GRE Physics Companion: Relative Motion

This companion is designed for rapid review after M01-09. Most GRE style relative motion problems become short once the observer labels are written explicitly before numbers are substituted.

1 Fast triage

Start from

|---------------------| vP ∕G = vP ∕B + vB ∕G. | ----------------------

(1)

Read the subscripts literally: “P relative to G equals P relative to B plus B relative to G.”

For two objects in the same frame,

|----------------| vA ∕B = vA − vB.| ------------------

(2)

In one dimension, choose a positive direction and keep the signs. In two dimensions, work in components before taking magnitudes or angles.

Figure 1. GRE speed triage for relative motion. Write the observer chain first, then use signed scalars in one dimension or components in two dimensions.

2 Common traps

Do not automatically add speeds. Same direction motions usually involve subtraction when one object is viewed from the other.

Do not subtract vector magnitudes when directions differ. Subtract the vectors component by component.

For river and wind problems, the velocity relative to the medium and the velocity of the medium relative to the ground are different vectors.

A frame moving at constant velocity changes measured velocity but not acceleration in Newtonian mechanics.

Figure 2. Common GRE traps in relative motion: sign errors in one dimension, subtracting magnitudes instead of vectors, and confusing velocity relative to a medium with velocity relative to the ground.

3 Worked GRE example 1: moving walkway

A person walks at 1.5 m/s relative to a walkway that moves east at 2.0 m/s relative to the floor. What is the person’s floor speed when walking east? What if the person walks west?

Eastward walking gives

vP∕F = 1.5 + 2.0 = 3.5 m∕s.

(3)

Westward walking gives v_P∕W = −1.5 m/s, so

vP∕F = − 1.5 + 2.0 = 0.5 m∕s.

(4)

The person still moves east relative to the floor in the second case.

4 Worked GRE example 2: river crossing

A boat moves due north at 6 m/s relative to the water while the current is 8 m/s east. Find the ground speed.

The components are perpendicular, so

√------- |vB ∕G| = 62 + 82 = 10 m ∕s.

(5)

The 6-8-10 triangle gives the magnitude immediately.

5 GRE speed questions

A train moves east at 20 m/s. A passenger walks east at 2 m/s relative to the train. The passenger’s ground speed is (A) 18 (B) 20 (C) 22 (D) 40 m/s.
Cars A and B move east at 30 m/s and 24 m/s. The speed of A relative to B is (A) 6 (B) 24 (C) 30 (D) 54 m/s.
Cars A and B move toward one another at 18 m/s and 12 m/s. Their closing speed is (A) 6 (B) 15 (C) 30 (D) 216 m/s.
A boat moves north at 3 m/s relative to the water while the current flows east at 4 m/s. Its ground speed is (A) 1 (B) 5 (C) 7 (D) 12 m/s.
Frame B moves at constant velocity relative to inertial frame G. A particle’s acceleration measured in G is 5 m/s². The acceleration measured in B is (A) 0 (B) less than 5 (C) 5 (D) greater than 5 m/s².
If v_A = 4e_x + 3e_y m/s and v_B = e_x − e_y m/s, then |v_A∕B| is (A) 3 (B) 4 (C) 5 (D) 7 m/s.

6 Answers and rationales

C. 20 + 2 = 22 m/s east.
A. Same-direction relative speed is 30 − 24 = 6 m/s.
C. With east positive, one velocity is negative, so subtraction gives 18 − (−12) = 30 m/s.
B. The perpendicular components form a 3-4-5 triangle.
C. Constant relative frame velocity has zero relative acceleration, so Galilean inertial frames measure the same acceleration.
C. v_A∕B = 3e_x + 4e_y m/s, whose magnitude is 5 m/s.

References

[1] PhysicsLibrary, M01-09, Relative Motion.

[2] S. J. Ling, J. Sanny, and W. Moebs, University Physics, Volume 1, OpenStax, 2016.

"GRE Physics Companion: Relative Motion" is owned by bloftin.
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