Two cars are moving on two perpendicular roads towards a crossing with uniform speeds of $72\text{…
Two cars are moving on two perpendicular roads towards a crossing with uniform speeds of $72\text{ kmh}^{-1}$ and $36\text{ kmh}^{-1}$. If first car blows horn of frequency $280\text{ Hz}$, then the frequency of horn heard by the driver of second car when line joining the cars make $45^\circ$ angle with the roads, will be
$321\text{ Hz}$
$298\text{ Hz}$
$289\text{ Hz}$
$280\text{ Hz}$
Solution
Velocity, $v_A = 72\text{ km h}^{-1} = 20\text{ ms}^{-1}$
Velocity, $v_B = 36\text{ km h}^{-1} = 10\text{ ms}^{-1}$
Frequency of horn heard by the driver,
[The diagram shows velocity vectors $v_A$ and $v_B$ at $45^\circ$ angles relative to the reference axis]
$f' = f \left( \frac{v + v_B \cos 45^\circ}{v - v_A \cos 45^\circ} \right)$
$= 280 \left( \frac{340 + 10 / \sqrt{2}}{340 - 20 / \sqrt{2}} \right) = 298\text{ Hz}$