The last soldier of an `80m` long marching troops runs from the end to its front, and then it runs back to the end with the same speed. During this, the marching troop covers a distance of `150m`. The distance covered by the soldier is

Options

A. `310m`
B. `250m`
C. `230m`
D. `160m`

Answer

To solve the problem step by step, we will analyze the motion of the last soldier and the marching troop.

### Step 1: Understand the scenario
The last soldier runs from the end of the troop to the front and then back to the end. The troop covers a distance of 150 meters during this time. The length of the troop is 80 meters.

### Step 2: Define variables
Let:
- \( V_1 \) = speed of the last soldier
- \( V_2 \) = speed of the marching troop
- \( t \) = total time taken for the soldier to run to the front and back to the end

### Step 3: Calculate the time taken by the troop
The marching troop covers a distance of 150 meters, so the time taken by the troop can be expressed as:
\[ t = \frac{150}{V_2} \]

### Step 4: Analyze the soldier's journey
1. **First leg (running to the front)**: The soldier runs 80 meters towards the front. The effective speed of the soldier relative to the troop is \( V_1 - V_2 \). The time taken for this leg is:
\[ t_1 = \frac{80}{V_1 - V_2} \]

2. **Second leg (running back to the end)**: The soldier runs back 80 meters to the end. The effective speed of the soldier relative to the troop is \( V_1 + V_2 \). The time taken for this leg is:
\[ t_2 = \frac{80}{V_1 + V_2} \]

### Step 5: Relate the total time
The total time taken by the soldier is the sum of the times for both legs:
\[ t = t_1 + t_2 = \frac{80}{V_1 - V_2} + \frac{80}{V_1 + V_2} \]

### Step 6: Substitute the expression for \( t \)
We can substitute \( t \) from the troop's distance equation:
\[ \frac{150}{V_2} = \frac{80}{V_1 - V_2} + \frac{80}{V_1 + V_2} \]

### Step 7: Solve the equation
To solve this equation, we can cross-multiply and simplify:
1. Multiply both sides by \( V_2(V_1 - V_2)(V_1 + V_2) \) to eliminate the denominators.
2. After simplification, we can find a relationship between \( V_1 \) and \( V_2 \).

### Step 8: Calculate the distance covered by the soldier
The distance covered by the soldier is given by:
\[ \text{Distance} = V_1 \times t \]
Substituting \( t \) from the earlier expression, we can find the distance covered by the soldier.

### Final Calculation
After solving the equations, we find that the distance covered by the soldier is:
\[ \text{Distance} = 250 \text{ meters} \]