Question

A radar operates at wavelength $50.0\, cm$. If the beat freqency between the transmitted singal and the singal reflected from aircraft $(\Delta v)$ is $1\, kHz$, then velocity of the aircraft will be :

• A. $800 \,km / hr$
• B. $900 \,km / hr$
• C. $1000\, km / hr$
• D. $1032\, km / hr$

Answer 1

Answer: (B)

When source is fixed and observer is moving towards it, by Doppler's effect the apparent frequency is given as
$v^{\prime}=\frac{c+a}{c} \cdot v$
When source is moving towards observer at rest, we use
$ v^{\prime \prime}=\frac{c}{c-a} v^{\prime}=\frac{c+a}{c-a} \cdot v=c\left[\frac{1+\frac{a}{c}}{1-\frac{a}{c}}\right] v$
$\Rightarrow v^{\prime \prime}=c\left[1+\frac{a}{c}\right]\left[1-\frac{a}{c}\right]^{-1} v \approx\left[1+\frac{2 a}{c}\right] v $
$\Rightarrow \Delta v=v^{\prime}-v=\frac{2 a v}{c}=\frac{2 a}{\lambda} $
$\Rightarrow a=\lambda \frac{\Delta v}{2}$
$=\frac{0.5 \times 1000}{2}=250 \,ms ^{-1}$
$\Rightarrow a=900\, km / hr$