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Q.
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?
Motion in a Straight Line
Solution:
$a=\alpha t^{3 / 2}$
$\int\limits_{u}^{v} d v=\int\limits_{0}^{t} a d t$
$\Rightarrow \int\limits_{u}^{v} d v=\int\limits_{0}^{t} \alpha t^{3 / 2} d t$
$\Rightarrow v\left|u^{v}=\alpha \frac{t^{3 / 2}+1}{\frac{3}{2}+1}\right|_{0}^{t}$
$\Rightarrow (v-u)=\alpha \times \frac{2}{5} \left(t^{5 / 2}-0\right)$
$\Rightarrow v-u=\frac{2}{5} \alpha t^{5 / 2}$
$\Rightarrow v=u+\frac{2}{5} \alpha t^{5 / 2}$