A swimmer has a swimming speed of 5.35 m/s in still water. When she heads directly across the river, she ends up travelling downstream at an angle of 30° (with respect to a line perpendicular to the shore). [
(a) What is the speed of the current?
(b) What heading would the boat need to have in order to reach a point directly across the river?
(c) If the stream is 15 m wide, how long would a trip directly across the river take?
(a) Let v be the speed of the current. The swimmer's velocity with respect to the ground can be broken into two components: one parallel to the current and one perpendicular to the current. The parallel component is unaffected by the current and has magnitude 5.35 m/s, while the perpendicular component is affected by the current and has magnitude v. Since the swimmer ends up travelling downstream at an angle of 30°, we can use trigonometry to write:
tan(30°) = v / 5.35
Solving for v, we get:
v = 1.84 m/s
Therefore, the speed of the current is 1.84 m/s.
(b) In order to reach a point directly across the river, the swimmer needs to swim at an angle that is the complement of the angle between her downstream velocity and a line perpendicular to the shore. This angle is 60°, so the swimmer needs to swim at an angle of:
90° - 60° = 30°
with respect to a line perpendicular to the shore.
(c) The time it takes to cross the river is given by the distance across the river divided by the component of the swimmer's velocity that is perpendicular to the current. The distance across the river is 15 m, and the perpendicular component of the swimmer's velocity is:
5.35 sin(30°) = 2.68 m/s
Therefore, the time it takes to cross the river is:
t = 15 m / 2.68 m/s ≈ 5.60 s