Ram Shyam are walking on two perpendicular tracks with speed `3ms^(-1)` and `4 ms^(-1)`, respectively. At a certain moment `(say t = 0s)`, Ram and Shyam are at 20 m and 40 m away from the intersection of tracks, respectively, and moving towards the intersection of the tracks.
During the motion the magnitude of velocity of ram with respect to Shyam is

Options

A. `1 ms^(-1)`
B. `4 ms^(-1)`
C. `5 ms^(-1)`
D. `7 ms^(-1)`

Explanation

To solve the problem, we need to find the magnitude of the velocity of Ram with respect to Shyam, given their speeds and directions.

### Step-by-Step Solution:

1. **Identify the Speeds and Directions**:
- Ram's speed \( V_r = 3 \, \text{m/s} \) (towards the intersection).
- Shyam's speed \( V_s = 4 \, \text{m/s} \) (towards the intersection).
- Ram and Shyam are moving on two perpendicular tracks.

2. **Set Up the Coordinate System**:
- Let's place the intersection of the tracks at the origin (0, 0).
- Ram starts at point A (20 m away from the intersection) along the x-axis, so his position can be represented as (20, 0).
- Shyam starts at point B (40 m away from the intersection) along the y-axis, so his position can be represented as (0, 40).

3. **Determine the Velocities in Vector Form**:
- Ram's velocity vector \( \vec{V_r} = (3, 0) \, \text{m/s} \) (moving along the x-axis).
- Shyam's velocity vector \( \vec{V_s} = (0, 4) \, \text{m/s} \) (moving along the y-axis).

4. **Calculate the Relative Velocity**:
- The relative velocity of Ram with respect to Shyam is given by:
\[ \vec{V_{r/s}} = \vec{V_r} - \vec{V_s} \]
- Substituting the values:
\[ \vec{V_{r/s}} = (3, 0) - (0, 4) = (3, -4) \, \text{m/s} \]

5. **Find the Magnitude of the Relative Velocity**:
- The magnitude of the relative velocity vector \( \vec{V_{r/s}} \) can be calculated using the Pythagorean theorem:
\[ |\vec{V_{r/s}}| = \sqrt{(3)^2 + (-4)^2} \]
- Calculating this gives:
\[ |\vec{V_{r/s}}| = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

6. **Conclusion**:
- The magnitude of the velocity of Ram with respect to Shyam is \( 5 \, \text{m/s} \).

### Final Answer:
The magnitude of the velocity of Ram with respect to Shyam is \( 5 \, \text{m/s} \).

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