$A$ rocket is moving at a speed of $200\; m s^{-1}$ towards a stationary target. While moving, it emits a wave of frequency $1000\; Hz$. Some of the sound reaching the target gets reflected back to the rocket as an echo. Calculate:
$(1)$ the frequency of the sound as detected by the target and
$(2)$ the frequency of the echo as detected by the rocket.

(N/A) $(1)$ The observer (target) is at rest $(v_o = 0)$ and the source (rocket) is moving towards the target with speed $v_s = 200\; m s^{-1}$. The speed of sound is $v = 330\; m s^{-1}$.
Using the Doppler effect formula for a moving source and stationary observer:
$f' = f \left( \frac{v}{v - v_s} \right)$
$f' = 1000\; Hz \times \left( \frac{330}{330 - 200} \right) = 1000 \times \left( \frac{330}{130} \right) \simeq 2538.46\; Hz$.
$(2)$ Now, the target acts as the source of the reflected sound (echo) and is stationary $(v_s = 0)$. The rocket acts as the observer moving towards the source with speed $v_o = 200\; m s^{-1}$. The frequency of the source is the frequency received by the target, $f' = 2538.46\; Hz$.
Using the Doppler effect formula for a stationary source and moving observer:
$f'' = f' \left( \frac{v + v_o}{v} \right)$
$f'' = 2538.46\; Hz \times \left( \frac{330 + 200}{330} \right) = 2538.46 \times \left( \frac{530}{330} \right) \simeq 4076.16\; Hz$.