Relative Velocity

The question

You swim straight across a river while the current pulls you sideways. Where do you land, and where should you aim?

Aiming straight across a 40 m river at 1.0 m/s in a 0.5 m/s current, you land 20 m downstream after 40 s: the current adds a velocity along the river but does not change the crossing time. To land exactly opposite, aim 30° upstream, so that the upstream part of your swimming velocity, 1.0 × sin 30° = 0.5 m/s, cancels the current. The crossing then takes 46 s.

In words, then in symbols

• To land opposite, the upstream part of your swimming velocity must equal the speed of the current.

  $v_{\text{swim}}\sin \varphi =v_{\text{current}}$

How to solve a relative velocity problem

1. Draw each velocity as an arrow.
2. Join them tip to tail; the resultant closes the triangle.
3. Use the part of the resultant across the river to find the crossing time.
4. Use the part along the river, over that same time, to find where you land.