Question

The frequency of a radar is $780MHz$ . After getting reflected from an approaching aeroplane, the apparent frequency is more than the actual frequency by $2.6kHz$ . The aeroplane has a speed of

• A. $2km/s$
• B. $1km/s$
• C. $0.5km/s$
• D. $0.25km/s$

Answer 1

The correct option is C $0.5km/s$

$f^{\prime }=\left(\frac{c+{\nu }_a}{c-{\nu }_a}\right)f$
where $c$ is the velocity of the radio wave , an electromagnetic wave , i.e., $c=3\times {10}^{8}m/s$ and ${\nu }_s$ is velocity of aeroplane .
$f^{\prime }-f=[\frac{c+{\nu }_a}{c-{\nu }_a}-1]f$
$\Rightarrow$ $\Delta f=\frac{2{\nu }_{a}f}{c-{\nu }_a}$
Since approaching aeroplane cannot have a speed comparable to the speed of electromagnetic wave , so $v_s\ll c.$
$\therefore$ $\Delta f=\frac{2{\nu }_{a}f}{c}$
$\Rightarrow$ $2.6\times {10}^3=\frac{2{\nu }_A(780\times {10}^6)}{3\times {10}^8}$
$\Rightarrow$ ${\nu }_A=0.5\times {10}^{3}m/s$
$=0.5km/s$