A particle starts moving along a straight line path with velocity `10m//s`. After `5s` the distance of the particle from the starting point is `50m`. Which of the following statements about the nature of motion of the particle are correct?

Options

A. The body may be speeding up with constant positive acceleration.
B. The motion may be moving with constant velocity.
C. The body may have constant negative acceleration.
D. The motion may be first accelerated and then retarded.

Explanation

To analyze the motion of the particle, let's break down the information given in the question step by step.

### Step 1: Identify the initial conditions
The particle starts with an initial velocity \( u = 10 \, \text{m/s} \).

### Step 2: Calculate the distance covered
After \( t = 5 \, \text{s} \), the distance covered by the particle is \( s = 50 \, \text{m} \).

### Step 3: Calculate the average velocity
The average velocity \( v_{\text{avg}} \) can be calculated using the formula:
\[ v_{\text{avg}} = \frac{s}{t} = \frac{50 \, \text{m}}{5 \, \text{s}} = 10 \, \text{m/s} \]

### Step 4: Analyze the nature of motion
Since the average velocity over the 5 seconds is equal to the initial velocity, we can conclude that the particle could be moving with constant velocity. However, we need to consider other possibilities as well.

### Step 5: Consider acceleration
1. If the particle is moving with constant velocity, it means there is no acceleration (\( a = 0 \)).
2. If the particle is accelerating, it could either be speeding up or slowing down, but still cover the same distance in the given time.

### Step 6: Possible scenarios
1. **Constant Velocity**: The particle moves with a constant velocity of \( 10 \, \text{m/s} \) for the entire duration.
2. **Constant Positive Acceleration**: The particle could start at \( 10 \, \text{m/s} \) and accelerate positively, covering the distance of \( 50 \, \text{m} \) in \( 5 \, \text{s} \).
3. **Constant Negative Acceleration**: The particle could start at \( 10 \, \text{m/s} \) and decelerate, but still cover \( 50 \, \text{m} \) in \( 5 \, \text{s} \).
4. **Variable Acceleration**: The particle could accelerate for a time and then decelerate, still resulting in the same distance.

### Conclusion
From the analysis, we can conclude that:
- The particle may be moving with constant velocity.
- The particle may have constant negative acceleration.
- The particle may first accelerate and then decelerate.

### Final Statements
The correct statements about the nature of motion of the particle are:
1. The body may be moving with constant velocity.
2. The body may have constant negative acceleration.
3. The body may first accelerate and then decelerate.

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