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 first swimmer (Swimmer A) gets back to the starting point first.

Step-by-step solution

Calculate the time taken by first swimmer: Path A

The first swimmer swims directly across the river and back. The swimmer's velocity, relative to the river, is given by: \(v_s = 5\frac{m}{s}\) Since the swimmer is swimming directly across the river, their velocity is perpendicular to the river's current velocity. To find the time it takes the swimmer to reach the other side, we can use the formula: Time = distance / speed The distance they need to travel across is equal to the width of the river (50 m). \(t_A =\frac{50 \mathrm{m}}{5\frac{m}{s}}= 10 \mathrm{s}\) Since the swimmer returns to their starting point, the time taken for the round trip is: \(t_A = 10 \mathrm{s} \times 2 = 20 \mathrm{s}\)

Calculate the time taken by second swimmer: Path B

In this case, the swimmer's speed relative to the water is \(5\frac{m}{s}\), and swimming upstream means swimming against the river's speed, which is \(3\frac{m}{s}\). We will first find the swimmer's velocity in the XY plane. \(v_x=5\frac{m}{s}-3\frac{m}{s}=2\frac{m}{s}\) Now we need to find the time taken to swim upstream and downstream. \(t_B=\frac{50 \mathrm{m}}{2\frac{m}{s}}= 25 \mathrm{s}\) (upstream) When swimming downstream, the swimmer's speed relative to the water will be added to the river's speed. \(v_x=5\frac{m}{s}+3\frac{m}{s}=8\frac{m}{s}\) \(t_B=\frac{50 \mathrm{m}}{8\frac{m}{s}}= 6.25 \mathrm{s}\) (downstream) Adding these times: \(t_B = 25 \mathrm{s}+ 6.25 \mathrm{s} = 31.25 \mathrm{s}\)

Compare the times taken by both swimmers

Comparing the times taken by both swimmers, we have: \(t_A = 20\mathrm{s}\) (swimmer A) \(t_B = 31.25\mathrm{s}\) (swimmer B) Since the first swimmer (swimming directly across the river and back) takes less time to return to the starting point: The first swimmer (Swimmer A) gets back to the starting point first.

Key concepts

Physics Optics Experiments

Understanding the physics of light is a significant field, primarily explored through optics experiments. One of the most crucial historical experiments in this domain is the Michelson-Morley experiment. This foundational experiment was designed to detect the presence of aether, a medium then thought to permeate space and facilitate the propagation of light. Through meticulous measurements of the speed of light traveling in perpendicular directions, the experiment revealed no significant difference, leading to the conclusion that aether did not exist and paving the way for Einstein's theory of relativity.

While the swimmers' race described in the textbook exercise isn't an optics experiment, it mirrors the essence of the Michelson-Morley setup by comparing two paths - a direct one across the stream and another along with it, akin to light's theoretical path through aether. By analyzing the swimmers' race, students can analogously grasp how relative motion can be used to investigate fundamental properties of our universe - just as Michelson and Morley did with light.

Relative Velocity in Physics

The concept of relative velocity is central to understanding motion from different reference frames. In our case, the textbook exercise involves swimmers moving in a river which has its own flow velocity. Relative velocity is the vector difference between the velocities of two objects, as observed in one of the object's frame of reference. Here, the exercise demonstrates that the swimmers' speeds are measured relative to the water, not the riverbank.

As shown in the exercise, when the swimmer moves across the river, her velocity is perpendicular to the flow. When she goes upstream or downstream, her relative velocity to the river - factoring in the river's flow speed - dictates her actual speed and thus the time taken for each segment of the race. Understanding how the direction of motion affects the relative velocity, and the subsequent impact on the speed of the swimmer in water, is essential for problem-solving in physics, particularly when dealing with vectors.

Problem-Solving in Physics

Problem-solving is at the heart of physics and involves applying concepts to resolve real-world scenarios. The exercise provided is a perfect example of applying theoretical knowledge about relative velocity and motion to solve a practical problem.

When attempting problems in physics, identifying the relevant concepts - such as relative velocity in this case - is crucial. Next is translating the physical scenario into mathematical terms, selecting appropriate formulas, and following a systematic approach to calculations. In the exercise, for instance, computing the time for two different paths taken by the swimmers requires separate calculations for upstream and downstream motion due to differing relative velocities. Adopting a step-by-step strategy for calculations, as shown, helps clarify the process and leads to a logical conclusion - much like in the scientific pursuit of understanding phenomena after careful experimentation.