Question

The inital 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 vaild?

• 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

The correct option is C $v=u+\frac{2}{5}\alpha t^{5/2}$

it is given that at

$t=0,u=u$

$a=\alpha t^{\frac{3}{2}}$

$\frac{dv}{dt}=\alpha t^{\frac{3}{2}}$

$dv=\alpha t^{\frac{3}{2}}dt$

$\underset{u}{\overset{v}{\int }}dv=\alpha \underset{0}{\overset{t}{\int }}{t}^{\frac{3}{2}}dt$

$(v-u)=\frac{2\alpha }{5}{t}^{\frac{5}{2}}$

$v=\frac{2\alpha }{5}{t}^{\frac{5}{2}}+u$