Answer
(a) The velocity relative to the earth is 4.7 m/s at an angle of $25.5^{\circ}$ south of east.
(b) It will take 119 seconds to cross the river.
(c) The arrival point on the opposite bank will be 238 meters south of the starting point.
Work Step by Step
(a) Let $v_x$ be the speed relative to the earth toward the east.
$v_x = 4.2~m/s$ to the east
Let $v_y$ be the speed relative to the earth toward the south.
$v_y = 2.0~m/s$ to the south
$v = \sqrt{(4.2~m/s)^2+(2.0~m/s)^2} = 4.7~m/s$
We can find the angle $\theta$ south of east.
$tan(\theta) = \frac{2.0}{4.2}$
$\theta = tan^{-1}(\frac{2.0}{4.2}) = 25.5^{\circ}$
The velocity relative to the earth is 4.7 m/s at an angle of $25.5^{\circ}$ south of east.
(b) We can find the time $t$ to cross the river.
$t = \frac{width}{v_x} = \frac{500~m}{4.2~m/s} = 119~s$
It will take 119 seconds to cross the river.
(c) $v_y~t = (2.0~m/s)(119~s) = 238~m$
The arrival point on the opposite bank will be 238 meters south of the starting point.
