A swimmer, capable of swimming at a speed of 1.01 m/s in still water (i.e., the swimmer can swim with a speed of 1.01 m/s relative to the water), starts to swim directly across a 1.47-km-wide river. However, the current is 1.01 m/s, and it carries the swimmer downstream. (a) How long does it take the swimmer to cross the river? (b) How far downstream will the swimmer be upon reaching the other side of the river?
To solve this problem, we'll break it down into two parts.
(a) How long does it take the swimmer to cross the river?
To cross the river, the swimmer needs to reach the other side, which is 1.47 km away. The swimmer's speed in still water is 1.01 m/s, but there is a current of the same speed (1.01 m/s) pushing the swimmer downstream.
Let's assume that the swimmer reaches the other side after time 't'. During this time, the swimmer will be pushed downstream due to the current. The distance covered downstream will be equal to the speed of the current (1.01 m/s) multiplied by the time (t). So, the distance covered downstream is given by:
Distance downstream = Speed of the current × Time
Distance downstream = 1.01 m/s × t
Now, the swimmer needs to cover the remaining distance to reach the other side of the river. The remaining distance is the width of the river minus the distance covered downstream:
Remaining distance = Width of the river - Distance downstream
Remaining distance = 1.47 km - (1.01 m/s × t)
Since the swimmer can swim at a speed of 1.01 m/s in still water, the time taken to cover the remaining distance is given by:
Time taken = Remaining distance / Speed in still water
Time taken = [1.47 km - (1.01 m/s × t)] / 1.01 m/s
Now, we can set up the equation to solve for 't':
[1.47 km - (1.01 m/s × t)] / 1.01 m/s = t
Solving this equation will give us the time it takes for the swimmer to cross the river.
(b) How far downstream will the swimmer be upon reaching the other side of the river?
To find out how far downstream the swimmer will be upon reaching the other side of the river, we can substitute the value of 't' (found in part (a)) into the equation we derived earlier:
Distance downstream = 1.01 m/s × t
Calculating this will give us the distance downstream.
Let's solve these equations step by step to find the answers.
To answer these questions, we can break down the swimmer's motion into two components: the swimming speed across the river and the speed due to the current carrying the swimmer downstream.
Let's call the time it takes to cross the river "t" and the distance downstream "D".
(a) To find the time it takes for the swimmer to cross the river, we can use the formula: time = distance / speed.
The distance to be crossed is the width of the river, which is 1.47 km. However, the swimmer will also be carried downstream by the current, so the effective velocity while crossing will be the difference between the swimmer's speed and the current's speed. In this case, it is 1.01 m/s (swimmer's speed) - 1.01 m/s (current speed) = 0 m/s.
Therefore, the effective speed of the swimmer across the river is 0 m/s, and the time it takes to cross the river can be calculated as follows:
time = distance / speed = 1470 m / 0 m/s = undefined.
Since the effective speed across the river is zero, it will take an infinitely long time to cross. This means the swimmer will not be able to cross the river if they swim directly across against the current.
(b) If the swimmer cannot cross the river directly, they will be carried downstream by the current. To determine how far downstream the swimmer will be upon reaching the other side of the river, we can calculate the distance traveled due to the current.
The time it takes for the swimmer to reach the other side can be approximated by dividing the distance of the width of the river by the effective speed of the swimmer:
time = distance / speed = 1470 m / (1.01 m/s - 1.01 m/s) = 1470 m / 0 m/s = undefined.
As previously established, it will take an infinitely long time for the swimmer to reach the other side while swimming directly across the river against the current. Therefore, the swimmer will be carried downstream for an infinite distance, which is not a physically meaningful answer.
In conclusion, due to the given values, it is not possible to precisely determine the time it takes for the swimmer to cross the river or the distance downstream they will be upon reaching the other side.