Two swimmers, Alan and Beth, start at the same point in a...

Two swimmers, Alan and Beth, start at the same point in a stream that flows with a speed $v$. Both move at the same speed $c(c>v)$ relative to the stream. Alan swims downstream a distance $L$ and then upstream the same distance. Beth swims such that her motion relative to the ground is perpendicular to the banks of the stream. She swims a distance $L$ in this direction and then back. The result of the motions of Alan and Beth is that they both return to the starting point. Which swimmer returns first? (Note: First guess at the answer.)

Key Concepts

Relative Velocity

This concept concerns the way an object's speed and direction relative to a moving medium is combined with the velocity of that medium to yield the object’s overall or ground velocity. It is crucial in understanding how swimmers, vehicles, or other objects move in environments where the surrounding medium (such as a river or wind) has its own motion.

Vector Addition of Velocities

In problems involving motion in a moving medium, velocities are vectors, meaning they have both magnitude and direction. The overall or effective velocity is found by vectorially adding the object's velocity relative to the medium with the velocity of the medium itself. This concept is vital for accurately determining paths and transit times when directions do not align with the flow of the medium.

Effective Speed in Different Directions

The effective speed of an object can vary depending on the direction of travel relative to the medium's movement. For instance, moving with or against the current (or wind) results in different ground speeds compared to moving perpendicular to it. Recognizing these differences is key in analyzing travel times and trajectories in such scenarios.

Time Calculation for Motion

Once the effective speeds are determined, calculating the time taken for a journey involves dividing the distance traveled by the effective speed for each segment of the trip. This concept highlights how even when distances are the same, varying effective speeds can result in different travel times, emphasizing the importance of careful analysis in problems involving relative motion.

Transcript

00:01 Hello.

00:02 In this problem, we have two swimmers that are swimming in a stream of water that already has a speed of v.

00:12 And both of them can swim relative to the water with speed c that is greater than the speed of the stream itself, which is v.

00:20 So here, the stream, let's assume that is upward.

00:24 The stream is always moving with v.

00:29 That's the speed of the stream itself.

00:31 And then we have alan.

00:33 And we have beth.

00:36 Alan moved with his speed, which is c relative to the stream, down the stream for speed for distance l, and then he turned back the same distant l, so he's back at his starting point.

00:51 For beth, she swam perpendicular to the stream according to someone who's standing in the frame of reference outside the stream.

01:00 So she swam l distance.

01:05 To the left and then she swam back to return again to her original point according to someone outside.

01:12 We're just asked to know which one of them returned first, beth or alan, given that both return to the same place.

01:23 So basically we're asked to calculate the time that it took for each swimmer to get back to his original position.

01:30 So let's start with alan.

01:34 In the first part, according to somebody else's frame of reference, alan was moving with velocity that is c plus the speed of the stream itself, which is v.

01:50 And in the second portion, he was moving counter the speed of the stream, so his velocity was c minus v.

01:59 So the time that it took alan to get back to his starting point is l.

02:06 Divided by c plus v for the first portion, and then plus l over c minus v for the second portion.

02:16 Working this out, so we have c squared minus v squared here, and we will just have two lc here.

02:26 You can do it step by step, just some algebra, and you will know that this is the answer, and this is the time that it takes alan to get back to.

02:36 His starting point.

02:38 Now let's move to beth.

02:43 Beth was always moving in order to counter the effect of the stream pushing upward on her.

02:50 Whether she was moving to the left or to the right, the stream is keeping to push up and she's countering this by expanding part of his velocity to just counter the effect of the stream...

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