A motorboat whose speed in still water is 3.00 m/s must aim upstream at an angle of 27.0^\circ (with respect to a line perpendicular to the shore) in order to travel directly across the stream. What is the speed of the current?
To find the speed of the current, we need to break down the motion of the boat relative to the river.
Let's assume that the speed of the current is represented by v_c (in m/s) and the speed of the boat in still water (also known as the speed of the boat without any current) is represented by v_b (in m/s).
When moving upstream, the boat's speed relative to the river is decreased by the speed of the current. In this case, since the boat is aiming directly across the stream at an angle of 27.0 degrees to the perpendicular line, the component of the boat's speed perpendicular to the river is counteracted by the current.
We can represent this situation mathematically as follows:
v_b - v_c = component of speed perpendicular to the river
We know that the component of the boat's speed perpendicular to the river is given by v_b * sin(27.0 degrees).
Therefore, we can rewrite the equation as:
v_b - v_c = v_b * sin(27.0 degrees)
Now, let's substitute the given value of v_b = 3.00 m/s into the equation:
3.00 m/s - v_c = 3.00 m/s * sin(27.0 degrees)
We can solve this equation to find the value of v_c, which represents the speed of the current.