A baseball coach uses a radar device to measure the speed of...

A baseball coach uses a radar device to measure the speed of an approaching pitched baseball. This device sends out electromagnetic waves with frequency $f_0$ and then measures the shift in frequency $\Delta{f}$ of the waves reflected from the moving baseball. If the fractional frequency shift produced by a baseball is $\Delta{f}/f_0$ = 2.86 $\times$ 10$^{-7}$, what is the baseball's speed in km/h ? ($Hint$: Are the waves Dopplershifted a second time when reflected off the ball?)

Key Concepts

Speed Conversion and Units

Calculating physical speeds often requires converting from one unit to another, such as from meters per second (m/s) to kilometers per hour (km/h). Understanding unit conversion is important for interpreting the measurements in appropriate real-world units appropriate to the context of the problem.

Double Doppler Shift (Reflection Doppler Effect)

When electromagnetic waves hit a moving object and are reflected, the Doppler shift occurs twice: once when the waves approach the moving object, and a second time when the waves are reflected back. This results in a frequency change that is effectively doubled compared to a single Doppler interaction, a critical factor in accurately determining the speed of the moving object.

Doppler Effect

The Doppler Effect refers to the change in the frequency of a wave as measured by an observer moving relative to the source of the wave. In the context of electromagnetic waves such as radar, it describes how the frequency of the wave shifts when the source and the reflector (or observer) are in relative motion.

Fractional Frequency Shift

The fractional frequency shift, represented as the ratio of the change in frequency to the original emitted frequency (?f/f?), is a dimensionless measure that directly relates the observed Doppler shift to the speed of the moving target relative to the speed of the wave propagation. For electromagnetic waves, this relationship involves the speed of light.

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