The acceleration (a)-time (t) graph for a par | vedclass

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(A) From the given acceleration $(a)$-time $(t)$ graph,it is evident that the acceleration is positive and decreasing linearly with time.Let the accel

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(A) From the given acceleration $(a)$-time $(t)$ graph,it is evident that the acceleration is positive and decreasing linearly with time.Let the acceleration be represented as $a = a_0 - kt$,where $a_0$ is the initial acceleration and $k$ is a positive constant.Since the particle starts from rest,the initial velocity $v(0) = 0$.We know that $a = \frac{dv}{dt}$,so $\frac{dv}{dt} = a_0 - kt$.Integrating both sides with respect to time $t$:$\int_{0}^{v} dv = \int_{0}^{t} (a_0 - kt) dt$$v = a_0 t - \frac{1}{2} kt^2$.This equation represents a downward-opening parabola starting from the origin $(0,0)$.As time $t$ increases,the slope of the velocity-time graph,which is the acceleration $a$,decreases. Therefore,the velocity-time graph should be a curve that starts with a positive slope and becomes flatter (slope decreases) as time approaches $T$. This corresponds to the graph shown in option $(A)$.

The acceleration $(a)$-time $(t)$ graph for a particle moving along a straight line starting from rest is shown in the figure. Which of the following graphs is the best representation of the variation of its velocity $(v)$ with time $(t)$?

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