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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 diagrams (see Section 2.6) and motion graphs (see Section 2.8). Coordinating these representations with words and equations can give you a powerful understanding of an object’s motion.
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

A tortoise at the top of a slide.
Figure 2.9.4. A tortoise at the top of a slide.

Activity 2.9.1. The Tortoise down the Slide.

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

(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.
Answer.
\(0.005 \mathrm{~m/s^2}\)

(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!
Seven dots distributed horizontally, closer together on the right side and on the left side than in the middle.
Figure 2.9.5. A strobe diagram.

Activity 2.9.2. The Rolling Ball.

The strobe diagram above 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 \(12 \mathrm{~m}\text{.}\)

(a) Practice Creating a Motion Diagram.

Use the strobe diagram to make a motion diagram with both velocity and acceleration vectors.

(b) Explanation.

Describe how the velocity of the ball is changing. Explain how you can tell from the strobe diagram.
Answer.
The velocity always points to the right, and the speed is first increasing and then decreasing.

(c) Calculation.

What is the average speed during the motion?
Tip.
Recall the steps for performing a calculation from Figure 1.5.9.
Answer.
\(v_{ave} = 2 \mathrm{~m/s}\)

(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.
Answer.
Greater than!

Activity 2.9.3. 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) Calculation.

Determine the average velocity of the tennis ball during this time interval.
Tip.
Recall the steps for performing a calculation from Figure 1.5.9.

(c) Sensemaking.

Average velocity is a vector quantity. Show that it is a vector with a reasonable direction.
Figure 2.9.6. A strobe diagram.

Activity 2.9.4. Changing Velocity.

A total of \(0.2 \mathrm{~s}\) passes between each successive dot. The distance between the leftmost and rightmost points is \(64 \mathrm{~cm}\text{.}\)
  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.

References References

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