Two particles `A` and `B` are initially `40 m`apart, `A` is behind `B`. Particle `A` is moving with uniform velocity of `10 m s^(-1)` towared `B`. Particle `B` starts moving away from `A` with constant acceleration of `2 m s^(-1)`.
The time which there is a minimum distance between the two is .

Options

A. `2s s`
B. `4 s`
C. `5 s`
D. `6 s`

Explanation

To solve the problem step by step, we need to analyze the motion of both particles A and B.

### Step 1: Understand the initial conditions
- Particle A is moving towards B with a uniform velocity of \(10 \, \text{m/s}\).
- Particle B is moving away from A with a constant acceleration of \(2 \, \text{m/s}^2\).
- The initial distance between A and B is \(40 \, \text{m}\).

### Step 2: Set up the equations of motion
1. **Position of Particle A**:
- Since A is moving with a constant velocity, its position as a function of time \(t\) can be given by:
\[ x_A(t) = 10t \]

2. **Position of Particle B**:
- B starts from rest and moves with constant acceleration. Its position as a function of time \(t\) is given by:
\[ x_B(t) = 40 + \frac{1}{2} \cdot 2t^2 = 40 + t^2 \]

### Step 3: Find the distance between A and B
- The distance \(d(t)\) between A and B at any time \(t\) is:
\[ d(t) = x_B(t) - x_A(t) = (40 + t^2) - (10t) = t^2 - 10t + 40 \]

### Step 4: Find the minimum distance
- To find the minimum distance, we need to minimize \(d(t)\). This can be done by taking the derivative of \(d(t)\) with respect to \(t\) and setting it to zero:
\[ \frac{d}{dt}(d(t)) = 2t - 10 = 0 \]
- Solving for \(t\):
\[ 2t - 10 = 0 \implies 2t = 10 \implies t = 5 \, \text{s} \]

### Step 5: Verify if A crosses B
- To check if A crosses B, we need to find the distance A travels in this time:
\[ \text{Distance traveled by A} = x_A(5) = 10 \cdot 5 = 50 \, \text{m} \]
- The position of B at \(t = 5\) is:
\[ x_B(5) = 40 + (5^2) = 40 + 25 = 65 \, \text{m} \]
- Since A travels \(50 \, \text{m}\) and B is at \(65 \, \text{m}\), A does not cross B.

### Conclusion
- The time at which the minimum distance occurs between the two particles is \(5 \, \text{s}\).