# Sarah can paddle a rowboat at 6 m/s in still water. She heads out across a 400 m river at an angle of 30 upstream. She reaches the other bank of the river 200 m downstream from the direct opposite point from where she started. Determine the river current?

May 27, 2018

Let us consider this as a projectile problem where there is no acceleration.
Let ${v}_{R}$ be river current. Sarah's motion has two components.

1. Across the river.
2. Along the river.
Both are orthogonal to each other and therefore can be treated independently.
3. Given is width of river $= 400 \setminus m$
4. Point of landing on the other bank $200 \setminus m$ downstream from the direct opposite point of start.
5. We know that time taken to paddle directly across must be equal to time taken to travel $200 \setminus m$ downstream parallel to the current. Let it be equal to $t$.

Setting up equation across the river

$\left(6 \cos 30\right) t = 400$
$\implies t = \frac{400}{6 \cos 30}$......(1)

Equation parallel to the current, she paddles upstream

$\left({v}_{R} - 6 \sin 30\right) t = 200$ .....(2)

Using (1) to rewrite (2) we get

$\left({v}_{R} - 6 \sin 30\right) \times \frac{400}{6 \cos 30} = 200$
$\implies {v}_{R} = \frac{200}{400} \times \left(6 \cos 30\right) + 6 \sin 30$
$\implies {v}_{R} = 2.6 + 3$
$\implies {v}_{R} = 5.6 \setminus m {s}^{-} 1$