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 is measuring it. An object’s velocity relative to the ground = its velocity relative to a moving frame (a train, a river, the air) + that frame’s velocity relative to the ground. Add them as vectors.
1. Along one line
Walking toward the front of a train going 20 m/s east, at 2 m/s: 20 + 2 = 22 m/s east relative to the ground.
Walking toward the back: 20 + (−2) = 18 m/s east.
Two cars heading toward each other at 15 m/s and 10 m/s close the gap at 15 + 10 = 25 m/s.
2. At right angles
A boat points straight across at 4.0 m/s relative to the water while the river flows at 3.0 m/s. Relative to the bank, the boat moves along the diagonal:
√(4.0² + 3.0²) = 5.0 m/s
Not 7.0 m/s: you can only add the numbers directly when the velocities point along the same line.
3. Example, then try it
The river in the figure is 80 m wide. How long does the boat take to cross, and how far downstream does it land?
Only the across part moves it to the far bank: t = 80 m ÷ 4.0 m/s = 20 s.
During those 20 s the river carries it downstream: 3.0 m/s × 20 s = 60 m.
A plane flies north at 120 m/s relative to the air. A wind blows east at 50 m/s. Find the plane’s speed relative to the ground and how far east it drifts in 60 s.
Set up: plane relative to ground = plane relative to air + air relative to ground. North and east are perpendicular.
Speed: √(120² + 50²) = √16,900 = 130 m/s
Drift: the east part of its velocity is 50 m/s, so in 60 s it drifts 50 × 60 = 3000 m east.
Answer: 130 m/s relative to the ground; it drifts 3.0 km east in one minute.
4. Common mistakes
A swimmer swims upstream at 1.5 m/s relative to the water. The river flows at 0.5 m/s. What is her speed relative to the bank?
Her velocity and the river’s point in opposite directions along the same line, so they subtract: 1.5 − 0.5 = 1.0 m/s upstream. The right-triangle method is only for perpendicular velocities.
Ready to practice?
5 questions on Relative Motion, each with a hint and an explanation for every wrong answer.