A gatekeeper in a colony observes a drop of 10% in the pitch of a motor car as it crosses him. Calculate the speed of motor car if velocity of sound is 332 m/s.

Answer

To solve the problem of calculating the speed of the motor car based on the observed drop in pitch, we will use the Doppler effect formula. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The problem states that a gatekeeper observes a 10% drop in the pitch (frequency) of a motor car as it crosses him. The speed of sound is given as 332 m/s. We need to find the speed of the motor car.

### Step 2: Define the Original and Apparent Frequencies
Let:
- \( f_0 \) = original frequency of the motor car (when stationary)
- \( f \) = apparent frequency heard by the gatekeeper

Since there is a 10% drop in pitch, we can express the apparent frequency as:
\[ f = f_0 - 0.1 f_0 = 0.9 f_0 \]

### Step 3: Use the Doppler Effect Formula
The Doppler effect formula for sound when the source is moving away from a stationary observer is given by:
\[ f = f_0 \frac{v}{v + v_s} \]
where:
- \( v \) = speed of sound (332 m/s)
- \( v_s \) = speed of the source (motor car)

Substituting the expression for \( f \):
\[ 0.9 f_0 = f_0 \frac{332}{332 + v_s} \]

### Step 4: Simplify the Equation
We can cancel \( f_0 \) from both sides (assuming \( f_0 \neq 0 \)):
\[ 0.9 = \frac{332}{332 + v_s} \]

### Step 5: Cross-Multiply to Solve for \( v_s \)
Cross-multiplying gives:
\[ 0.9(332 + v_s) = 332 \]
Expanding this:
\[ 298.8 + 0.9 v_s = 332 \]

### Step 6: Isolate \( v_s \)
Now, isolate \( v_s \):
\[ 0.9 v_s = 332 - 298.8 \]
\[ 0.9 v_s = 33.2 \]
\[ v_s = \frac{33.2}{0.9} \approx 36.89 \, \text{m/s} \]

### Final Answer
The speed of the motor car is approximately **36.89 m/s**.

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