# In what direction should you swim across a river if you want to minimize the time spent in the water?

Feb 25, 2018

Perpendicular to the edge of the river

#### Explanation:

By this way, the distance will be minimized so it becomes the smallest possible distance traveled (essentially it is the displacement), and if the distance is minimized the time will be minimized according to the equation

$t = \frac{d}{v}$

Where $t$ is the time, $d$ is the distance and $v$ is the velocity.

Mar 8, 2018

See explanation...

#### Explanation:

Seek to swim perpendicular to the bank, but do not adjust to compensate for the water carrying you downstream.

That way, your swimming is not wasted in fighting against the current, but all the motion contributed by your swimming will be effective.

If the width of the river is $w$, your swimming speed is ${v}_{s}$ and the speed of the river is ${v}_{r}$, then you will cross the river in time $\frac{w}{v} _ s$, ending up at a position $w {v}_{r} / {v}_{s}$ downstream from where you started, but on the opposite bank.

So if you wanted a point to aim for on the opposite bank, then you could aim for a point at angle ${\tan}^{- 1} \left({v}_{r} / {v}_{s}\right)$ downstream from straight across.

If you instead attempted to swim the shortest distance across the river, then you would actually be in the water somewhat longer.

One way of thinking about this problem is to consider the frame of reference of the river, rather than the land.