Problem 87 - Pearson Physics

Question

A train moving with a speed of \(31.8 \mathrm{~m} / \mathrm{s}\) sounds a 136 -Hz horn. What frequency is heard by an observer standing near the tracks as the train approaches?

Step-by-Step Solution

Answer

The frequency heard is approximately 149.9 Hz.

Step 1: Understand the Doppler Effect Formula

To find the frequency heard by an observer, we use the Doppler Effect formula: \( f' = \frac{f \cdot (v + v_0)}{v - v_s} \), where \( f' \) is the observed frequency, \( f \) is the source frequency (136 Hz), \( v \) is the speed of sound in air (approximately 343 m/s at 20°C), \( v_0 \) is the speed of the observer (0 m/s since the observer is stationary), and \( v_s \) is the speed of the source (31.8 m/s for the train).

Step 2: Substitute the Known Values

Substitute the known values into the formula: \( f' = \frac{136 \cdot (343 + 0)}{343 - 31.8} \). This simplifies since the observer is stationary, so \( v_0 = 0 \).

Step 3: Calculate the Denominator

Calculate the denominator: \( 343 - 31.8 = 311.2 \). Substitute this into the equation to get: \( f' = \frac{136 \cdot 343}{311.2} \).

Step 4: Calculate the Numerator

Calculate the numerator: \( 136 \cdot 343 = 46648 \). Substitute this value back into the formula to find the observed frequency.

Step 5: Compute the Observed Frequency

Divide the values to compute the observed frequency: \( f' = \frac{46648}{311.2} \approx 149.9 \) Hz. Therefore, the frequency heard by an observer is approximately 149.9 Hz.

Key Concepts

Frequency

Frequency refers to the number of waves that pass a given point per second. It's measured in hertz (Hz). For sound, this is how many sound wave cycles are produced or heard per second.
In the context of the exercise, we analyze how the frequency of a train's horn sounds different to a stationary observer. This change is due to the Doppler Effect, which alters the frequency as the source of sound approaches or moves away from the observer.

Sound Wave

A sound wave is an oscillation of pressure through a medium like air. It consists of areas of compression and rarefaction that move through the air and allow us to hear sound. While sound waves originated from the train's horn in the example, they're perceived differently by the observer due to their interaction with the environment and speed of the moving source.
Sound waves change in frequency when there is relative motion between the source and observer, as demonstrated in our observation of the train's horn.

Speed of Sound

The speed of sound is how fast a sound wave travels through a medium. At 20°C in the air, the speed of sound is approximately 343 meters per second. This value is used in the Doppler Effect formula for calculations in sound-based problems.
Knowing the speed of sound is vital for accurately determining the observed frequency. Variations in temperature, humidity, and air density can affect this speed slightly, which may, in turn, affect sound detection.

Observer

In physics, an observer is a point or person detecting or measuring a phenomenon. In the Doppler Effect scenario, the observer is stationary, standing near the tracks, and measures the frequency of the approaching train's horn.
The observer's relative speed compared to the source impacts the frequency heard. Since the observer is not moving, the equation simplifies, only considering the train's speed and the speed of sound.

Source Frequency

The source frequency is the initial frequency emitted by the sound source, in this case, the train horn at 136 Hz. This value is used in the Doppler equation to calculate the frequency observed by someone receiving the sound.
The source frequency remains constant, but the relative motion between the source and observer alters how the frequency is perceived, leading to it sounding higher or lower, depending on the movement.