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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 2.9 Representations of Motion

You now have two major representations for describing motion: Motion Diagram and Motion Graph. Coordinating these representations with words and equations can give you a powerful understanding of an object’s motion.

Example 2.9.1. Constant and Changing Speed.

The simplest possible motion is motion along a straight-line path with a constant, unvarying speed. Let’s imagine a car that is driving to the right at a constant speed. Suppose you take images at a regular time interval, once every second:
Six pictures of a car arranged horizontally. A blue arrow above each car points to the right.
Figure 2.9.2. A stroboscopic diagram of a car moving at constant speed.
You can apply the particle model and represent the car as a point object, creating a motion diagram showing the position of the car at each instant of time and the corresponding displacement vectors.
Six horizontally spaced dots, each dot separated by a labeled arrow.
Figure 2.9.3. A motion diagram of a car moving at constant speed.
Notice that the displacement vectors are equal in length when the car moves at constant speed.

Exercises Practice Activities

Activity 2.9.1. The Tortoise down the Slide.

It takes \(40 \mathrm{~s}\) for a tortoise to travel down the slide shown in the figure below. Its initial speed is \(0 \mathrm{~m/s}\) and its final speed is \(0.2 \mathrm{~m/s}\) down the ramp. Assume the slide is \(4 \mathrm{~m}\) long and the given angle is equal to \(22^o\text{.}\)
A tortoise at the top of a slide.
Figure 2.9.4. A tortoise at the top of a slide.

(a) Practice Creating a Strobe Diagram.

Sketch a qualitatively accurate strobe diagram for the tortoise using the given information. Your diagram should have about 4-6 points.

(b) Calculation.

Determine the magnitude of the tortoise’s average acceleration. What direction does it point?
Answer.
\(0.005 \mathrm{~m/s^2}\) down the ramp

(c) Key Skill: Breaking a Vector into Components.

Determine the horizontal (\(x\)) component of the tortoise’s average acceleration
Tip.
Draw a right triangle where the hypotenuse is the average acceleration vector and the legs represent the horizontal and vertical components. Be sure to label the given angle \(\theta\) carefully in your triangle!

Activity 2.9.2. Up and Down the Ramp.

Draw a motion diagram for the ball rolling up and then down the ramp..
Figure 2.9.5. A strobe diagram of a ball rolling up and then down a ramp.
Start by sketching velocity vectors for each instant.
Then use the steps below to help you sketch acceleration vectors
  1. Draw a vector for Δv between instants 1 and 2
  2. Use this to draw a vector for a between instants 1 and 2
  3. Repeat the above steps for instants 7 and 8
  4. What was different about these instants?
  5. Repeat the above steps for the instants around the turnaround point at the top of the ramp
  6. What was different about these instants?

Activity 2.9.3. Four Strobe Diagrams.

The initial position in each strobe diagram is labeled \(t_o\text{.}\)
Figure 2.9.6. Four strobe diagrams.
Using a standard reference frame for each case:
  1. Sketch velocity vectors for each point
  2. Sketch acceleration vectors for each point
  3. Identify whether your vectors are positive or negative
  4. Sketch a set of motion graphs

Activity 2.9.4. Motion Diagrams for Two Cars.

Below are strobe diagrams for two cars.
Figure 2.9.7. Two strobe diagrams.
  1. Draw and label \(x\) vs. \(t\) graphs for cars A and B on the same set of axes.
  2. Draw and label \(v\) vs. \(t\) graphs for cars A and B on the same set of axes.
  3. Do the cars ever have the same position at one instant in time?
  4. Do the cars ever have the same velocity at one instant in time?
  5. Do you think it would be appropriate to use a graphing calculator for this activity?

Activity 2.9.5. Changing Velocity.

A total of \(0.2 \mathrm{~s}\) passes between each successive dot in the strobe diagram below. The distance between the leftmost and rightmost points is \(64 \mathrm{~cm}\text{.}\)
Figure 2.9.8. A strobe diagram.
  1. Describe how the velocity of the ball is changing. Explain how you can tell from the strobe diagram.
  2. Make a motion diagram with velocity and acceleration vectors.
  3. What is the average speed during the motion?
  4. Is the instantaneous speed at the point in the middle greater than, less than, or equal to the average speed? Explain your reasoning.
  5. Sketch motion graphs for the ball showing position, velocity, and acceleration vs. time. Confirm that your graph agrees with the derivative relationships between position, velocity, and acceleration.

Activity 2.9.6. Metacognition.

Why do you think working with so many different representations is useful in physics? Cite some specific activities where different representations have led you to understand something differently.

Activity 2.9.7. Relative Bicycles.

You and a friend are each riding a bicycle. You are moving \(8 \mathrm{~m/s}\) to the west relative to the ground. Your friend is moving \(12 \mathrm{~m/s}\) to the west relative to the ground.
  1. Explain why the information given above is not enough for tell whether your friend’s position is east, west, or the same as your position.
  2. Suppose at \(t = 0\) your friend is located \(400 \mathrm{~m}\) east of you. Draw position vs. time graphs for each bicycle.
  3. Use your graphs to write equations that describe the position of each bicycle as a function of time.
  4. Determine when the bicycles are at the same position.
  5. Choose a different reference frame; how would your answers to the above questions change, if at all?

Activity 2.9.8. The Rolling Ball.

The strobe diagram below shows a ball moving from left to right. A total of \(1 \mathrm{~s}\) has passed between each successive dot. The distance between the leftmost and rightmost points is \(12 \mathrm{~m}\text{.}\)
Seven dots distributed horizontally, closer together on the right side and on the left side than in the middle.
Figure 2.9.9. A strobe diagram.

(a) Practice Creating a Motion Diagram.

Use the strobe diagram to make a motion diagram with both velocity and acceleration vectors.
Solution.
Figure 2.9.10. A motion diagram with velocity vectors (middle, red) and acceleration vectors (bottom, blue).
The instantaneous velocity vectors (middle, red) can be found by looking at the change in position on either side of each dot. The average velocity vectors (bottom, blue) can be found by looking at the change in velocity at the start and end of each interval. This diagram assumes that the speed at the start and end is zero, and that the acceleration is zero during the middle.

(b) Explanation.

Describe how the velocity of the ball is changing. Explain how you can tell from the strobe diagram.
Solution.
The velocity always points to the right, and the speed is first increasing and then decreasing. This can be seen by using the definition of velocity, which is the change in position divided by the change in time.

(c) Calculation.

What is the average speed during the motion?
Solution.
\(v_{x,ave} = \frac{\Delta x}{\Delta t} = \frac{12 \mathrm{~m}}{1 \mathrm{~s}} = 2 \mathrm{~m/s}\)

(d) Explanation.

Is the instantaneous speed at the point in the middle greater than, less than, or equal to the average speed? Explain your reasoning.
Solution.
The instantaneous speed is the magnitude of the change in position divided by the change in time. The change in time is the same for each successive instant in a strobe diagram, so you only have to look at the change in position. Given that the objects are draw farther apart in the middle, the change in position has a bigger magnitude there, and therefore the instantaneous speed is also greater!

Activity 2.9.9. The Thrown Ball.

You have a tennis ball that you throw directly upward with initial speed \(v\text{.}\) The ball rises to height \(h\text{,}\) then falls back down. At time \(t\text{,}\) you catch the ball when it is moving downward with the same speed \(v\text{.}\)

(a) Practice Creating a Motion Graph.

Use the given information to make a graph of the ball’s velocity vs. time. What assumptions did you make about the ball’s motion?

(b) Calculations.

Calculate (1) the average velocity of the tennis ball during this time interval and (2) the change in velocity of the tennis ball over this time interval.
Tip.
Recall the Calculation Steps.

(c) Sensemaking.

Average velocity and change in velocity are both vectors. Show that they are vectors with reasonable directions.

References References

[1]
Tortoise image based on tortoise by vector quipo from Noun Project (CC BY 3.0)
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