The displacement of a body is proporticonal to the cube of time elapsed. What is the nature of the acceleration of the body ?

Options

A. increasing with time
B. decreasing with time
C. constant but not zero
D. zero

Explanation

To solve the problem, we need to analyze the relationship between displacement, velocity, and acceleration based on the given information that the displacement of a body is proportional to the cube of time elapsed.

### Step-by-Step Solution:

1. **Understanding the Displacement**:
- The problem states that the displacement \( x \) is proportional to the cube of time \( t \). We can express this mathematically as:
\[ x \propto t^3 \]
- This can be rewritten as:
\[ x = k t^3 \]
where \( k \) is a constant of proportionality.

2. **Finding Velocity**:
- Velocity \( v \) is defined as the rate of change of displacement with respect to time. Therefore, we differentiate \( x \) with respect to \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(k t^3) = 3k t^2 \]
- This shows that the velocity is proportional to \( t^2 \).

3. **Finding Acceleration**:
- Acceleration \( a \) is defined as the rate of change of velocity with respect to time. We differentiate \( v \) with respect to \( t \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(3k t^2) = 6k t \]
- This indicates that acceleration is proportional to \( t \).

4. **Analyzing the Nature of Acceleration**:
- From the expression \( a = 6k t \), we see that as time \( t \) increases, the acceleration \( a \) also increases (since \( k \) is a constant).
- Therefore, we conclude that the acceleration of the body is increasing with time.

5. **Final Conclusion**:
- The nature of the acceleration of the body is that it is **increasing with time**.