Question

Two trains are traveling toward each other in still air at \(25.0 \mathrm{~m} / \mathrm{s}\) relative to the ground. One train is blowing a whistle at \(300 .\) Hz. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). a) What frequency is heard by a man on the ground facing the whistle-blowing train? b) What frequency is heard by a man on the other train?

Short answer

Answer: The frequency heard by a man on the ground is approximately 323.90 Hz, and the frequency heard by a man on the other train is approximately 347.80 Hz.

Step-by-step solution

Calculate the frequency heard by a man on the ground

First, consider the man standing on the ground. Since he is not moving, his speed (\(v_0\)) will be 0. The train with the whistle is approaching the observer, so its speed (\(v_s\)) is -25.0 m/s. Using the given values and the Doppler effect equation: $$f' = 300 \frac{343}{343 + (-25)}$$ Now, solve the equation $$f' = 300 \frac{343}{318} \approx 323.90 \,\text{Hz}$$

Calculate the frequency heard by a man on the other train

Now let's find the frequency heard by the man on the other train. This time, both \(v_0\) and \(v_s\) will be the same because both the observer and the source are moving toward each other at equal speed. So, \(v_0\) is 25 m/s and \(v_s\) is -25 m/s. Plugging these values into the Doppler effect equation: $$f' = 300 \frac{343 + 25}{343 + (-25)}$$ Now, solve the equation $$f' = 300 \frac{368}{318} \approx 347.80 \,\text{Hz}$$ To conclude: a) The frequency heard by a man on the ground facing the whistle-blowing train is approximately 323.90 Hz. b) The frequency heard by a man on the other train is approximately 347.80 Hz.

Key concepts

Sound Waves

Imagine a ripple spreading out in a pond when you drop a stone into the water. Similarly, sound waves spread out in all directions from the source of the sound. These waves are vibrations of air molecules that our ears interpret as sound. In technical terms, sound waves are longitudinal waves, which means they cause the particles of the medium to vibrate parallel to the direction of wave travel.

Sound waves have specific characteristics such as wavelength, frequency, and amplitude. The frequency of a sound wave, measured in hertz (Hz), is particularly important because it determines the pitch of the sound we hear. The higher the frequency, the higher the pitch will seem. It's crucial to understand that sound waves need a medium like air, water or solids to travel through; they cannot propagate in a vacuum where there are no molecules to vibrate.

Relative Motion

The concept of relative motion is essential when we try to understand the movement of objects from different reference points. It's the difference in motion between two objects, as observed from a specific point of view. For example, if you see a train moving past you while standing still, it has a certain speed in relation to you. However, if you are moving alongside the train at the same speed, from your point of view, the train appears to be still, even though both of you are in motion relative to the ground.

Relative motion plays a vital role in the Doppler effect, where the movement of the source of sound waves and the observer affect the frequency heard. If they are moving towards each other, the frequency increases; if they are moving apart, the frequency decreases. Remember that it's not the actual speed that matters, but the speed relative to each other.

Frequency Change

The concept of 'frequency change' is closely tied to the Doppler effect. When there is relative motion between a source of sound and an observer, the frequency of the sound perceived by the observer changes. In the case of the exercise, the frequency changes because of the movement of the trains towards each other.

The Doppler effect can cause a sound to appear higher in pitch (frequency) if the source is approaching the observer or lower in pitch if the source is moving away. The equation used in the exercise represents how to calculate this effect quantitatively. It's crucial to note that the frequency change observed depends not only on the speed of the source and the observer but also on the speed of sound in the medium through which it travels. In our daily life, we experience the Doppler effect when we hear the siren of an ambulance change in pitch as it drives by us.