Two particles starts simultaneously from a point `O` and move along line `OP` and `OQ`, one with a uniform velocity `10m//s` and other from rest with a constant acceleration `5m//s^(2)` respectively. Line `OP` makes an angle `37^(@)` with the line `OQ`. Find the time at which they appear to each other moving at minimum speed?

Answer

To solve the problem step by step, we need to analyze the motion of both particles and determine the time at which they appear to be moving at minimum speed relative to each other.

### Step-by-Step Solution:

1. **Understand the Motion of the Particles**:
- Particle A moves along line OP with a uniform velocity of \(10 \, \text{m/s}\).
- Particle B moves along line OQ from rest with a constant acceleration of \(5 \, \text{m/s}^2\).

2. **Set Up the Coordinate System**:
- Let point O be the origin of our coordinate system.
- Line OP makes an angle of \(37^\circ\) with line OQ.

3. **Break Down the Velocity of Particle A**:
- The velocity of particle A can be broken down into components:
- \(V_{Ax} = 10 \cos(37^\circ)\)
- \(V_{Ay} = 10 \sin(37^\circ)\)
- Using the trigonometric values:
- \(\cos(37^\circ) \approx 0.8\)
- \(\sin(37^\circ) \approx 0.6\)
- Therefore:
- \(V_{Ax} = 10 \times 0.8 = 8 \, \text{m/s}\)
- \(V_{Ay} = 10 \times 0.6 = 6 \, \text{m/s}\)

4. **Determine the Motion of Particle B**:
- Since particle B starts from rest, its initial velocity \(u = 0\).
- The acceleration \(a = 5 \, \text{m/s}^2\).
- The velocity of particle B after time \(t\) is given by:
\[ V_B = u + at = 0 + 5t = 5t \, \text{m/s} \]

5. **Relative Velocity**:
- To find when they appear to be moving at minimum speed relative to each other, we consider the relative velocity of particle B with respect to particle A.
- The relative velocity in the x-direction (assuming right is positive) is:
\[ V_{rel} = V_B - V_{Ax} = 5t - 8 \]

6. **Setting Relative Velocity to Zero**:
- To find the time when they appear to be moving at minimum speed, we set the relative velocity to zero:
\[ 5t - 8 = 0 \]
- Solving for \(t\):
\[ 5t = 8 \implies t = \frac{8}{5} = 1.6 \, \text{seconds} \]

### Final Answer:
The time at which the two particles appear to be moving at minimum speed relative to each other is \(t = 1.6 \, \text{seconds}\).