Can Two Runners Have the Same Average Velocity on Different Routes?

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1D Kinematics: Two Runners, Same Average Velocity

1D Kinematics: Two Runners, Same Average Velocity

PREDICTB’s average velocity, compared with A’s, is…
Detour Dm
Back speedm/s
Race clock 0.00 s of 10.00 s
PHYSICS INSIGHTS

Average velocity sees only the start, the finish and the clock. vavg = Δx ÷ Δt. B’s detour takes B past the mark and back again, but it changes neither Δx nor Δt, so B’s average velocity is A’s. Runner A is that average made real: one steady velocity from start to finish, and on the x–t graph A’s line is the chord from B’s first point to B’s last.

Average speed counts every meter of the route. average speed = d ÷ Δt. Distance never decreases, so d ≥ |Δx| and average speed ≥ |vavg|, equal only when the runner never turns back. It is weighted by time, not by leg: out at 6 m/s and back at 2 m/s averages 3 m/s, not 4, because B spends three times as long on the slow leg. In the shuttle run, A runs at B’s average speed.

On a v–t graph, area is displacement. Area above the time axis counts +, area below counts −, and the net area is Δx. Counted all as +, the shaded area is the distance d. The velocity changes sign where B’s line crosses the axis: that instant is the turning point, the peak on the x–t graph. Each turn here is instant, the usual idealisation.

Simulation by The Science Cube — https://www.thesciencecube.com/

Does a Detour Change Average Velocity?

In short: Average velocity depends only on the start, the finish and the time taken: v_avg = Δx/Δt. A runner who takes a detour but starts and finishes at the same places in the same time has the same average velocity as a runner who goes straight. The detour only adds distance, so the detouring runner has the higher average speed.

Common doubts

Can two runners with different speeds have the same average velocity? Yes. In the simulation, runner B runs 5 m past the mark and back, so B's average speed (3.00 m/s) is higher than A's (2.00 m/s). Both finish 20 m from the start after 10 s, so both have an average velocity of 2.00 m/s.

Does turning around change average velocity? Not by itself. A turn changes the distance covered, not the displacement. Only the start point, the end point and the elapsed time matter.

When are average speed and the size of average velocity equal? Only when the runner never turns back. Any turn adds distance without adding displacement, so average speed ≥ |v_avg| always.

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