The speed of a boat is 5Km/h in still water. It crosses a river of width 1km along the shortest path in 15 minutes.

Question

Velocity of the river is?

Since it covers 1km in 15 mins, the relative velocity of the boat with respect to the river will be $$\frac 14 V_{br}=1$$ $$V_{br}=4km/h$$ So $$V_b=V_{br} + V_r$$ $$V_r=1km/h$$ The right answer is 3km/h, so what am I doing wrong?

Answer 1

The only way to go along the shortest path is by tilting the boat against the river current. A good picture helps:

You're given $V_{BW} = 5kmph$ and $V_{BS} = 1km ~in~15min$ .
You can find $V_{WS}$

Answer 2

You should use the equation

$$V_b^2=V_{br}^2+ V_r^2$$

instead of

$$V_b=V_{br} + V_r$$

because the three velocity vectors are not in the same direction. Rather, they form a right triangle.

Then, you get

$$V_r = \sqrt{ V_b^2 - V_{br}^2} = \sqrt{ 5^2 - 4^2}= 3 \text{km/h}$$