A person is attempting to swim directly across a river. They are able to swim at a speed of 0.500 m/s relative to the water. The river is 25.0 m wide and the person ends up 50.0 m downstream from their starting point. How fast is the current in the river?
The person swims at 0.5m/s.
If the current takes him 50m downstream in the same time it took him to swim 25m across (with or without current), what would be the speed of the current?
To solve this problem, we need to break it down into two components - the person's speed relative to the ground and the speed of the river current.
First, let's consider the person's speed relative to the ground. This can be calculated by using the Pythagorean theorem. We know the person's speed through the water is 0.500 m/s, and they end up 50.0 m downstream from their starting point. From this information, we can determine that their speed relative to the ground is:
speed relative to the ground = √((speed through water)^2 + (distance downstream)^2)
speed relative to the ground = √((0.500 m/s)^2 + (50.0 m)^2)
speed relative to the ground = √(0.250 m^2/s^2 + 2500 m^2)
speed relative to the ground ≈ √2500.250 m^2
speed relative to the ground ≈ 50.02 m/s
Next, let's determine the speed of the river current. We know that the person swam directly across the river, which means they aimed to swim perpendicular to the current. The person ended up 50.0 m downstream, so it took them a certain amount of time to swim directly across. During this time, the river current carried them downstream.
The speed of the river current can be calculated by dividing the distance downstream by the time it took:
speed of current = distance downstream / time
time can be calculated by using the formula:
time = distance across river / speed through water
time = 25.0 m / 0.500 m/s
time ≈ 50 s
Now, we can calculate the speed of the river current:
speed of current = 50.0 m / 50 s
speed of current = 1.00 m/s
Therefore, the speed of the river current is approximately 1.00 m/s.