The water in a river flows uniformly at a constant speed of 2.53 m/s between parallel banks 69.8 m apart. You are to deliver a package directly across the river, but you can swim only at 1.74 m/s.

Question

(a) If you choose to minimize the time you spend in the water, in what direction should you head? ______° from the direction of the stream

(b) How far downstream will you be carried? ______m

(c) If you choose to minimize the distance downstream that the river carries you, in what direction should you head? ______° from the direction of the stream

(d) How far downstream will you be carried? ______m

Any help as to how to start this or which formula to apply would be great. I'm really confused with motion in two dimensions.

Answer 1

I suppose you have done vectors to do this exercise.

Basically, you'd need to represent each velocity in magnitude and direction, and proceed to add them up verctorially to get the resultant.

For the first part, to swim across in minimum time, I would aim right across the river (90° to the flow), and let the river take me wherever I end up. Any other direction will cost me a component to fight the current or help the current. The time is exactly the distance divided by the swimmer's still water speed.

The velocities can be represented by a triangle ABC, in which AB(1.74) is the swimmer's still water velocity across the river. BC(2.53) is the river's current along the river towards downstream. ∠ABC is a right angle. AC represents the resultant velocity and can be obtained by Pythagoras theorem. (not required for answer).

For parts (C) and (D), it is a little more tricky because the swimmer's speed is slower than the current's. He will not be able to make it straight across, but he can minimize the distance carried downstream by making the angle BCA as big as possible. I will leave it to you to find the solution.