Question

On acceleration-time graph the area under the curve equals the:

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Solution

Step 1: AccelerationAcceleration is the time rate of change in velocity. It is a vector quantity.The acceleration is defined by the form, $a=\frac{v}{t}$, where, v is the velocity of the body and t is the time.Step 2: DiagramStep 3: Finding the area under the acceleration-time graphLet us consider the triangle on the above graph.The base of the triangle indicates the time interval, suppose it is $∆t$ and the height of the triangle indicates the change in acceleration, suppose it is $∆a$.Now, the area of the triangle is $A=\frac{1}{2}∆a×∆t$. …………………(1)Again we know, $changeinvelocity=acceleration×time$Therefore, from equation (1),$\frac{1}{2}∆a×∆t=\frac{1}{2}∆v\phantom{\rule{0ex}{0ex}}or∆a×∆t=∆v.....................\left(2\right)$Where $∆v$ refers to the change in velocity.In integral form it is written as, $v={\int }_{t=0}^{t=t}adt$ Therefore, on the acceleration-time graph, the area under the curve at a particular time interval will give the rate of change in velocity.

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