Two engines pass each other moving in opposite directions with uniform speed of 30 m/s. One of them is blowing a whistle of frequency 540 Hz. Calculate the frequency heard by driver of second engine before they pass each other. Speed of sound is 330 m/sec:

We begin by recalling the Doppler effect formula for sound when both the source and the observer are in motion:

$$f' = f \left(\dfrac{v + v_o}{\,v - v_s\,}\right)$$

Here,

$$f'$$ is the frequency heard by the observer,

$$f$$ is the actual frequency emitted by the source,

$$v$$ is the speed of sound in air,

$$v_o$$ is the speed of the observer relative to the medium,

$$v_s$$ is the speed of the source relative to the medium.

When the observer and the source move toward each other, the numerator $$v + v_o$$ is used because the observer moves against the wave fronts, effectively increasing the speed of sound for him, and the denominator $$v - v_s$$ is used because the source moves toward the observer, compressing the wave fronts and thus decreasing the apparent wavelength.

Now we substitute the given data step by step. The numerical values are:

$$f = 540\ \text{Hz}$$ (whistle frequency),

$$v = 330\ \text{m s}^{-1}$$ (speed of sound),

$$v_o = 30\ \text{m s}^{-1}$$ (speed of the second engine, the observer),

$$v_s = 30\ \text{m s}^{-1}$$ (speed of the first engine, the source).

Substituting these values into the Doppler formula, we have

$$ f' \;=\; 540 \left(\dfrac{330 + 30}{330 - 30}\right). $$

First, carry out the additions and subtractions in the fraction:

$$330 + 30 = 360,$$

$$330 - 30 = 300.$$

So the fraction becomes

$$\dfrac{360}{300}.$$

Next, simplify this fraction:

$$\dfrac{360}{300} \;=\; \dfrac{36}{30} \;=\; \dfrac{6}{5} \;=\; 1.2.$$

Now multiply this factor by the original frequency:

$$ f' \;=\; 540 \times 1.2. $$

Performing the multiplication,

$$ 540 \times 1.2 = 648. $$

Hence, the frequency heard by the driver of the second engine is

$$f' = 648\ \text{Hz}.$$

Among the given options, 648 Hz corresponds to Option D.

Hence, the correct answer is Option D.