Ride Along

Riding the ground

Frames of reference

Ride along

Two trains and a walker inside one of them. Pick who you are riding with: the scene is redrawn as that observer sees it, and every velocity is measured from there.

Changing who watches slides every velocity by the same amount — the whole line of marks shifts until your own mark sits on zero. The gaps between the marks never move, which is why every observer measures the same velocity of A relative to B. v_AB = v_A − v_BNo observer is the right one. The ground is just another passenger.

What every observer agrees on

v_A − v_B, A measured from B—

All values

Velocities as this observer measures them

A the purple train—

B the blue train—

W the walker, inside A—

G the ground—

The same from every frame

v_W − v_A, the walker measured from A—

v_W − v_B, the walker measured from B—

the gap between the trains, now—

A relative to B

Over the ground

frame you are riding

train A over the ground

train B over the ground

walker over the ground

train A

train B

the walker, along the aisle of A

leave trails show the velocity line

Guided investigation — ages 16–18 physics

Measured from where?

Every task asks for a written prediction first. Relative motion is the topic where the answer you feel and the answer you get come apart most often, so the value is in recording what you believed before the stopwatch settled it.

1Frames

Nobody is standing still

Mode: Ride along · 15 min

Predict

Train A does 25 m s⁻¹ and train B does 18 m s⁻¹, both the same way. You climb from the ground into train B. Write down what happens to: the velocity of A, the velocity of the ground, and the velocity of A measured from B.

Read all four velocities off the velocity line from each of the four frames in turn. Tabulate them.
Calculate v_A − v_B in every frame. State what you notice.
The walker strolls forward at 1.4 m s⁻¹ along train A. Determine the walker's velocity over the ground, and from train B.
Explain why the ground is not a special frame, using your table rather than words alone.
Set both trains to the same velocity. Describe what a passenger in A sees B doing, and explain what that would look like on the velocity line.

2Closing speed

Why overtaking takes so long

Mode: Close the gap · 20 min

Predict

Two cars approach each other, each at 30 m s⁻¹, starting 600 m apart. State how long until they meet. Then the same two cars drive the same way, 30 and 28 m s⁻¹, 600 m apart. State how long now.

Measure the time to meet for both cases with the stopwatch. Compare with gap ÷ closing speed.
Calculate how far each car travels over the ground in the head-on case, and show that the two distances add to the gap.
Set the cars to the same speed in the same direction. Explain what the gap does and what the closing speed is.
A lorry does 25 m s⁻¹ and you overtake at 27 m s⁻¹. It is 18 m long and you need 40 m of clear road. Determine how long the overtake takes, and comment on what that means on a single carriageway.

3Vectors

The ferry and the river

Mode: Cross it · 25 min

Predict

The ferry points straight across at 4.0 m s⁻¹ and the current runs at 2.5 m s⁻¹. The current then doubles. State what happens to the crossing time and to where the ferry lands.

With the heading straight across, record the crossing time and the drift at three currents. State which one changes.
Explain the result using the across-component and the downstream-component separately.
Determine the heading that lands the ferry straight opposite, and check it against sin⁻¹(v_wg / v_bw).
Turn the current above the ferry's own speed. Explain why no heading lands it straight opposite, and state the best it can do.
Compare the crossing time for the straight-across heading with the one that lands opposite, and discuss which a ferry master would choose.

The physicsGalilean velocity addition, exact in every readout: v_PA = v_PB + v_BA, so a change of frame subtracts one velocity from every other. Speeds here are everyday ones, where this is accurate to better than one part in 10¹⁴.

The modelConstant velocities, no acceleration, no wind, no shallows. The trains run on straight parallel track; the river runs straight with the same current at every point, which a real river does not: it is fastest mid-stream and nearly still at the banks.

The apparatusA 4.0 m mast on train A drops a steel ball; two 90 m trains on parallel track with a 250 m stretch in view; a 600 m measured road with a stopwatch reading to 0.01 s; an 80 m river crossed by a 4.0 m s⁻¹ ferry. Distances on the drawing are to scale within each mode.

What it is built onThe documented difficulty that students treat one frame as the true one and cannot transform between frames, and that they add or subtract speeds without regard to direction: Panse, Ramadas & Kumar (1994), Int. J. Sci. Educ. 16, 63 — 73% of 111 undergraduates agreed a ship is stationary for an observer walking along its deck; Ramadas, Barve & Kumar (1996), same journal — 57% of 102 senior undergraduates said a swimmer going upstream beside a drifting boat takes longer, with the distance measured on the boat; Saltiel & Malgrange (1980), Eur. J. Phys. 1, 73, on speed treated as something a body owns; McCloskey, Washburn & Felch (1983), J. Exp. Psychol. Learn. Mem. Cogn. 9, 636, on the straight-down belief in 177 undergraduates.