Equation of a progressive wave is given by $y = a \sin \pi \left[ \frac{t}{2} - \frac{x}{4} \right]$, where…
Equation of a progressive wave is given by $y = a \sin \pi \left[ \frac{t}{2} - \frac{x}{4} \right]$, where $t$ is in second and $x$ in metre. The distance through which the wave travels in $8\text{ s}$ (in metre) is
(a) $8$
(b) $16$
(c) $2$
(d) $4$
Solution
Velocity, $v_A = 72\text{ km h}^{-1} = 20\text{ ms}^{-1}$
Velocity, $v_B = 36\text{ km h}^{-1} = 10\text{ ms}^{-1}$
(A vector diagram shows $v_A$ pointing down-left at $45^\circ$ and $v_B$ pointing vertically downward.)
Frequency of horn heard by the driver,
$f' = f \left(\frac{v + v_B \cos 45^\circ}{v - v_A \cos 45^\circ}\right)$
$= 280 \left(\frac{340 + 10 / \sqrt{2}}{340 - 20 / \sqrt{2}}\right) = 298\text{ Hz}$