A rocket is going towards moon with a speed v. The astronaut in the rocket sends signals of frequency v towards the moon and receives them back on reflection from the moon. What will be the frequency of the signal received by the astronaut (Take v << c)

$\frac{c}{c-2 v} v$

In this case, we can assume as if both the source and the observer are moving towards each other with speed v. Hence $v'=\frac{c-u_o}{c-u_s} v=\frac{c-(-v)}{c-v} v=\frac{c+v}{c-v} v$

$=\frac{(c+v)(c-v)}{(c-v)^2} v=\frac{c^2-v^2}{c^2+v^2-2 v c} v$

Since v << c, therefore $v'=\frac{c^2}{c^2-2 v c}=\frac{c}{c-2 v} v$