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 determine the solution, we utilize the relationship between acceleration and velocity. The acceleration-time graph depicts acceleration's change over time, and the velocity-time graph is derived by integrating acceleration with respect to time.
Step 1: Understanding the Acceleration-Time Graph From the given acceleration-time graph: 1. Acceleration is constant for the first interval (from \( t = 0 \) to \( t = 6 \)). 2. Acceleration is also constant for the second interval (from \( t = 6 \) to a higher value).
Step 2: Velocity-Time Graph from Acceleration-Time Graph Velocity is the integral of acceleration over time. Since acceleration is constant in each interval, the velocity-time graph will show a linear increase during intervals with non-zero acceleration. - In the first interval (constant acceleration), velocity increases linearly. - In the second interval (constant acceleration), velocity continues to increase linearly, with a potentially different rate based on the acceleration value.
Step 3: Analyzing the Options - Option (A): A trapezium-shaped graph implies a non-linear increase, which is incorrect due to constant acceleration. - Option (B): A right-angle triangle-shaped graph is also incorrect as it suggests a sharp, right-angled slope. - Option (C): An L-shape suggests abrupt velocity changes, inconsistent with constant acceleration. - Option (D): This option shows a linear velocity increase, consistent with constant acceleration.
Step 4: Conclusion The correct velocity-time graph displays a linear velocity increase over time due to constant acceleration. Thus, the correct answer is: \[ \boxed{(D)} \]