# Why two doppler shifts when reflecting from a moving object?

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 $\frac{\Delta f}{f_0}=2.86\times10^{-7}$, what is the baseball's speed? (Hint: Are the waves Doppler-shifted a second time when reflected off the ball?)

I know that because the ball has a non-relativistic velocity, there are certain terms of the Doppler effect equation that can be simplified, such that $\frac{\Delta f_0}{f}=\frac{u}{c}$, but I do not understand the relation that exists with the waves reflected in the ball.

The answer to the problem is $u=\frac{\Delta f_0}{2f}(c)=\frac{2.86\times 10^{-7}}{2}(3\times10^8\,{\rm m})=42.9\,{\rm m}\,{\rm s}^{-1}=154\,{\rm km}\,{\rm h}^{-1}$

Where does the "$2$" in $u=\frac{\Delta f_0}{2f}(c)$ of the answer come from?

When the ball moves a distance $x$ towards the real detector, the virtual detector also moves a distance $x$ towards the ball, or a distance $2x$ towards the real gun.
So, after using the standard Doppler shift equation to find the velocity of the virtual source towards the detector, you need to divide the answer by $2$ to find the velocity of the ball...
There is a factor of two because there are two doppler shifts: one when the ball sees the wave from the coach, and one when the coach sees the waves reflected by the moving ball. The baseball sees a fractional frequency shift of $u/c$ owing to the balls own motion. Let's say for concreteness that the radar gun emitted a flash of light every second, and $u/c$ is $10\%$. Then the ball sees flashes of light every $0.9$ seconds, since it is moving towards the coach. Therefore the ball is relecting light with a frequency shifted up by $u/c$, so every $0.9$ seconds.
But now the coach is seeing this signal coming from a moving source, so there is another doppler shift owing to the fact that the baseball is a moving source. The fractional shift is again $u/c$, so even though the ball is flashing every $0.9$ seconds, the coach sees a flash every $0.8$ seconds, so you get a total fractional shift of $2u/c$, so $\Delta f/f_0=2u/c$.