Question

The displacement-time graph of a particle moving along a straight line is shown in the figure. The acceleration-time graph of this particle is:

• A. Increasing linearly with time

  

• B. Constant acceleration

  undefined

• C. First increases then decreases

  undefined

• D. Increasing non-linearly with time

  

Hint:

If \(x \propto t^2\), then acceleration is constant.

Answer 1

The Correct Option is B

The problem presents a displacement-time graph of a particle moving along a straight line. The task is to determine the nature of the acceleration-time graph for the same particle.

To analyze this, we first need to understand the relationship between displacement, velocity, and acceleration:

1. Displacement-Time Graph: The given graph shows that displacement \(x\) is proportional to \(t^2\) (as illustrated by \(x = 9t^2\)).
2. Velocity-Time Graph: Velocity \(v\) is the derivative of displacement with respect to time. We calculate: \(v = \frac{dx}{dt} = \frac{d}{dt}(9t^2) = 18t\).
3. Acceleration-Time Graph: Acceleration \(a\) is the derivative of velocity with respect to time. We find: \(a = \frac{dv}{dt} = \frac{d}{dt}(18t) = 18\).

From the above steps, it is evident that the acceleration is a constant value, \(18\). This means the acceleration-time graph is a constant horizontal line.

Therefore, the correct option is: Constant acceleration.