How Does Skier Speed Affect Wave Frequency and Wavelength?

[mmiller39]

I am trying to work through this problem to no avail.

Homework Statement

A water-skier is moving at a speed of 12.5 m/s. When she skis in the same direction as a traveling wave, she springs upward every 0.498 s because of the wave crests. When she skis in the direction opposite to that in which the wave moves, she springs upward every 0.305 s in response to the crests. The speed of the skier is greater than the speed of the wave. Determine (a) the speed and (b) the wavelength of the wave.

Homework Equations

we know that v = wavelength/ period

and that wavelength equals V/f

The Attempt at a Solution

I am having trouble deriving period from the given data. I know that the speed of the wave is propotional to the speed of the water skier, but I am not sure about the correct ratio to use.

any help is appreciated.

-Matt

 

[berkeman]

The speed of the wave has nothing to do with the speed of the water skier. The rate at which the skier hits the wave peaks is related to both the speed of the skier and the wave speed. What is the apparent wavelength of the water wave from the skier's point of view ("reference frame") in each direction? And how can you use those wavelengths to get you your answers?

 

[kyle8921]

Hello Matt,

I can understand your confusion with this problem. Let's break it down step by step.

First, we know that the speed of the water skier (12.5 m/s) is greater than the speed of the wave. This means that the skier is catching up to the wave, causing her to spring upward every 0.498 s when skiing in the same direction as the wave. In this case, the wave crests are occurring more frequently due to the skier's speed.

Next, when the skier is skiing in the opposite direction of the wave, she is moving away from the wave and therefore the wave crests are occurring less frequently, causing her to spring upward every 0.305 s.

Now, we can use the equation v = λ/T, where v is the speed of the wave, λ is the wavelength, and T is the period. We can also use the equation λ = v/f, where f is the frequency of the wave. Since we know that the skier is skiing at a constant speed of 12.5 m/s, we can use this value for v in both equations.

For the first scenario (skier skiing in the same direction as the wave), we can use T = 0.498 s and solve for λ using the equation v = λ/T. This gives us a wavelength of approximately 6.28 m.

For the second scenario (skier skiing in the opposite direction of the wave), we can use T = 0.305 s and solve for λ using the equation v = λ/T. This gives us a wavelength of approximately 3.82 m.

So, the speed of the wave in both scenarios is the same (since the skier is moving at a constant speed), but the wavelength changes depending on the direction of the skier's movement.

I hope this helps you understand the problem better. Good luck with your studies!

Best,