hphys unit 03 capm packet 2012

Name: ________________________________

Constant Acceleration ModelPhysical Quantity Description Symbol Units

Honors Physics / Unit 03 / CAPM

– 1 – from Modeling Workshop Project © 2006

Ramp and Cart Experiment

Sketch and label the experiment setup:

What could we measure? How could we measure it?

Make sketches of the graphs that the motion sensor and Data Studio make. Do this only after your teacher gives you the OK upon checking your computer screen. Remember to do several trials before asking your teacher to check.


from Modeling Workshop Project © 2006 ! – 2 –

Ramp & Cart Experiment (Post-Lab)

Unit IConstant Velocity Model

Unit III Constant Acceleration Model



Walk-A-GraphFrom Minds-On Physics

(1) A marble is rolled at constant speed along a horizontal surface toward the origin. The marble is released at a distance of 1 meter away from the origin.

(2) A block sits at rest on a table 1 meter above the floor. Take the origin to be the level of the floor.(3) A ball is dropped from a height of 2 meters above the floor. Take the origin to be the point from which the ball is released.(4) A ball is rolled along a horizontal surface. The ball strikes a wall and rebounds toward the origin.(5) A car is parked on a steep hill.

(1) A block is dropped from rest with a height of 1 meter above the floor. Take the origin to be at the level of the floor.(2) A marble is released from the top of an inclined plane. Assume that positive x is measured down the plane.(3) A ball is thrown straight up into the air. Take the origin to be at the level of the floor.(4) A ball rolls along a horizontal surface without changing speed. The ball strikes a wall and rebounds toward the origin at

approximately the same speed as before.(5) A marble rolls on to a piece of felt, eventually stopping.



Worksheet 1: Graphs of Motion with Changing Velocity

1. Consider the velocity-vs-time graphs and describe the motion of the objects.

Object A Object B

8 1 2 3 4 5 6 7

16

-16

-12

-8

-4

0

4

8

12

t (s)

v (m

/s)

8 1 2 3 4 5 6 7

16

-16

-12

-8

-4

0

4

8

12

t (s)

v (m

/s)

Determine the displacement between 4 and 8 seconds. Show work!

Determine the average acceleration during the first 3 seconds.Show work!

Describe the motion in words.

Sketch a motion map.Be sure to include both velocity and acceleration vectors.



2. Use the velocity-vs-time graph to analyze the motion of the object.

a. Give a written description of the motion.

b. Sketch a motion map. Be sure to include both velocity and acceleration vectors.

c. Determine the displacement of the object from t = 0 s to t = 4 s.

d. Determine the displacement of the object from t = 4 s to t = 8 s.

e. Determine the displacement of the object from t = 2 s to t = 6 s.

f. Determine the object’s acceleration at t = 4 s.

g. Sketch a possible position-vs-time graph for the motion of the object. Explain why your graph is only one of many possible graphs.



Reading: Two Methods to Find the Slope



ClockReading Position

(s) (m)0.00 0.0000.42 0.0180.84 0.0601.26 0.1251.68 0.2102.10 0.3252.52 0.4602.94 0.6203.36 0.805

� ��

�

�

�

�

�

�

0 1 2 3 40.0

0.2

0.4

0.6

0.8

time �s�

position�m�

Figure 1: The position vs. time graph madefrom the data in the table above. Note thatthe slope of this graph is not constant!

� ��

�

�

�

�

�

�

�x�0.58 m

�t�2.0 s

v��0.58 m ��2.0 s� �0.29 m �s

0 1 2 3 40.0

0.2

0.4

0.6

0.8

time �s�

position�m�

Figure 2: To find the velocity of the puckat 2.0 s, we draw a tangent line that justtouches the curve at 2.0 s. Then we findthe slope of that line using any convenientpoints along the tangent line.

Consider a puck for which we take the position and time data shownon the right. A quick glance at the data table tells us that the puck is mov-ing farther in each equal time interval of 0.42 s. A graph of the data, shownjust below the table allows us to say even more about the motion: the dis-tance the puck moves in equal intervals of time is increasing smoothly.While we can’t find a straight segment of the curve in order to find a slope,we can estimate the velocity of the puck right at a certain time, even if thepuck is changing velocity. We go about making this estimate two ways:

1. We can, using the graphed position and time data, draw a good es-timate of the tangent line to the data at the instant of interest. Forexample, let’s say we want to estimate the velocity of the puck at 2.0s. The slope of this tangent line, shown drawn the the lower figure,is then the estimated velocity of the object at that instant. We find theslope of the tangent line at t = 2.0 s to be 0.29 m/s. This graphicalmethod is called the tangent slope method, and it relies on beingable to draw a reasonably accurate tangent line. Clearly, the velocitydetermined this way is an estimate.

2. We can use the position and time data table to calculate the slopeof the smallest segment of graph surrounding a particular instant ofinterest. We do this by taking the data from one time interval beforeand one time interval after the instant for which we are estimating thevelocity. Such a calculation, called the double interval method, isshown in the table below, using the original data table for our puck’smotion. The first two rows show the full calculation (take close noteof these calculations) and the other rows show only the results. Notethat this method is also an estimate of the velocity.

ClockReading Position Velocity

(s) (m) (m/s)0.00 0.000 —0.42 0.018 (.060−.000) m

(0.84−0.0) s = 0.07 m/s

0.84 0.060 (.125−.018) m(1.26−0.42) s = 0.127 m/s

1.26 0.125 0.1781.68 0.210 0.2382.10 0.325 0.2982.52 0.460 0.3512.94 0.620 0.4103.36 0.805 —

The tangent slope method is so-named because one must first draw atangent, then find the tangent line’s slope. The double interval methodis so-named because one uses a duration twice that of the data intervalto calculate the velocity at each time. That is, if position data were takenevery one second, you would use time intervals of two seconds to calculatethe velocity at each point.

Worksheet 2: Changing Velocities

3. The table shows some position and time data.

a. Use the double interval method to calculate the velocity at t = 0.030 s. Show your calculation below.

b. Use the double interval method to calculate the velocity at each time and fill in the rest of the table.

4. Consider the velocity-vs-time graph.

a. During which time interval(s) is the acceleration positive? During which time interval(s) is the acceleration negative? How do you know?

b. At what time or times is the acceleration zero? How do you know?

c. Use the tangent slope method to calculate the acceleration at time t = 10.0s.

d. Describe the motion of the object in words. In your complete sentences, you might want to use phrases like speeding up, slowing down, in the positive direction, in the negative direction, reverses direction, starting from rest.

Time (s) Position (cm) Velocity (cm/s)

0.000 0.0 —

0.030 1.2

0.060 2.2

0.090 3.0

0.120 6.0

0.150 8.1 —

47

(e) Describe the motion in words. Choose from among the following phrases to con-struct your complete sentence description.

speeding up in the positive direction reverses directionslowing down in the negative direction starting from rest

20 40 60 80 100

Time !s"

5

10

15

20

25

30

Velocity!m#s"

10. The table below represents the velocity and time coordinates for the motion depictedin the graph of problem 9. Compute the accelerations at each time using the doubleinterval method. Do the accelerations approximately agree with our answers for partsa, b, and c of problem 9?

Time Velocity Acceleration(s) (m/s) (m/s2)0 0.05 5.0 0.9810 9.815 13.720 17.0 0.6025 19.7 0.4830 21.835 23.3 0.2540 24.3 0.1445 24.7 0.0350 24.6 -0.1555 23.2 -0.4760 19.965 18.2 -0.2570 17.475 17.1 -0.0180 17.3 0.0985 18.090 20.0



5. The acceleration-vs-time graph for a cart moving along a straight-line track is shown below.

a. Calculate the change in velocity over the first 3.0 s of motion.

b. Calculate the change in velocity over the entire 8.0 s of motion.

c. Given that the cart starts with an initial velocity of +2.0 m/s, plot the velocity-vs-time graph for this motion. (Yep, this is a bit tricky… go for it!! Break the a-t graph into useful parts.)

48

Section 3.4

11. The acceleration versus time graph for a cart moving along a straight-line track isshown below.

1 2 3 4 5 6 7 8

Time !s"

!2

!1

0

1

2

3

Acceleration!cm#s2"

(a) Calculate the change in velocity over the first 3.0 s of motion.(b) Calculate the change in velocity over the entire 8.0 s of motion.(c) Given that the cart starts with an initial velocity of + 2.0 m/s plot the velocity

vs. time graph for this motion and describe the motion in words.

1 2 3 4 5 6 7 8

Time !s"

!2

0

2

4

6

8

Velocity!cm#s"

(d) Plot the velocity vs. time graph if the cart’s initial velocity were +5.0 m/s.Describe this motion in words.

d. Describe the motion of the object in words based on the velocity graph that you drew.

48

Section 3.4

11. The acceleration versus time graph for a cart moving along a straight-line track isshown below.

1 2 3 4 5 6 7 8

Time !s"

!2

!1

0

1

2

3

Acceleration!cm#s2"

(a) Calculate the change in velocity over the first 3.0 s of motion.(b) Calculate the change in velocity over the entire 8.0 s of motion.(c) Given that the cart starts with an initial velocity of + 2.0 m/s plot the velocity

vs. time graph for this motion and describe the motion in words.

1 2 3 4 5 6 7 8

Time !s"

!2

0

2

4

6

8

Velocity!cm#s"

(d) Plot the velocity vs. time graph if the cart’s initial velocity were +5.0 m/s.Describe this motion in words.



Stacks of Kinematics CurvesGiven the following position vs time graphs, sketch the corresponding velocity vs time and acceleration vs time graphs. For each problem (or part of problem), tell whether the forces on the object must be balanced or unbalanced. If they are unbalanced, say whether they are unbalanced in the positive or negative direction.

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t

Balanced or Unbalanced? (+ / –)




x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t







For the following velocity vs time graphs, draw the corresponding position vs time and acceleration vs time graphs. For each problem (or part of problem), tell whether the forces on the object must be balanced or unbalanced. If they are unbalanced, say whether they are unbalanced in the positive or negative direction.

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a

t





x

t

v

t

a

t

x

t

v

t

a

t

x

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a

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x

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a

t







Worksheet 3: CAPM Ranking Tasks!

v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(a)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(b)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(c)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(d)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(e)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(f)

! v (m/s)

t (s)

2

1

0

-1

-2

-3

-4

3

1 2 3 4 5

(g)

! v (m/s)

t (s)

2

1

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-2

-3

-4

3

1 2 3 4 5

(h)

! v (m/s)

t (s)

2

1

0

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-2

-3

-4

3

1 2 3 4 5

(i)

6. Rank the situations (a through i) based on the total distance traveled for the object during the time shown on the graph. Write your answers on a single line using the > and = signs to show the relationships.

Explain the reason for your ranking. Try to make a single, clear statement that applies to every case rather than enumerating the work for each case.

7. Rank the situations based on the maximum absolute value of acceleration during the time shown on the graph.




The following situations illustrate the position of two different balls at different times. The first ball (a through f) rolls with constant velocity across a horizontal surface, while the second ball (g through l) rolls with constant acceleration down an inclined ramp. Both objects are at position zero at time = 0, and both are at position = d at time = 6 s.!

(a) t = 0s, pos = 0m !

(b) t = 2s

!

(c) t = 4s

!(d) t = 6s, pos = d

!(e) t = 8s

!(f) t = 10s

!

(g) t = 0, x = 0, v = 0

!

(h) t = 2s

!

(i) t = 4s

!

(j) t = 6s, pos = d

!

(k) t = 8s

!

(l) t = 10s

8. Rank each situation (a through l… yes, all 12 together, not two separate lists) according to the position along the surface of the ball at the indicated time. Write your answer on a single line, using the > and = signs to show the relationships. NOTE: The pictures are not drawn to scale, so you cannot rely on them to show which ball is ahead.


Rank each situation (a through l) according to the instantaneous velocity of the ball at the indicated time. Write your answer on a single line, using the > and = signs to show the relationships. NOTE: The pictures are still not necessarily drawn to scale.




Worksheet 4: Apply the Model

9. Read the following three problems and consider if the Constant Velocity Particle Model (CVPM) applies.I. A Mac Truck starts from rest and reaches a speed of 8.5 m/s in 20 seconds.II. A dune buggy travels for 20 seconds at a speed of 8.5 m/s. III. A driver sees a deer in the road ahead and applies the brakes. The car slows to a stop from 8.5 m/s in 20 seconds.

a. For each of the three above problems, say whether CVPM applies, whether BFPM applies, whether CAPM applies, or whether none of those models apply, and explain your reasoning.

b. For each problem where CAPM applies, draw at least three diagrams and/or graphs to illustrate the situation. Choose the diagrams and graphs that you find most useful.

c. Using the constant acceleration particle model, solve for any unknown quantities. Show your work and use units.



For the following problems, a complete solution will consist of (a) at least three diagrams and/or graphs to represent the situation. (Use the ones you find most useful.)(b) a determination of the quantities for which it is possible to solve.(c) a clear presentation of the procedure used to produce a numerical answer for each unknown quantity, with units.

10. A car whose initial speed is 30 m/s slows uniformly to 10 m/s in 5 seconds.

11. A bear spies some honey 10 m away and takes off from rest, accelerating at a rate of 2.0 m/s2.

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!

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CAPM Model Summary



hphys unit 03 capm packet 2012

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