Question

Acceleration-time (\( a \) vs. \( t \)) graph of a body is shown in the figure. Corresponding velocity-time (\( v \) vs. \( t \)) graph is:

• A. A shape resembling a trapezium
• B. A shape resembling a right-angle triangle
• C. A shape resembling an L-shape
• D. A shape resembling a linearly increasing curve

Hint:

When given an acceleration-time graph, the velocity-time graph can be obtained by integrating the acceleration. A constant acceleration results in a linear increase in velocity.

Answer 1

The Correct Option is D

To solve this, we need to use the relationship between acceleration and velocity. The acceleration-time graph represents how acceleration changes over time, and the velocity-time graph can be obtained by integrating the acceleration with respect to time.
Step 1: Understanding the Acceleration-Time Graph From the given acceleration-time graph: 1. The acceleration is constant for the first interval (from \( t = 0 \) to \( t = 6 \)). 2. The acceleration is also constant for the second interval (from \( t = 6 \) to some higher value).
Step 2: Velocity-Time Graph from Acceleration-Time Graph The velocity is the integral of acceleration with respect to time. Since acceleration is constant during each interval, the velocity-time graph will show a straight-line increase during the time intervals where acceleration is non-zero. - In the first interval, where the acceleration is constant, the velocity will increase linearly. - In the second interval, where the acceleration remains constant, the velocity will continue to increase linearly, but the rate of increase may be different based on the value of acceleration.
Step 3: Analyzing the Options - Option (A): A trapezium-shaped graph suggests a non-linear increase, which is not the case here because the acceleration is constant. - Option (B): A right-angle triangle-shaped graph is also incorrect, as the graph will not have a sharp, right-angled slope. - Option (C): An L-shape would imply sudden changes in velocity, which is not consistent with constant acceleration. - Option (D): This option shows a graph where the velocity increases linearly, which is consistent with constant acceleration.
Step 4: Conclusion The correct velocity-time graph is the one where the velocity increases linearly over time due to constant acceleration. Thus, the correct answer is: \[ \boxed{(D)} \]