A group of soldiers are marching with a speed of 5 m/s. The distance between the first and the last row of soldiers is 100 m. A dog starts running from the last row and moves towards the first row, turns and comes back to the last row. If the dog has travelled 400 m, the speed of the dog is

Options

A. `5sqrt(2)m//s`
B. `3sqrt(5)m//s`
C. `6sqrt(5)m//s`
D. `6sqrt(2)m//s`

Answer

To solve the problem step by step, we start by analyzing the information given:

1. **Understanding the Problem**:
- Soldiers are marching at a speed of 5 m/s.
- The distance between the first and last row of soldiers is 100 m.
- A dog runs from the last row to the first row and back, covering a total distance of 400 m.

2. **Setting Up the Variables**:
- Let the speed of the dog be \( x \) m/s.
- The time taken by the dog to run from the last row to the first row and back should be the same for both segments of the journey.

3. **Calculating Time for Each Segment**:
- **Time taken to run from last row to first row**:
- The distance is 100 m.
- The effective speed of the dog when moving towards the soldiers is \( (x - 5) \) m/s (since the soldiers are moving in the opposite direction).
- Time taken \( t_1 = \frac{100}{x - 5} \).

- **Time taken to run from first row back to last row**:
- The distance is again 100 m.
- The effective speed of the dog when moving back is \( (x + 5) \) m/s.
- Time taken \( t_2 = \frac{100}{x + 5} \).

4. **Setting Up the Equation**:
- Since the total distance covered by the dog is 400 m, we can express the total time taken as:
\[ t_1 + t_2 = \frac{400}{x} \]
- Therefore, we have:
\[ \frac{100}{x - 5} + \frac{100}{x + 5} = \frac{400}{x} \]

5. **Finding a Common Denominator**:
- The common denominator for the left-hand side is \( (x - 5)(x + 5) \):
\[ \frac{100(x + 5) + 100(x - 5)}{(x - 5)(x + 5)} = \frac{400}{x} \]
- Simplifying the left side:
\[ \frac{100x + 500 + 100x - 500}{(x - 5)(x + 5)} = \frac{400}{x} \]
\[ \frac{200x}{(x - 5)(x + 5)} = \frac{400}{x} \]

6. **Cross Multiplying**:
- Cross multiplying gives:
\[ 200x^2 = 400(x^2 - 25) \]
- Expanding the right side:
\[ 200x^2 = 400x^2 - 10000 \]

7. **Rearranging the Equation**:
- Bringing all terms to one side:
\[ 400x^2 - 200x^2 - 10000 = 0 \]
\[ 200x^2 - 10000 = 0 \]
\[ 200x^2 = 10000 \]
\[ x^2 = 50 \]
\[ x = \sqrt{50} = 5\sqrt{2} \text{ m/s} \]

8. **Conclusion**:
- The speed of the dog is \( 5\sqrt{2} \) m/s.