Question

Acceleration of the particle in a rectilinear motion is given as $a=2t-6$ in $\left({\text{m/s}}^2\right)$. The displacement as a function of time $t$ respectively is
(Given, at $t=0,u=2\text{m/s},x=2\text{m}$)

• A. $\frac{2t^3}{3}-3t^2+2t+2$
• B. $\frac{t^3}{3}-3t^2+2t$
• C. $\frac{t^3}{3}-3t^2+2t+2$
• D. $\frac{t^3}{3}-3t^2$

Answer 1

The correct option is B $\frac{t^3}{3}-3t^2+2t$

Given, $a\left(t\right)=2t-6$
As we know that the velocity as a fuction of time is given by
${\int }_u^{v}dv={\int }_0^t(2t-6)dt$
$\Rightarrow [v{]}_u^v=[t^2-6t{]}_0^t$
$\Rightarrow v-u=t^2-6t$
$\Rightarrow v-2=t^2-6t\Rightarrow v\left(t\right)=t^2-6t+2$
Now, displacement is given by
$x\left(t\right)={\int }_0^{t}v\left(t\right)dt={\int }_0^t(t^2-6t+2)dt$
$\Rightarrow x\left(t\right)={[\frac{t^3}{3}-3t^2+2t]}_0^t$
$\Rightarrow x\left(t\right)=(\frac{t^3}{3}-3t^2+2t)$