Relative Motion
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Velocity depends on who’s measuring. The velocity of A relative to the ground is its velocity relative to the medium plus the medium’s velocity: vA/ground = vA/medium + vmedium/ground, added as vectors.
1. Adding velocities
2. River crossings
Head straight across: fastest crossing. Time = width ÷ vboat. The current pushes you downstream by vcurrent × time.
The current doesn’t change the crossing time, because it’s perpendicular to the motion across.
Land directly opposite: aim partly upstream so the upstream part of your velocity cancels the current: vboat sin θ = vcurrent. You cross more slowly, at √(vboat² − vcurrent²).
3. Example, then try it
Car A drives east at 25 m/s and car B drives east at 15 m/s. What is B’s velocity relative to A? What if B drives west at 15 m/s?
vB/A = 15 − 25 = −10 m/s: 10 m/s west (A sees B falling behind).
B westbound: −15 − 25 = −40 m/s: 40 m/s west.
A river is 120 m wide and flows at 3.0 m/s. A boat moves at 5.0 m/s relative to the water. (a) It heads straight across: how long does it take, and how far downstream does it land? (b) It aims upstream to land directly opposite: at what angle, and how long does it take?
(a) Time: 120 ÷ 5.0 = 24 s. Drift: (3.0)(24) = 72 m downstream.
(b) Angle: sin θ = 3.0/5.0 = 0.60, so θ ≈ 37° upstream from straight across.
Crossing speed: √(5.0² − 3.0²) = 4.0 m/s, so the time is 120 ÷ 4.0 = 30 s.
Answer: (a) 24 s and 72 m downstream; (b) aim 37° upstream, taking 30 s.
4. Common mistakes
You walk at 1.5 m/s toward the back of a train that moves forward at 20 m/s. What is your velocity relative to the ground?
Taking forward as +: v = +20 + (−1.5) = +18.5 m/s.
Ready to practice?
5 questions on Relative Motion, each with a hint and an explanation for every wrong answer.