Question

A spaceship is approaching the moon with a uniform velocity $\nu$ ($\nu <<c$), emitting microwaves of frequency $n$. Then the apparent change in the frequency observed by the astronaut for the waves reflected back from the surface of the moon will be:

• A. increased by $\nu n/c$
• B. decreased by $\nu n/c$
• C. increased by $2\nu n/c$
• D. decreased by $2\nu n/c$

Answer 1

The correct option is C increased by $2\nu n/c$

This is similar to source moving towards a wall or hill. But remember that microwaves are Em waves. First treating moon's surface as observer, the frequency of waves reaching moon's surface is:
$n^{\prime }=(1+\frac{\nu }{c})n$
This is also the frequency of reflected wave with moon's surface as source. Then, for the approaching astronaut (observer), the apparent frequency will be:
$n"=(1+\frac{\nu }{c})n^{\prime }=(1+\frac{\nu }{c}{)}^{2}n\cong (1+\frac{2\nu }{c})n$
Thus, the apparent change is:
$\Delta n=n"-n=\left(\frac{2\nu n}{c}\right)$