Question

The initial velocity of a particle is $u$ (at $t=0$ ) and the acceleration a is given by $\alpha t^{3/2}$. Which of the following relations is valid?

• A. $v=u+\alpha t^{3/2}$
• B. $v=u+\frac{3\alpha t^3}{2}$
• C. $v=u+\frac{2}{5}\alpha t^{5/2}$
• D. $v=u+\alpha t^{5/2}$

Answer 1

Answer: (C)

$a=\alpha t^{3/2}$
$\underset{u}{\overset{v}{\int }}dv=\underset{0}{\overset{t}{\int }}adt$
$\Rightarrow \underset{u}{\overset{v}{\int }}dv=\underset{0}{\overset{t}{\int }}\alpha t^{3/2}dt$
$\Rightarrow v{\left|u^v=\alpha \frac{t^{3/2}+1}{\frac{3}{2}+1}\right|}_0^t$
$\Rightarrow (v-u)=\alpha \times \frac{2}{5}\left(t^{5/2}-0\right)$
$\Rightarrow v-u=\frac{2}{5}\alpha t^{5/2}$
$\Rightarrow v=u+\frac{2}{5}\alpha t^{5/2}$