Question

If the acceleration-time diagram is represented by a horizontal straight line, then the displacement is:

• A. Zero
• B. Straight line
• C. Parabolic curve
• D. Cubic curve

Hint:

For motion with constant acceleration, the displacement is always a parabolic curve. This is because the integral of constant acceleration gives a linear velocity, and the integral of linear velocity gives a parabolic displacement.

Answer 1

The Correct Option is C

Step 1: Analyze the acceleration-time diagram.
Given that the acceleration-time diagram is a horizontal straight line, this means that the acceleration is constant over time. Mathematically, acceleration \( a(t) \) can be expressed as a constant value \( a_0 \). Thus, \[ a(t) = a_0 \quad \text{(constant acceleration)}. \] Step 2: Determine the velocity-time relationship.
The velocity is the integral of acceleration with respect to time: \[ v(t) = \int a(t) \, dt = \int a_0 \, dt = a_0 t + v_0, \] where \( v_0 \) is the initial velocity. This gives a linear relationship between velocity and time, meaning the velocity-time diagram is a straight line with slope \( a_0 \). Step 3: Determine the displacement-time relationship.
The displacement is the integral of velocity with respect to time: \[ s(t) = \int v(t) \, dt = \int (a_0 t + v_0) \, dt = \frac{1}{2} a_0 t^2 + v_0 t + s_0, \] where \( s_0 \) is the initial displacement. This equation represents a parabolic curve in the displacement-time diagram. The displacement-time graph is a parabola because the second derivative of the displacement is constant acceleration. Conclusion:
Therefore, when the acceleration is constant, the displacement follows a parabolic curve. Final Answer: The displacement is a parabolic curve.