The distance travelled by a particle is proportional to the squares of time, then the particle travels with

Options

A. uniform acceleration
B. uniform velocity
C. Both of these
D. speed changes

Explanation

To solve the problem, we need to analyze the relationship between distance, time, velocity, and acceleration based on the information given in the question.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The problem states that the distance \( s \) traveled by a particle is proportional to the square of time \( t \). Mathematically, we can express this as:
\[ s \propto t^2 \]
This means that there exists a constant \( k \) such that:
\[ s = k t^2 \]

2. **Finding Velocity**:
Velocity \( v \) is defined as the rate of change of distance with respect to time. We can find the velocity by differentiating the distance equation with respect to time:
\[ v = \frac{ds}{dt} = \frac{d}{dt}(k t^2) \]
Using the power rule of differentiation, we get:
\[ v = k \cdot 2t = 2kt \]
This shows that velocity \( v \) is directly proportional to time \( t \).

3. **Analyzing Velocity**:
Since \( v = 2kt \), we can see that as time increases, velocity also increases. This indicates that the particle does not travel with uniform velocity.

4. **Finding Acceleration**:
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. We can find the acceleration by differentiating the velocity equation with respect to time:
\[ a = \frac{dv}{dt} = \frac{d}{dt}(2kt) \]
Differentiating gives:
\[ a = 2k \]
Here, \( 2k \) is a constant, indicating that the acceleration is uniform.

5. **Conclusion**:
Based on the analysis:
- The particle travels with **uniform acceleration** because the acceleration \( a = 2k \) is constant.
- The velocity is not uniform since it changes with time.

Thus, the correct answer is that the particle travels with **uniform acceleration**.