} refers to the instant just after the ball has been launched, so it is still at ground level (x_0 = y_0= 0\;\rm{m}). However, it is already moving with initial velocity \texttip{\vec{v}_{\rm 0}}{v_0_vec}, whose magnitude is v_0 = 30.0\;{\rm m/s} and direction is \theta = 60.0\;{\rm degrees} counterclockwise from the positive x direction. \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 1}}{v_1} = \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 1}}{v_1} > \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 Typesetting math: 15% Part B What are the values of the intial velocity vector components \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{0,x}}{a_0, x} and \texttip{a_{0,y}}{a_0, y} (both in \rm{m/s^2})? Here the subscript 0 means “at time \texttip{t_{\rm 0}}{t_0}.” Hint 1. Determining components of a vector that is aligned with an axis If a vector points along a single axis direction, such as in the positive x direction, its x component will be its full magnitude, whereas its y component will be zero since the vector is perpendicular to the y direction. If the vector points in the negative x direction, its x component will be the negative of its full magnitude. Hint 2. Calculating the components of the initial velocity Notice that the vector \texttip{\vec{v}_{\rm 0}}{v_0_vec} points up and to the right. Since “up” is the positive y axis direction and “to the right” is the positive x axis direction, \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} will both be positive. As shown in the figure, \texttip{v_{0,x}}{v_0, x}, \texttip{v_{0,y}}{v_0, y}, and \texttip{v_{\rm 0}}{v_0} are three sides of a right triangle, one angle of which is \texttip{\theta }{theta}. Thus \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} can be found using the definition of the sine and cosine functions given below. Recall that v_0 = 30.0\;\rm{m/s} and \theta = 60.0\;\rm{degrees} and note that \large{\sin(\theta) = \frac{\rm{length\;of\;opposite\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, y}}{v_0}}, \large{\cos(\theta) = \frac{\rm{length\;of\;adjacent\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, x}}{v_0}.} What are the values of \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y}? Enter your answers numerically in meters per second separated by a comma. ANSWER: ANSWER: 15.0,26.0 \rm{m/s} Typesetting math: 15% Correct Also notice that at time \texttip{t_{\rm 2}}{t_2}, just before the ball lands, its velocity components are v_{2, x} = 15\;\rm{m/s} (the same as always) and v_{2, y} = – 26.0\;\rm{m/s} (the same size but opposite sign from \texttip{v_{0,y}}{v_0, y} by symmetry). The acceleration at time \texttip{t_{\rm 2}}{t_2} will have components (0, -9.80 \rm{m/s^2}), exactly the same as at \texttip{t_{\rm 0}}{t_0}, as required by Rule 2. The peak of the trajectory occurs at time \texttip{t_{\rm 1}}{t_1}. This is the point where the ball reaches its maximum height \texttip{y_{\rm max}}{y_max}. At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. Part C What are the values of the velocity vector components \texttip{v_{1,x}}{v_1, x} and \texttip{v_{1,y}}{v_1, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{1,x}}{a_1, x} and \texttip{a_{1,y}}{a_1, y} (both in \rm{m/s^2})? Here the subscript 1 means that these are all at time \texttip{t_{\rm 1}}{t_1}. ANSWER: 30.0, 0, 0, 0 0, 30.0, 0, 0 15.0, 26.0, 0, 0 30.0, 0, 0, -9.80 0, 30.0, 0, -9.80 15.0, 26.0, 0, -9.80 15.0, 26.0, 0, +9.80 Typesetting math: 15% Correct At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directed change in velocity, so the ball is accelerating downward even at the peak. The flight time refers to the total amount of time the ball is in the air, from just after it is launched (\texttip{t_{\rm 0}}{t_0}) until just before it lands (\texttip{t_{\rm 2}}{t_2}). Hence the flight time can be calculated as t_2 – t_0, or just \texttip{t_{\rm 2}}{t_2} in this particular situation since t_0 = 0. Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. Part D If a second ball were dropped from rest from height \texttip{y_{\rm max}}{y_max}, how long would it take to reach the ground? Ignore air resistance. Check all that apply. Hint 1. Kicking a ball of cliff; a related problem Consider two balls, one of which is dropped from rest off the edge of a cliff at the same moment that the other is kicked horizontally off the edge of the cliff. Which ball reaches the level ground at the base of the cliff first? Ignore air resistance. Hint 1. Comparing position, velocity, and acceleration of the two balls Both balls start at the same height and have the same initial y velocity (v_{0,y} = 0) as well as the same acceleration (\vec a = g downward). They differ only in their x velocity (one is 0, 0, 0, 0 0, 0, 0, -9.80 15.0, 0, 0, 0 15.0, 0, 0, -9.80 0, 26.0, 0, 0 0, 26.0, 0, -9.80 15.0, 26.0, 0, 0 15.0, 26.0, 0, -9.80 Typesetting math: 15% zero, the other nonzero). This difference will affect their x motion but not their y motion. ANSWER: ANSWER: Correct In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground. The range \texttip{R}{R} of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as x_2 – x_0, or just \texttip{x_{\rm 2}}{x_2} in this particular situation since x_0 = 0. Range can be calculated as the product of the flight time \texttip{t_{\rm 2}}{t_2} and the x component of the velocity \texttip{v_{\mit x}}{v_x} (which is the same at all times, so v_x = v_{0,x}). The value of \texttip{v_{\mit x}}{v_x} can be found from the launch speed \texttip{v_{\rm 0}}{v_0} and the launch angle \texttip{\theta }{theta} using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from \texttip{v_{\rm 0}}{v_0} and \texttip{\theta }{theta} using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position (x_0 = y_0 = 0) at time t_0 = 0 with initial speed \texttip{v_{\rm 0}}{v_0} and launch angle \texttip{\theta }{theta} measured from the horizontal. As was the case above, \texttip{t_{\rm 2}}{t_2} refers to the flight time and \texttip{R}{R} refers to the range of the projectile. flight time: \large{t_2 = \frac{2 v_{0, y}}{g} = \frac{2 v_0 \sin(\theta)}{g}} range: \large{R = v_x t_2 = \frac{v_0^2 \sin(2\theta)}{g}} The ball that falls straight down strikes the ground first. The ball that was kicked so it moves horizontally as it falls strikes the ground first. Both balls strike the ground at the same time. \texttip{t_{\rm 0}}{t_0} t_1 – t_0 \texttip{t_{\rm 2}}{t_2} t_2 – t_1 \large{\frac{t_2 – t_0}{2}} Typesetting math: 15% In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Part E Which of the following changes would increase the range of the ball shown in the original figure? Check all that apply. ANSWER: Correct A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object does not land at the same height from which is was launched, the rules given in the introduction will still be useful. Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influenced by air. The value of the gravitational free-fall acceleration \texttip{g}{g} is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers, like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable. A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \;{\rm m}/{\rm s}^{2}. Part A How much horizontal force \texttip{F}{F} must a sprinter of mass 64{\rm kg} exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Increase \texttip{v_{\rm 0}}{v_0} above 30 \rm{m/s}. Reduce \texttip{v_{\rm 0}}{v_0} below 30 \rm{m/s}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to 45 \rm{degrees}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to less than 30 \rm{degrees}. Increase \texttip{\theta }{theta} from 60 \rm{degrees} up toward 90 \rm{degrees}. Typesetting math: 15% Hint 1. Newton’s 2nd law of motion According to Newton’s 2nd law of motion, if a net external force \texttip{F_{\rm net}}{F_net} acts on a body, the body accelerates, and the net force is equal to the mass \texttip{m}{m} of the body times the acceleration \texttip{a}{a} of the body: F_{\rm net} = ma. ANSWER: Co

Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: xbus(t) a t x(t) = x(0) + v(0)t + (1/2)at2 vbus(0) = 0 xbus = 1 a 2 t2 tcatch tcatch Typesetting math: 15% Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? xman(tcatch) > xbus(tcatch) xman(tcatch) = xbus(tcatch) xman(tcatch) < xbus(tcatch) c = a  tcatch xman(t) xbus(t) xman(tcatch) = xbus(tcatch) −b+ct = a catch 1 2 t2 catch 1 a −c +b = 0 2 t2 catch tcatch c tcatch cmin c a b 1 a − c + b = 0 2 t2 catch tcatch tcatch Typesetting math: 15% Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint 1. What is the general quadratic equation? The general quadratic equation is , where , \texttip{B}{B}, and \texttip{C}{C} are constants. Depending on the value of the discriminant, \Delta = c^2-2ab, the equation may have Ax2 + Bx + C = 0 x = −B±B2−4AC 2A  = cc − 2ab tcatch c cmin cmin = (2ab) −−−−  c > cmin Ax2 + Bx + C = 0 A Typesetting math: 15% two real valued solutions 1. if \Delta > 0, 2. one real valued solution if \Delta = 0, or 3. two complex valued solutions if \Delta < 0. In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A No; there is no chance he is going to get aboard. Yes; he will get a second chance Typesetting math: 15% Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: C and E E and F A and F C and D B and D Typesetting math: 15% Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: C and D A and F E and F A and B E and D A and F A and E D and B C and D E and F Typesetting math: 15% To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector \texttip{\vec{A}}{A_vec} is decomposed into component vectors \texttip{\vec{A}_{\mit x}}{A_vec_x} and \texttip{\vec{A}_{\mit y}}{A_vec_y} parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector \texttip{\vec{A}}{A_vec}, denoted \texttip{A_{\mit x}}{A_x} and \texttip{A_{\mit y}}{A_y}. TACTICS BOX 3.1 Determining the components of a vector The absolute value |A_x| of the x component \texttip{A_{\mit x}}{A_x} is the magnitude of the component vector \texttip{\vec{A}_{\1. mit x}}{A_vec_x}. The sign of \texttip{A_{\mit x}}{A_x} is positive if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the positive x direction; it is negative if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the negative x direction. 2. 3. The y component \texttip{A_{\mit y}}{A_y} is determined similarly. Part A What is the magnitude of the component vector \texttip{\vec{A}_{\mit x}}{A_vec_x} shown in the figure? Express your answer in meters to one significant figure. ANSWER: Correct |A_x| = 5 \rm m Typesetting math: 15% Part B What is the sign of the y component \texttip{A_{\mit y}}{A_y} of vector \texttip{\vec{A}}{A_vec} shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, \texttip{B_{\mit x}}{B_x} and \texttip{B_{\mit y}}{B_y}, of vector \texttip{\vec{B}}{B_vec} shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. positive negative Typesetting math: 15% ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, \texttip{v_{\mit x}}{1. v_x}, is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by \texttip{g}{g}, is equal to 9.80 \rm{m/s^2} near the surface of the earth. Hence, the y component of its velocity, \texttip{v_{\mit y}}{v_y}, changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time \texttip{t}{t} the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t_0 = 0\;\rm{s} corresponds to the moment just after the ball is launched from position x_0 = 0\;\rm{m} and y_0 = 0\;\rm{m}. Its launch velocity, also called the initial velocity, is \texttip{\vec{v}_{\rm 0}}{v_vec_0}. Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time \texttip{t_{\rm 1}}{t_1} with velocity \texttip{\vec{v}_{\rm 1}}{v_1_vec}. Its position at this moment is denoted by (x_1, y_1) or (x_1, y_{\max}) since it is at its maximum \texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = -2,-5 \rm m, \rm m Typesetting math: 15% The other point, at time \texttip{t_{\rm 2}}{t_2} with velocity \texttip{\vec{v}_{\rm 2}}{v_2_vec}, corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is (x_2, y_2), also known as (x_{\max}, y_2) since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence y_2 = y_0 = 0\;\rm{m}. Part A How do the speeds \texttip{v_{\rm 0}}{v_0}, \texttip{v_{\rm 1}}{v_1}, and \texttip{v_{\rm 2}}{v_2} (at times \texttip{t_{\rm 0}}{t_0}, \texttip{t_{\rm 1}}{t_1}, and \texttip{t_{\rm 2}}{t_2}) compare? ANSWER: Correct Here \texttip{v_{\rm 0}}{v_0} equals \texttip{v_{\rm 2}}{v_2} by symmetry and both exceed \texttip{v_{\rm 1}}{v_1}. This is because \texttip{v_{\rm 0}}{v_0} and \texttip{v_{\rm 2}}{v_2} include vertical speed as well as the constant horizontal speed. Consider a diagram of the ball at time \texttip{t_{\rm 0}}{t_0}. Recall that \texttip{t_{\rm 0}}{t_0} refers to the instant just after the ball has been launched, so it is still at ground level (x_0 = y_0= 0\;\rm{m}). However, it is already moving with initial velocity \texttip{\vec{v}_{\rm 0}}{v_0_vec}, whose magnitude is v_0 = 30.0\;{\rm m/s} and direction is \theta = 60.0\;{\rm degrees} counterclockwise from the positive x direction. \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 1}}{v_1} = \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 1}}{v_1} > \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 Typesetting math: 15% Part B What are the values of the intial velocity vector components \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{0,x}}{a_0, x} and \texttip{a_{0,y}}{a_0, y} (both in \rm{m/s^2})? Here the subscript 0 means “at time \texttip{t_{\rm 0}}{t_0}.” Hint 1. Determining components of a vector that is aligned with an axis If a vector points along a single axis direction, such as in the positive x direction, its x component will be its full magnitude, whereas its y component will be zero since the vector is perpendicular to the y direction. If the vector points in the negative x direction, its x component will be the negative of its full magnitude. Hint 2. Calculating the components of the initial velocity Notice that the vector \texttip{\vec{v}_{\rm 0}}{v_0_vec} points up and to the right. Since “up” is the positive y axis direction and “to the right” is the positive x axis direction, \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} will both be positive. As shown in the figure, \texttip{v_{0,x}}{v_0, x}, \texttip{v_{0,y}}{v_0, y}, and \texttip{v_{\rm 0}}{v_0} are three sides of a right triangle, one angle of which is \texttip{\theta }{theta}. Thus \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} can be found using the definition of the sine and cosine functions given below. Recall that v_0 = 30.0\;\rm{m/s} and \theta = 60.0\;\rm{degrees} and note that \large{\sin(\theta) = \frac{\rm{length\;of\;opposite\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, y}}{v_0}}, \large{\cos(\theta) = \frac{\rm{length\;of\;adjacent\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, x}}{v_0}.} What are the values of \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y}? Enter your answers numerically in meters per second separated by a comma. ANSWER: ANSWER: 15.0,26.0 \rm{m/s} Typesetting math: 15% Correct Also notice that at time \texttip{t_{\rm 2}}{t_2}, just before the ball lands, its velocity components are v_{2, x} = 15\;\rm{m/s} (the same as always) and v_{2, y} = – 26.0\;\rm{m/s} (the same size but opposite sign from \texttip{v_{0,y}}{v_0, y} by symmetry). The acceleration at time \texttip{t_{\rm 2}}{t_2} will have components (0, -9.80 \rm{m/s^2}), exactly the same as at \texttip{t_{\rm 0}}{t_0}, as required by Rule 2. The peak of the trajectory occurs at time \texttip{t_{\rm 1}}{t_1}. This is the point where the ball reaches its maximum height \texttip{y_{\rm max}}{y_max}. At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. Part C What are the values of the velocity vector components \texttip{v_{1,x}}{v_1, x} and \texttip{v_{1,y}}{v_1, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{1,x}}{a_1, x} and \texttip{a_{1,y}}{a_1, y} (both in \rm{m/s^2})? Here the subscript 1 means that these are all at time \texttip{t_{\rm 1}}{t_1}. ANSWER: 30.0, 0, 0, 0 0, 30.0, 0, 0 15.0, 26.0, 0, 0 30.0, 0, 0, -9.80 0, 30.0, 0, -9.80 15.0, 26.0, 0, -9.80 15.0, 26.0, 0, +9.80 Typesetting math: 15% Correct At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directed change in velocity, so the ball is accelerating downward even at the peak. The flight time refers to the total amount of time the ball is in the air, from just after it is launched (\texttip{t_{\rm 0}}{t_0}) until just before it lands (\texttip{t_{\rm 2}}{t_2}). Hence the flight time can be calculated as t_2 – t_0, or just \texttip{t_{\rm 2}}{t_2} in this particular situation since t_0 = 0. Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. Part D If a second ball were dropped from rest from height \texttip{y_{\rm max}}{y_max}, how long would it take to reach the ground? Ignore air resistance. Check all that apply. Hint 1. Kicking a ball of cliff; a related problem Consider two balls, one of which is dropped from rest off the edge of a cliff at the same moment that the other is kicked horizontally off the edge of the cliff. Which ball reaches the level ground at the base of the cliff first? Ignore air resistance. Hint 1. Comparing position, velocity, and acceleration of the two balls Both balls start at the same height and have the same initial y velocity (v_{0,y} = 0) as well as the same acceleration (\vec a = g downward). They differ only in their x velocity (one is 0, 0, 0, 0 0, 0, 0, -9.80 15.0, 0, 0, 0 15.0, 0, 0, -9.80 0, 26.0, 0, 0 0, 26.0, 0, -9.80 15.0, 26.0, 0, 0 15.0, 26.0, 0, -9.80 Typesetting math: 15% zero, the other nonzero). This difference will affect their x motion but not their y motion. ANSWER: ANSWER: Correct In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground. The range \texttip{R}{R} of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as x_2 – x_0, or just \texttip{x_{\rm 2}}{x_2} in this particular situation since x_0 = 0. Range can be calculated as the product of the flight time \texttip{t_{\rm 2}}{t_2} and the x component of the velocity \texttip{v_{\mit x}}{v_x} (which is the same at all times, so v_x = v_{0,x}). The value of \texttip{v_{\mit x}}{v_x} can be found from the launch speed \texttip{v_{\rm 0}}{v_0} and the launch angle \texttip{\theta }{theta} using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from \texttip{v_{\rm 0}}{v_0} and \texttip{\theta }{theta} using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position (x_0 = y_0 = 0) at time t_0 = 0 with initial speed \texttip{v_{\rm 0}}{v_0} and launch angle \texttip{\theta }{theta} measured from the horizontal. As was the case above, \texttip{t_{\rm 2}}{t_2} refers to the flight time and \texttip{R}{R} refers to the range of the projectile. flight time: \large{t_2 = \frac{2 v_{0, y}}{g} = \frac{2 v_0 \sin(\theta)}{g}} range: \large{R = v_x t_2 = \frac{v_0^2 \sin(2\theta)}{g}} The ball that falls straight down strikes the ground first. The ball that was kicked so it moves horizontally as it falls strikes the ground first. Both balls strike the ground at the same time. \texttip{t_{\rm 0}}{t_0} t_1 – t_0 \texttip{t_{\rm 2}}{t_2} t_2 – t_1 \large{\frac{t_2 – t_0}{2}} Typesetting math: 15% In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Part E Which of the following changes would increase the range of the ball shown in the original figure? Check all that apply. ANSWER: Correct A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object does not land at the same height from which is was launched, the rules given in the introduction will still be useful. Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influenced by air. The value of the gravitational free-fall acceleration \texttip{g}{g} is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers, like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable. A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \;{\rm m}/{\rm s}^{2}. Part A How much horizontal force \texttip{F}{F} must a sprinter of mass 64{\rm kg} exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Increase \texttip{v_{\rm 0}}{v_0} above 30 \rm{m/s}. Reduce \texttip{v_{\rm 0}}{v_0} below 30 \rm{m/s}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to 45 \rm{degrees}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to less than 30 \rm{degrees}. Increase \texttip{\theta }{theta} from 60 \rm{degrees} up toward 90 \rm{degrees}. Typesetting math: 15% Hint 1. Newton’s 2nd law of motion According to Newton’s 2nd law of motion, if a net external force \texttip{F_{\rm net}}{F_net} acts on a body, the body accelerates, and the net force is equal to the mass \texttip{m}{m} of the body times the acceleration \texttip{a}{a} of the body: F_{\rm net} = ma. ANSWER: Co

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