Difference in Speeds of Two Approaching Cars from Reflected Sound Frequencies

To find the difference in the speeds of the two cars, we use the Doppler effect for sound waves.

Let:

• $f_0$ be the original frequency emitted by the stationary source,
• $c = 330\text{ ms}^{-1}$ be the speed of sound in air,
• $v$ be the speed of an approaching car ($v \ll c$).

Step 1: Frequency Received by the Approaching Car

The car acts as a moving observer approaching a stationary source:
$f' = f_0 \left(\frac{c + v}{c}\right)$

Step 2: Frequency of the Reflected Sound

The car now acts as a moving source reflecting the sound back towards the stationary observer:
$f'' = f' \left(\frac{c}{c - v}\right) = f_0 \left(\frac{c + v}{c - v}\right)$

Since $v \ll c$, we use the binomial approximation:
$f'' = f_0 (c + v)(c - v)^{-1} = f_0 \left(1 + \frac{v}{c}\right)\left(1 - \frac{v}{c}\right)^{-1} \approx f_0 \left(1 + \frac{v}{c}\right)\left(1 + \frac{v}{c}\right) \approx f_0 \left(1 + \frac{2v}{c}\right)$

Step 3: Difference in Reflected Frequencies

For two cars moving at speeds $v_1$ and $v_2$ ($v_1 > v_2$), the reflected frequencies are:
$f''_1 \approx f_0 \left(1 + \frac{2v_1}{c}\right)$
$f''_2 \approx f_0 \left(1 + \frac{2v_2}{c}\right)$

The difference in the reflected frequencies is:
$\Delta f = f''_1 - f''_2 = \frac{2 f_0 (v_1 - v_2)}{c} = \frac{2 f_0 \Delta v}{c}$

Step 4: Calculation of the Difference in Speeds

We are given that the difference in frequencies is $1.2\%$ of $f_0$:
$\Delta f = \frac{1.2}{100} f_0 = 0.012 f_0$

Equating the two expressions for $\Delta f$:
$0.012 f_0 = \frac{2 f_0 \Delta v}{c}$

Solving for $\Delta v$:
$\Delta v = \frac{0.012 \times c}{2} = \frac{0.012 \times 330}{2} = 1.98\text{ ms}^{-1}$

Converting the speed difference to kilometers per hour ($\text{km h}^{-1}$):
$\Delta v = 1.98 \times \frac{18}{5} = 1.98 \times 3.6 = 7.128\text{ km h}^{-1}$

Rounding to the nearest integer, we get:
$\Delta v \approx 7\text{ km h}^{-1}$