Velocity v. Time
The displacement of an object is equal to the area under a velocity-time graph.
True or False?
Answer (2)
The displacement of an object is indeed equal to the area under a velocity-time graph. This relationship arises because integrating the velocity function over time gives the total displacement. Therefore, the statement is true.
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Displacement is the change in position of an object.
Velocity is the rate of change of displacement with respect to time, v = d t d s .
Integrating velocity with respect to time gives displacement: s = ∫ v ( t ) d t .
The area under the velocity-time graph represents the displacement, so the statement is T r u e .
Explanation
Problem Analysis The question states: The displacement of an object is equal to the area under a velocity-time graph. We need to determine if this statement is true or false.
Understanding Displacement and Velocity Displacement is defined as the change in position of an object. Velocity is the rate of change of displacement with respect to time. Mathematically, velocity v is the derivative of displacement s with respect to time t , i.e., v = d t d s .
Relationship between Displacement and Integral of Velocity To find the displacement, we integrate the velocity function with respect to time: s = ∫ v ( t ) d t . The definite integral of the velocity function over a time interval [ t 1 , t 2 ] gives the displacement during that time interval: Δ s = ∫ t 1 t 2 v ( t ) d t .
Conclusion The definite integral ∫ t 1 t 2 v ( t ) d t represents the area under the velocity-time graph between times t 1 and t 2 . Therefore, the displacement of an object is indeed equal to the area under the velocity-time graph.
Final Answer The statement 'The displacement of an object is equal to the area under a velocity-time graph' is true.
Examples
Imagine you are tracking a race car's velocity over time. The area under the velocity-time graph tells you exactly how far the car has moved from its starting point. This is crucial for race strategists to plan pit stops and optimize the car's performance. Similarly, in robotics, understanding the area under a robot's velocity-time graph helps in precisely controlling its movements and ensuring it reaches its destination accurately.
