You have a boat with a motor that propels it at vboat = 4.5 m/s relative to the water. You point it directly across the river and find that when you reach the other side, you have traveled a total distance of 27 m (indicated by the dotted line in the diagram) and wound up 14 m downstream. What is the speed of the current?
To find the speed of the current, we can use the concept of vector addition.
Let's denote the speed of the boat in still water as v_boat and the speed of the current as v_current. Since the boat moves directly across the river, it is perpendicular to the current.
From the information given in the problem, the boat's speed relative to the water (v_boat) is 4.5 m/s, and the total distance traveled (dotted line in the diagram) is 27 m. The boat is also displaced downstream by 14 m.
To understand how to find the speed of the current, let's break the boat's motion into two components: one parallel to the current (downstream) and one perpendicular to the current (across the river).
The distance traveled downstream can be found by multiplying the time taken to cross the river (t) by the speed of the current (v_current):
Distance downstream = v_current * t
Given that the boat was displaced 14 m downstream, we have:
14 m = v_current * t ---(1)
Now, let's consider the distance traveled across the river. Since the boat moves directly across the river, the distance across is the total distance (27 m) minus the distance downstream (14 m):
Distance across the river = 27 m - 14 m = 13 m
The time taken to cross the river can be calculated by dividing the distance across the river by the speed of the boat relative to the water:
t = distance across the river / v_boat
t = 13 m / 4.5 m/s = 2.89 s
Substituting this value of t into equation (1):
14 m = v_current * 2.89 s
Now, we can solve for the speed of the current:
v_current = 14 m / 2.89 s
Calculating this gives us:
v_current ≈ 4.84 m/s
Therefore, the speed of the current is approximately 4.84 m/s.