A soldier fired two bullets at an interval of 335 seconds moving at a uniform speed `v_1`. A terrorist who was running ahead of the soldier in the same direction, hears the two shots at an interval of 330 seconds. If the speed of sound is 1188 km/h, then who is the faster and by how much?

Options

A. soldier, 22 km/h
B. Terrorist, 25 km/h
C. soldier, 18 km/h
D. Terrorist, 20 km/h

Answer

To solve the problem step by step, we will analyze the information given and apply the relevant formulas.

### Step 1: Understand the problem
The soldier fires two bullets at an interval of 335 seconds. The terrorist hears these shots at an interval of 330 seconds. We need to determine who is faster between the soldier and the terrorist.

### Step 2: Define the given values
- Interval between bullets fired by the soldier, \( t_s = 335 \) seconds
- Interval between shots heard by the terrorist, \( t_t = 330 \) seconds
- Speed of sound, \( v_s = 1188 \) km/h

### Step 3: Convert the speed of sound to meters per second
To work with consistent units, we convert the speed of sound from kilometers per hour to meters per second:
\[ v_s = 1188 \text{ km/h} = \frac{1188 \times 1000 \text{ m}}{3600 \text{ s}} = 330 \text{ m/s} \]

### Step 4: Calculate the difference in time
The difference in time between the intervals is:
\[ \Delta t = t_s - t_t = 335 \text{ seconds} - 330 \text{ seconds} = 5 \text{ seconds} \]

### Step 5: Use the formula for relative speed
We know that the speed of sound divided by the relative speed equals the time taken divided by the difference in time:
\[ \frac{v_s}{R} = \frac{t_t}{\Delta t} \]
Where \( R \) is the relative speed between the soldier and the terrorist.

### Step 6: Substitute the known values into the formula
Substituting the known values into the equation:
\[ \frac{330 \text{ m/s}}{R} = \frac{330 \text{ seconds}}{5 \text{ seconds}} \]

### Step 7: Simplify the equation
\[ \frac{330}{R} = 66 \]
Now, solving for \( R \):
\[ R = \frac{330}{66} = 5 \text{ m/s} \]

### Step 8: Determine the speed of the soldier
Since the soldier is moving faster than the terrorist, we can express the speed of the soldier \( v_1 \) as:
\[ v_1 = v_t + R \]
Where \( v_t \) is the speed of the terrorist. We need to find \( v_t \).

### Step 9: Calculate the speed of the soldier
To find the speed of the soldier, we need to convert the relative speed from meters per second to kilometers per hour:
\[ R = 5 \text{ m/s} = 5 \times \frac{3600}{1000} = 18 \text{ km/h} \]
Thus, the speed of the soldier \( v_1 \) is:
\[ v_1 = v_t + 18 \text{ km/h} \]

### Step 10: Conclusion
Since the soldier is faster than the terrorist by 18 km/h, we conclude that the soldier is the faster one.

### Final Answer
The soldier is faster than the terrorist by 18 km/h.

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