Question

Two swimmers with a soft spot for physics engage in a peculiar race that models a famous optics experiment: the Michelson-Morley experiment. The race takes place in a river \(50.0 \mathrm{~m}\) wide that is flowing at a steady rate of \(3.00 \mathrm{~m} / \mathrm{s} .\) Both swimmers start at the same point on one bank and swim at the same speed of \(5.00 \mathrm{~m} / \mathrm{s}\) with respect to the stream. One of the swimmers swims directly across the river to the closest point on the opposite bank and then turns around and swims back to the starting point. The other swimmer swims along the river bank, first upstream a distance exactly equal to the width of the river and then downstream back to the starting point. Who gets back to the starting point first?

Short answer

Answer: The swimmer who swims across the river and back arrives at the starting point first.

Step-by-step solution

Time taken by the swimmer swimming across the river and back

To determine the time taken by the swimmer swimming across the river and back, we must determine the time taken for each leg of the journey separately. The swimmer swims in a direction perpendicular to the flow of the river, so we must find the "resultant" swimming speed in the direction across the river. We know the swimmer's speed with respect to the stream, \(5.00\,\mathrm{m/s}\), and the stream's speed \(3.00\,\mathrm{m/s}\). We can use the Pythagorean theorem to find the resultant speed $$v = \sqrt{(\text{swimmer's speed})^2 - (\text{stream's speed})^2}.$$ Next, we can find the time taken to swim across the river $$t_{\text{across}} = \frac{\text{width of river}}{v}$$ and back $$t_{\text{back}} = \frac{D}{v}.$$ The total time taken will be the sum of \(t_{\text{across}}\) and \(t_{\text{back}}.\)

Time taken by the swimmer swimming upstream and downstream

For the swimmer swimming along the river bank, the time taken for each portion of the journey is found differently. To swim upstream, the swimmer swims against the flow of the river, so their speed relative to the stationary bank will be $$v_{\text{upstream}} = \text{swimmer's speed} - \text{stream's speed}.$$ The time taken to swim this distance is $$t_{\text{upstream}} = \frac{\text{length of journey}}{v_{\text{upstream}}}.$$ To swim downstream, the swimmer swims with the flow of the river, so their speed relative to the stationary bank is $$v_{\text{downstream}} = \text{swimmer's speed} + \text{stream's speed}.$$ The time taken to swim this distance is $$t_{\text{downstream}} = \frac{\text{length of journey}}{v_{\text{downstream}}}.$$ The total time is the sum of \(t_{\text{upstream}}\) and \(t_{\text{downstream}}.\) Now that we have calculated the time taken for each swimmer, we can compare the two times and determine who gets back to the starting point first.

Calculation and comparison of times

Calculate the time taken by the swimmer swimming across the river and back: $$ v = \sqrt{(5.00\,\mathrm{m/s})^2 - (3.00\,\mathrm{m/s})^2} = 4\,\mathrm{m/s}. $$ $$ t_{\text{across}} = t_{\text{back}} = \frac{50.0\,\mathrm{m}}{4\,\mathrm{m/s}} = 12.5\,\mathrm{s}. $$ Total time is \(2 \times 12.5\,\mathrm{s} = 25.0\,\mathrm{s}\). Calculate the time taken by the swimmer swimming upstream and downstream: $$ v_{\text{upstream}} = 5.00\,\mathrm{m/s} - 3.00\,\mathrm{m/s} = 2.00\,\mathrm{m/s}. $$ $$ t_{\text{upstream}} = \frac{50.0\,\mathrm{m}}{2.00\,\mathrm{m/s}} = 25.0\,\mathrm{s}. $$ $$ v_{\text{downstream}} = 5.00\,\mathrm{m/s} + 3.00\,\mathrm{m/s} = 8.00\,\mathrm{m/s}. $$ $$ t_{\text{downstream}} = \frac{50.0\,\mathrm{m}}{8.00\,\mathrm{m/s}} = 6.25\,\mathrm{s}. $$ Total time is \(25.0\,\mathrm{s} + 6.25\,\mathrm{s} = 31.25\,\mathrm{s}\). Now, compare the times: the swimmer swimming across the river and back took a total time of \(25.0\,\mathrm{s}\), whereas the swimmer swimming upstream and downstream took \(31.25\,\mathrm{s}\). The swimmer who swims across the river and back arrives at the starting point first.

Key concepts

Relative Velocity

When we talk about relative velocity, we refer to the velocity of an object as observed from another moving object's frame of reference. Imagine you're sitting on a train moving at a constant speed and another train on a parallel track starts to pass you. The speed at which the other train appears to move past you is a result of both trains' velocities combined; this effect is precisely what's known as relative velocity.

In our swimmers' scenario, one of them swims across a river with a flowing current. The swimmer's velocity relative to the shore is not just the speed of their swimming but it combines with the speed of the river's flow. It is crucial to consider both these speeds for calculating the time it will take for the swimmer to reach the other side and back, using the relative velocity concept.

Pythagorean Theorem

A cornerstone of geometry, the Pythagorean theorem, states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is written as \( a^2 + b^2 = c^2 \), where \( c \) represents the length of the hypotenuse, and \( a \) and \( b \) represent the lengths of the triangle's other two sides.

Utilizing this theorem helps the swimmer across the river calculate the resultant speed needed to counter the river's current. Since the river's current creates a scenario where the swimmer's velocity and the river's velocity form a right-angled triangle, the Pythagorean theorem becomes an invaluable tool for finding the effective swimming speed directly across the river.

Resultant Speed

Resultant speed is the combined effect of two or more velocity vectors. It's the total speed and direction in which an object moves relative to a point of reference. The concept is essential in physics and in explaining everyday phenomena.

The swimmer swimming across a flowing river is subjected to both the speed of their swimming and the river's current. To find the swimmer’s resultant speed across the river, the Pythagorean theorem is applied. For the other swimmer, the resultant speed is different when moving upstream compared to when moving downstream because the river's current either opposes or aids the swimmer's efforts.

Resultant Speed when Swimming Across

For the swimmer moving across the river, we calculate the resultant speed using the swimmer's speed perpendicularly to the flow of the river and the river's current speed. Both these velocities at a right angle create a scenario where the vector sum, which constitutes the true velocity across the river, is found using the Pythagorean Theorem.

Resultant Speed when Swimming Along

In contrast, the other swimmer's resultant speeds upstream and downstream are linear combinations of swimming speed and current speed. The upstream resultant speed is the difference between the swimming speed and the current because they're in opposite directions, while the downstream resultant speed is their sum because they're in the same direction.