A police on duty detects a drop of 10% in the pitch of the horn of a motor car as it crosses hi. Calculate the speed of car

To determine the speed of the car based on the drop in pitch of its horn, we can use the Doppler effect, which describes how the frequency of sound changes for an observer moving relative to the source of the sound. In this case, the police officer is the observer, and the car is the source of the sound. Let's break down the problem step by step.

Understanding the Doppler Effect

The Doppler effect occurs when there is relative motion between a sound source and an observer. If the source is moving towards the observer, the frequency of the sound increases, and if it is moving away, the frequency decreases. The formula for the observed frequency (f') when the source is moving away from the observer is given by:

f' = f (v / (v + v_s))

Where:

• f' = observed frequency
• f = actual frequency of the sound
• v = speed of sound in air (approximately 343 m/s at room temperature)
• v_s = speed of the source (the car)

Calculating the Speed of the Car

In this scenario, we know that the pitch of the horn drops by 10%. This means the observed frequency is 90% of the actual frequency:

f' = 0.9f

Substituting this into the Doppler effect formula gives us:

0.9f = f (v / (v + v_s))

We can simplify this equation by dividing both sides by f (assuming f is not zero):

0.9 = v / (v + v_s)

Next, we can rearrange this equation to solve for the speed of the car (v_s):

0.9(v + v_s) = v

Expanding this gives:

0.9v + 0.9v_s = v

Now, isolating v_s:

0.9v_s = v - 0.9v

0.9v_s = 0.1v

Thus, we find:

v_s = (0.1v) / 0.9

Substituting the speed of sound (v = 343 m/s):

v_s = (0.1 * 343) / 0.9

v_s ≈ 38.11 m/s

Final Thoughts

The speed of the car is approximately 38.11 meters per second. This calculation illustrates how the Doppler effect can be applied to real-world scenarios, such as determining the speed of vehicles based on sound frequency changes. Understanding these principles can be quite useful in various fields, including law enforcement and physics.