A train of length L move with a constant speed `V_(t)`. A person at the back of the train fires a bullet at time t = 0 towards a target which is at a distance of D ( at time t = 0 ) from the front of the train ( on the same direction of motion ). Another person at the front of the train fires another bullet at time t = T towards the same target. Both bullets reach the target at the same time. Assuming the speed of the bullets `V_(b)` are same, the length of the train is

Options

A. `T ( V_(b)xx 2V_(t) )`
B. `T ( V_(b)+V_(t) )`
C. `2T ( V_(b)+2V_(t) )`
D. `2T ( V_(b)-2V_(t) )`

Answer

To solve the problem, we need to analyze the motion of the bullets fired from the train towards the target.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the length of the train be \( L \).
- The speed of the train is \( V_t \).
- The speed of the bullets is \( V_b \).
- The distance from the front of the train to the target at \( t = 0 \) is \( D \).

2. **Bullet Fired from the Back of the Train**:
- At \( t = 0 \), the bullet is fired from the back of the train.
- The distance the bullet needs to cover to reach the target is \( L + D \) (the length of the train plus the distance to the target).
- The time taken for this bullet to reach the target can be expressed as:
\[ t_1 = \frac{L + D}{V_b} \]

3. **Bullet Fired from the Front of the Train**:
- The second bullet is fired from the front of the train at \( t = T \).
- The distance this bullet needs to cover to reach the target is simply \( D \).
- The time taken for this bullet to reach the target is:
\[ t_2 = \frac{D}{V_b} \]
- However, since this bullet is fired after a time \( T \), the total time from the moment the first bullet is fired until the second bullet reaches the target is:
\[ t_1 = T + t_2 = T + \frac{D}{V_b} \]

4. **Setting the Times Equal**:
- Since both bullets reach the target at the same time, we can set the two time equations equal to each other:
\[ \frac{L + D}{V_b} = T + \frac{D}{V_b} \]

5. **Simplifying the Equation**:
- Multiply through by \( V_b \) to eliminate the denominator:
\[ L + D = T V_b + D \]
- Subtract \( D \) from both sides:
\[ L = T V_b \]

6. **Final Expression**:
- Thus, the length of the train \( L \) can be expressed as:
\[ L = T V_b \]

### Conclusion:
The length of the train \( L \) is given by \( L = T V_b \).