Question

A motorcycle's horn is blowing out a $3015\text{Hz}$ sound as it moves eastward towards a stationary observer. A train sounds its horn at $855\text{Hz}$ as it heads westward away from that same observer. If the observer hears the train as $850\text{Hz}$ and the motorcycle's horn as $3060\text{Hz}$, what is the speed of the motorcycle relative to the train (in $\text{m/s}$) ? Use $340\text{m/s}$ as the speed of sound in air.

Answer 1

For motor cycle the observed frequency when it is moving closer to the observer.
$f=f_0\left(\frac{V}{V-V_S}\right)$
$\Rightarrow 3060=3015\left(\frac{340}{340-V_M}\right)$
$V_m\to$ Velocity of motorcycle
$\Rightarrow 340-V_m=\frac{3015}{3060}\times 340$
$\Rightarrow V_m=340-335$
$\Rightarrow V_m=5\text{m/s}$
For train; the observed frequency when it is moving away from the observer.
$f=f_0\left(\frac{V}{V+V_S}\right)$
$V_T\to$ Velocity of train
$V_T\to 340+V_T=\frac{340\times 855}{850}$
$\Rightarrow V_T=342-340$
$\Rightarrow V_T=2\text{m/s}$
$\Rightarrow V_{relative}=V_M-V_T$
$\Rightarrow V_{rel}=5-(-2)=7\text{m/s}$