Question

A radar operates at wavelength $50.0\, cm$ If the beat frequency between the transmitted signal and the signal reflected from aircraft $\Delta v$ is $1 \,kHz$, then velocity(in $km \,h ^{-1}$ ) of the aircraft will be

Answer 1

When source is fixed and observer is moving towards it
$v^{\prime}=\frac{c+a}{c} \times v$
When source is moving towards observer at rest
$v^{\prime \prime}=\frac{c}{c-a} v^{\prime}=\frac{c+a}{c-a} \times v=\frac{1+\frac{a}{c}}{1-\frac{a}{c}} v$
$=1+\frac{a}{c} 1-{\frac{a^{-1}}{c}}^{-1} \approx 1+\frac{2 a}{c} v$
$\therefore \Delta v=v^{\prime \prime} v=\frac{2 a v}{c}=\frac{2 a}{\lambda}$
$\therefore a =\lambda \frac{\Delta v}{2}=\frac{0.5 \times 1000}{2}=250 \,m s ^{-1}$
$=900\, km\, h ^{-1}$