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Learning Introductory Physics with Activities

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

Section 2.4 Relative Motion

In Position Vectors, you defined position vectors in terms of a Coordinate System with a specified origin and axes. In order to describe other vectors, including velocity, you will generalize this idea into a reference frame that also specifies how your coordinate system is moving through space.
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Definition 2.4.1. Reference Frame.

A reference frame is a coordinate system (a set of axes with an origin) for which the velocity is specified. Typically, the velocity is specified to be 0 relative to some object whose motion is known. For example, you might work with the reference frame for the ground or for a car moving at 30 mph down a level street. An example of how to draw different possible reference frames is shown below
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Figure 2.4.2. Two Cartesian reference frames, one moving to the right relative to the other.
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Example 2.4.4. Walking Toward an Airplane.

You are stationary on the ground, watching as a person walks past you traveling east at \(\vec{v}_{p,g} = +5 \hat{x} \mathrm{~km/hr}\) toward the front of an airplane which is traveling west at \(\vec{v}_{a,g} = -150 \hat{x} \mathrm{~km/hr}\text{.}\) The first subscript (\(p\) and \(a\)) represents the object, while the second (\(g\)) indicates that these velocities are given relative to the ground’s reference frame. Since this also happens to be your reference frame, your velocity is \(\vec{v}_{y,g} = 0 \mathrm{~km/hr}\text{.}\)
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From the walking person’s perspective, you are traveling with velocity \(\vec{v}_{y,p} = \vec{v}_{y,g} - \vec{v}_{p,g} = -5 \hat{x} \mathrm{~km/hr}\text{,}\) while the airplane is traveling with velocity \(\vec{v}_{a,p} = \vec{v}_{a,g} - \vec{v}_{p,g} = -155 \hat{x} \mathrm{~km/hr}.\)
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See if you can determine your velocity and the walking person’s velocity relative to the airplane!
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Example 2.4.5. A Thrown Ball in Different Reference Frames.

Imagine you threw and caught a ball while you were sitting on a train moving at a constant velocity past a train station. To you, the ball appears to simply travel vertically up and then down. However, to an observer who stood on the station platform the ball would appear to travel in a parabola, with a constant horizontal component of velocity equal to the velocity of the train.
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Historical Note 2.4.6. Inertial Reference Frames.

The laws of physics privilege reference frames that are not accelerating: such a reference frame is known as an inertial reference frame. The fact that the laws of physics are the same in all inertial frames is known as Galilean invariance, and asserts that there is no experiment you can do to determine which reference frame you are in. Einstein used this hypothesis to devise the theory of Special Relativity.
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Example 2.4.7. Relative Motion with a Cannonball.

The video below illustrates relative motion with a precise and powerful example.
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Exercises Activities

1.

Suppose you are running down a straight track at a speed of \(8 \mathrm{~m/s}\) (relative to the track). Two friends, one in front of you and one behind, each throw a baseball horizontally toward you at a speed of \(24 \mathrm{~m/s}\) (relative to the track). Relative to you, which baseball, if either, has a greater speed? Explain your reasoning.
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Tip.
Relative motion problems can be tricky to visualize, so it often helps to draw a diagram!
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2.

Now suppose you are running down a straight track at a speed of \(8 \mathrm{~m/s}\) (relative to the track). Two friends, one in front of you and one behind, each throw a baseball horizontally toward you. Relative to you, each baseball has a speed of \(24 \mathrm{~m/s}\text{.}\) Relative to the track, which baseball has a greater speed? Explain your reasoning.
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Tip.
The difference between this activity and the previous one is subtle, but impactful!
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References References

[1]
Relative Velocity by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
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[2]
Soccer Ball Shot from Truck by Mythbusters.
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