A water-skier is moving at a speed of 12.0 m / s. When she...

A water-skier is moving at a speed of $12.0 \mathrm{~m} / \mathrm{s}$. When she skis in the same direction as a traveling wave she springs upward every $0.600 \mathrm{~s}$ because of the wave crests. When she skis in the direction opposite to the direction in which the wave moves, she springs upward every $0.500 \mathrm{~s}$ 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.

A water-skier is moving at a speed of $12.0 \mathrm{~m} / \mathrm{s}$. When she skis in the same direction as a traveling wave she springs upward every $0.600 \mathrm{~s}$ because of the wave crests. When she
skis in the direction opposite to the direction in which the wave moves, she springs upward every $0.500 \mathrm{~s}$ 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.

Key Concepts

Relative Motion

This concept involves understanding that the observed intervals between wave crests are affected by the motion of the observer relative to the wave. The effective speed at which the crests are encountered is the difference (or sum) of the observer's speed and the wave's speed, depending on whether they are moving in the same or opposite directions.

Wave Frequency and Period

The frequency of a wave is the number of crests that pass a point per unit time, and the period is the time interval between successive crests. In this context, the measured period is modified by the relative speed between the skier and the wave, leading to different apparent frequencies.

Wavelength

Wavelength is the distance between successive wave crests. It is related to the wave's speed and its intrinsic frequency. In problems like this, knowing the relationship between the time interval (period), the observer's relative speed, and the wavelength allows one to determine the actual wave's properties.

Frame of Reference

Analyzing the situation from the framework of different frames of reference, such as the moving water-skier versus the stationary medium, is crucial. This concept explains how the relative motion between observer and wave alters the perception of time intervals between wave crests and is key to setting up the equations for solving the problem.

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