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**Section 2 Extra Questions**

Preview Section 1 Acceleration Section 2 Extra Questions

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**What do you think? Which of the following cars is accelerating?**

A car shortly after a stoplight turns green A car approaching a red light A car with the cruise control set at 80 km/h A car turning a curve at a constant speed Based on your answers, what is your definition of acceleration? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students will often only choose the first option as an accelerating vehicle. They think of the others as decelerating and constant velocity.

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**Acceleration Rate of change in velocity What are the units?**

SI Units: (m/s)/s or m/s2 Other Units: (km/h)/s or (mi/h)/s Acceleration = 0 implies a constant velocity (or rest) Have students analyze the equation before providing the answer to the units. Stress that m/s2 are a short way of saying (m/s)/s. It is a good idea to keep saying (m/s)/s in order to emphasize the fact that acceleration is the change in velocity (m/s) over a period of time (s).

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**Classroom Practice Problem**

Find the acceleration of an amusement park ride that falls from rest to a velocity of 28 m/s downward in 3.0 s. Answer: 9.3 m/s2 downward For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow them some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.

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**Direction of Acceleration**

Describe the motion of an object with vi and a as shown to the left. Moving right as it speeds up Moving right as it slows down Moving left as it speeds up Moving left as it slows down Vi a + - To make the situations more concrete, use an automobile as an example. For example, the first combination would be a car moving to the right (v is +) and accelerating to the right (a is +), so the speed will increase. Some students may be confused by the latter two examples, thinking that a negative acceleration corresponds to slowing down and a positive acceleration corresponds to speeding up. Emphasize that the directions of velocity and acceleration must both be taken into account. In the third example, the velocity and acceleration are in the same direction, so the object is speeding up. In the fourth case, they are in opposite directions, so the object is slowing down.

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Acceleration Click below to watch the Visual Concept. Visual Concept

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Graphing Velocity The slope (rise/run) of a velocity/time graph is the acceleration. Rise is change in v Run is change in t This graph shows a constant acceleration. Average speed is the midpoint. The equation for vavg is only valid if the velocity increases uniformly (a straight line in a velocity-time graph) or, in other words, if the acceleration is constant.

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**Graph of v vs. t for a train**

Describe the motion at points A, B, and C. Answers A: accelerating (increasing velocity/slope) to the right B: constant velocity to the right C: negative acceleration (decreasing velocity/slope) and still moving to the right Students may think “B” is at rest and “C” is moving backwards or to the left. If so, they are confusing position-time graphs with velocity-time graphs. Ask them to look at “B” and think about what it means if velocity stays the same or look at “C” and ask them what it means if velocity is decreasing. A good exercise for the students at this time would be the use of the Phet web site: NOTE: These simulations are downloadable so you can avoid the need for internet access after a onetime download. If you choose the “Motion” simulations and then choose the “Moving man” option, the students can observe the motion of a man (constant velocity, speeding up, slowing down, at rest) and see the graphs of position-time, velocity-time and acceleration-time. You might start with “at rest” and ask them to predict the shape of each graph before running the simulation. Then ask them how each would change if he moved forward with a constant speed. Follow this with other changes, such as changing the starting position or accelerating the walker.

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Useful Equations 1. 2. 3. 4. 5. Equations (1) and (2) are the definitions of velocity and acceleration. Equations (3), (4), and (5) are only valid for uniform acceleration. Show students how to derive equation (4) by combining (1), (2), and (3). Then allow students some time to derive (5) from (1), (2), and (3) by eliminating time. Since (4) and (5) are derived from the first three, there are no problems that can be solved with them that could not have been solved by using the first three equations. It might be easier to use (4) and (5) but it is not necessary. They do not represent any “new” rules.

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**Classroom Practice Problems**

A bicyclist accelerates from 5.0 m/s to 16 m/s in 8.0 s. Assuming uniform acceleration, what distance does the bicyclist travel during this time interval? Answer: 84 m An aircraft has a landing speed of 83.9 m/s. The landing area of an aircraft carrier is 195 m long. What is the minimum uniform acceleration required for safe landing? Answer: m/s2 For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow them some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful. After using equation (4) to solve the first problem, show the students that they would obtain the same answer by using equation (3) followed by equation (1). Similarly, the second problem can be solved with equation (5) or by using (3), then (1), then (2).

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**Now what do you think? Which of the following cars is accelerating?**

A car shortly after a stoplight turns green A car approaching a red light A car with the cruise control set at 80 km/h A car turning a curve at a constant speed Based on your answers, what is the definition of acceleration? How is acceleration calculated? What are the SI units for acceleration? The car is accelerating in each example except for the cruise control. The first is positive acceleration, the second is negative acceleration, and the fourth is accelerating because direction is changing (and thus velocity is changing, even though speed is constant). Centripetal acceleration will be covered in a later chapter but it is good to introduce the idea here, so students realize that acceleration is any change in velocity (either a change in the magnitude of velocity, or a change in the direction of velocity, or both).

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**What do you think? Observe a metal ball being dropped from rest.**

Describe the motion in words. Sketch a velocity-time graph for this motion. Observe the same ball being tossed vertically upward and returning to the starting point. When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students may not see that the ball is accelerating uniformly and, as a result, they may draw a graph that rises and then starts to level off. You might also ask them how their answers would change if you dropped a similar but lighter ball. Then, drop the two simultaneously and let them observe the motion.

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Graphing Free Fall Based on your present understanding of free fall, sketch a velocity-time graph for a ball that is tossed upward (assuming no air resistance). Is it a straight line? If so, what is the slope? Compare your predictions to the graph to the right. Now students are asked to graph the motion they just observed. This graph should match the answers to the chart on the last slide. Remind them that they are graphing velocity, but acceleration is the slope of the velocity-time graph. Student graphs may have a different initial velocity and a different x-intercept (the time at which the velocity reaches zero), but their graphs should have the same shape and slope as the one given on the slide. Point out that the velocity is zero at the peak (t = 1.1 s for this graph) while the acceleration is never zero because the slope is always negative. Help them get an approximate slope for the graph shown on the slide. It should be close to (m/s)/s.

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**Velocity and Acceleration of an Object at its High Point**

Click below to watch the Visual Concept. Visual Concept

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Preview Multiple Choice Short Response Extended Response

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**Multiple Choice, continued**

Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6. 5. What is the squirrel’s displacement at time t = 3.0 s? A. –6.0 m B. –2.0 m C m D m

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**Multiple Choice, continued**

Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6. 5. What is the squirrel’s displacement at time t = 3.0 s? A. –6.0 m B. –2.0 m C m D m

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**Multiple Choice, continued**

Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6. 6. What is the squirrel’s average velocity during the time interval between 0.0 s and 3.0 s? F. –2.0 m/s G. –0.67 m/s H. 0.0 m/s J m/s

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**Multiple Choice, continued**

Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6. 6. What is the squirrel’s average velocity during the time interval between 0.0 s and 3.0 s? F. –2.0 m/s G. –0.67 m/s H. 0.0 m/s J m/s

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**Multiple Choice, continued**

7. Which of the following statements is true of acceleration? A. Acceleration always has the same sign as displacement. B. Acceleration always has the same sign as velocity. C. The sign of acceleration depends on both the direction of motion and how the velocity is changing. D. Acceleration always has a positive sign.

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**Multiple Choice, continued**

7. Which of the following statements is true of acceleration? A. Acceleration always has the same sign as displacement. B. Acceleration always has the same sign as velocity. C. The sign of acceleration depends on both the direction of motion and how the velocity is changing. D. Acceleration always has a positive sign.

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**Multiple Choice, continued**

8. A ball initially at rest rolls down a hill and has an acceleration of 3.3 m/s2. If it accelerates for 7.5 s, how far will it move during this time? F. 12 m G. 93 m H. 120 m J. 190 m

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**Multiple Choice, continued**

8. A ball initially at rest rolls down a hill and has an acceleration of 3.3 m/s2. If it accelerates for 7.5 s, how far will it move during this time? F. 12 m G. 93 m H. 120 m J. 190 m

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**Short Response, continued**

12. For an object moving with constant negative acceleration, draw the following: a. a graph of position vs. time b. a graph of velocity vs. time For both graphs, assume the object starts with a positive velocity and a positive displacement from the origin.

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**Short Response, continued**

12. For an object moving with constant negative acceleration, draw the following: a. a graph of position vs. time b. a graph of velocity vs. time For both graphs, assume the object starts with a positive velocity and a positive displacement from the origin. Answers:

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**Short Response, continued**

13. A snowmobile travels in a straight line. The snowmobile’s initial velocity is +3.0 m/s. a. If the snowmobile accelerates at a rate of m/s2 for 7.0 s, what is its final velocity? b. If the snowmobile accelerates at the rate of –0.60 m/s2 from its initial velocity of +3.0 m/s, how long will it take to reach a complete stop?

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**Short Response, continued**

13. A snowmobile travels in a straight line. The snowmobile’s initial velocity is +3.0 m/s. a. If the snowmobile accelerates at a rate of m/s2 for 7.0 s, what is its final velocity? b. If the snowmobile accelerates at the rate of –0.60 m/s2 from its initial velocity of +3.0 m/s, how long will it take to reach a complete stop? Answers: a m/s b. 5.0 s

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Extended Response 14. A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0 s. a. What is the car’s average acceleration? b. What is the car’s average velocity? c. How far did the car move while accelerating? Show all of your work for these calculations.

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Extended Response 14. A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0 s. a. What is the car’s average acceleration? b. What is the car’s average velocity? c. How far did the car move while accelerating? Answers: a. 1.6 m/s2 eastward b. 24 m/s c. 240 m

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**Extended Response, continued**

15. A ball is thrown vertically upward with a speed of 25.0 m/s from a height of 2.0 m. a. How long does it take the ball to reach its highest point? b. How long is the ball in the air? Show all of your work for these calculations.

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**Extended Response, continued**

15. A ball is thrown vertically upward with a speed of 25.0 m/s from a height of 2.0 m. a. How long does it take the ball to reach its highest point? b. How long is the ball in the air? Show all of your work for these calculations. Answers: a s b s

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