Which of the following graphs correctly represents a particle moving in a horizontal circle, if its magnitude of tangential acceleration is directly proportional to time and particle starts from rest?

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Solution

The correct option is B
Given, $a_{t} \propto t$
or, $a_{t} = k t$ , where $k$ is proportionality constant.
As we know that
$a_{t} = \frac{d v}{d t} = k t$
$\Rightarrow \int_{0}^{v} d v = k \int_{0}^{t} t d t \Rightarrow v = \frac{k t^{2}}{2}$

Hence, $v \propto t^{2}$ and $at t = 0, v = 0$
Also, as $t \to \infty, v \to \infty$
Thus, the $v - t$ curve will be parabolic and increasing.

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Which of the following graphs correctly represents a particle moving in a horizontal circle, if its magnitude of tangential acceleration is directly proportional to time and particle starts from rest?

The correct option is B Given, at∝t or, at=kt, where k is proportionality constant. As we know that at=dvdt=kt ⇒ ∫v0dv=k∫t0tdt⇒ v=kt22 Hence, v∝t2 and at t=0,v=0 Also, as t→∞,v→∞ Thus, the v−t curve will be parabolic and increasing.