A steel tube of length 1.00 m is struck at one end. A person with his ear close to the other end hears the sound of the blow twice one travelling through the body of the tube and the other through the air in the tube. Find the time gap between the two hearings.

Explanation

To solve the problem of finding the time gap between the two sounds heard by a person at the end of a steel tube when it is struck, we can follow these steps:

### Step 1: Identify the speeds of sound in different mediums
- The speed of sound in air is approximately \( v_{air} = 330 \, \text{m/s} \).
- The speed of sound in steel is approximately \( v_{steel} = 5200 \, \text{m/s} \).

### Step 2: Determine the length of the tube
- The length of the steel tube is given as \( L = 1.00 \, \text{m} \).

### Step 3: Calculate the time taken for sound to travel through air
- The time taken for sound to travel through air can be calculated using the formula:
\[ t_{air} = \frac{L}{v_{air}} = \frac{1.00 \, \text{m}}{330 \, \text{m/s}} \approx 0.00303 \, \text{s} \]

### Step 4: Calculate the time taken for sound to travel through steel
- The time taken for sound to travel through steel can be calculated using the formula:
\[ t_{steel} = \frac{L}{v_{steel}} = \frac{1.00 \, \text{m}}{5200 \, \text{m/s}} \approx 0.000192 \, \text{s} \]

### Step 5: Calculate the time gap between the two sounds
- The time gap \( \Delta t \) between the two sounds is given by:
\[ \Delta t = t_{air} - t_{steel} \]
Substituting the values we calculated:
\[ \Delta t = 0.00303 \, \text{s} - 0.000192 \, \text{s} \approx 0.002838 \, \text{s} \]

### Step 6: Convert the time gap to milliseconds
- To express the time gap in milliseconds, we multiply by 1000:
\[ \Delta t \approx 0.002838 \, \text{s} \times 1000 \approx 2.838 \, \text{ms} \]

### Final Answer
The time gap between the two hearings is approximately \( 2.84 \, \text{ms} \).

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