A Friecracker exploding on the surface of a lake is heard as two sounds a time interval t apart by a man on a boat close to water surface. Sound travels with a speed u in water and a sped v in air. The distance from the exploding firecracker to the boat is

Options

A. `(uvt)/(u+v)`
B. `(t(u+v))/(uv)`
C. `(t(u-v))/(uv)`
D. `(uvt)/(u-v)`

Answer

To solve the problem of finding the distance \( D \) from the exploding firecracker to the boat, we can follow these steps:

### Step 1: Understand the problem
The firecracker explodes on the surface of a lake, and a man on a boat hears two sounds at a time interval \( T \). The sound travels faster in water with speed \( u \) and slower in air with speed \( v \). We need to find the distance \( D \) from the firecracker to the boat.

### Step 2: Set up the equations
Let \( T_1 \) be the time taken for the sound to travel through water, and \( T_2 \) be the time taken for the sound to travel through air. According to the problem, the time difference between the two sounds is given as:
\[ T_2 - T_1 = T \]

### Step 3: Express distances in terms of time and speed
The distance \( D \) can be expressed in two ways:
1. When the sound travels through water:
\[ D = u \cdot T_1 \]
2. When the sound travels through air:
\[ D = v \cdot T_2 \]

### Step 4: Relate \( T_2 \) to \( T_1 \)
From the time difference equation, we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2 = T_1 + T \]

### Step 5: Substitute \( T_2 \) in the distance equation
Now, substituting \( T_2 \) into the distance equation for air:
\[ D = v \cdot (T_1 + T) \]

### Step 6: Set the two expressions for \( D \) equal to each other
Since both expressions represent the same distance \( D \), we can set them equal:
\[ u \cdot T_1 = v \cdot (T_1 + T) \]

### Step 7: Solve for \( T_1 \)
Expanding the right side gives:
\[ u \cdot T_1 = v \cdot T_1 + v \cdot T \]
Rearranging terms to isolate \( T_1 \):
\[ u \cdot T_1 - v \cdot T_1 = v \cdot T \]
Factoring out \( T_1 \):
\[ T_1 (u - v) = v \cdot T \]
Thus, we can solve for \( T_1 \):
\[ T_1 = \frac{v \cdot T}{u - v} \]

### Step 8: Substitute \( T_1 \) back to find \( D \)
Now, substituting \( T_1 \) back into the equation for \( D \):
\[ D = u \cdot T_1 = u \cdot \left(\frac{v \cdot T}{u - v}\right) \]
This simplifies to:
\[ D = \frac{u \cdot v \cdot T}{u - v} \]

### Final Answer
The distance \( D \) from the firecracker to the boat is:
\[ D = \frac{u \cdot v \cdot T}{u - v} \]