First slide

Rectilinear Motion

Question

A particle starts from rest and moves with acceleration a which varies with time t as a=kt where k is a constant. The displacement s of the particle at time t is

Moderate

A

$\frac{1}{2} k t^{2}$
B

$\frac{1}{2} a t^{2}$
C

$\frac{1}{6} a t^{2}$
D

$\frac{1}{6} k t^{2}$

Solution

Initial velocity, $u = 0$
Acceleration $a = k t$ (Where k is constant)
$\Rightarrow \frac{d v}{d t} = k t (∵ a = \frac{d v}{d t})$
$\Rightarrow \int_{0}^{v} d v = \int_{0}^{t} k t d t$
Integrating both sides, we get
$\begin{array}{l} v = \frac{k t^{2}}{2} (∵ v = \frac{d x}{d t}) \\ \Rightarrow \frac{d x}{d t} = \frac{k t^{2}}{2} \\ \Rightarrow \int_{0}^{x} d x = \int_{0}^{t} \frac{k}{2} t^{2} d t \end{array}$
Again integrating both sides, we get
$x = \frac{k}{6} t^{3} = \frac{1}{6} k t . t^{2}$
$∴ x = \frac{1}{6} a t^{2}$

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