Final answer:
The speed of the river current with respect to the ground is 0.8 m/s. To find the swimmer's speed relative to a friend on the shore, we combine the swimmer's velocity across the river (0.5 m/s) with the river's current speed (0.8 m/s) to get approximately 0.94 m/s.
Step-by-step explanation:
To answer how fast the water in the river is flowing with respect to the ground and the speed of the swimmer with respect to a friend on the ground, we have to use concepts of relative motion and vector addition.
The athlete swims perpendicularly to the current, which means the velocity of the swimmer relative to the water is pointing directly across the river. However, the current adds a downstream component to this motion. Since the swimmer ends up 40 m downstream upon crossing the 25 m width, we can calculate the speed of the river's current (vcurrent) using the Pythagorean theorem.
The horizontal distance covered downstream: 40 m
The vertical distance covered across the river: 25 m
The swimmer's speed across the river (perpendicular to the current): 0.5 m/s
Now, let's calculate the time (t) it takes for the swimmer to cross the river: t = 25 m / 0.5 m/s = 50 s.
The river current's velocity can then be calculated: vcurrent = 40 m / 50 s = 0.8 m/s.
To determine the swimmer's speed relative to an observer on the shore, we use vector addition:
The vertical component (across the river) remains at 0.5 m/s.
The horizontal component (downstream) is the current's speed of 0.8 m/s.
We find the resultant speed by using the Pythagorean theorem: Speedrelative = √(0.52 + 0.82) m/s ≈ 0.94 m/s.
This is the speed of the swimmer relative to the friend at rest on the ground.