Question

Given below are two statements:
Statement I: Area under velocity- time graph gives the distance travelled by the body in a given time.
Statement II: Area under acceleration-time graph is equal to the change in velocity- in the given time.
In the light of given statements, choose the correct answer from the options given below.

• A. Statement I is incorrect but Statement II is True.
• B. Both Statement I and Statement II are False.
• C. Statement I is correct but Statement II is false.
• D. Both Statement I and Statement II are true.

Answer 1

The Correct Option is D

Step 1: Analyze Statement I. The velocity \( \vec{v} \) is given by: \[ \vec{v} = \frac{d\vec{s}}{dt}. \] Integrating both sides with respect to time: \[ \int d\vec{s} = \int \vec{v} \, dt. \] The area under the velocity-time (\( \vec{v} \)-\( t \)) graph gives displacement. Hence, Statement I is true. 
Step 2: Analyze Statement II. The acceleration \( \vec{a} \) is given by: \[ \vec{a} = \frac{d\vec{v}}{dt}. \] Integrating both sides with respect to time: \[ \int d\vec{v} = \int \vec{a} \, dt. \] The area under the acceleration-time (\( \vec{a} \)-\( t \)) graph gives the change in velocity. Hence, Statement II is also true. 
Final Answer: Both statements are: \[ \boxed{\text{True.}} \]