Q. The acceleration $a$ of a particle starting from rest varies with time according to relation $a = \alpha t + \beta.$ The velocity of the particle after a time $t$ will be
A) $\frac{\alpha t^{2}}{2} + \beta$
B) $\frac{\alpha t^{2}}{2} + \beta t$
C) $\alpha t^{2} + \frac{1}{2}\beta t$
D) $\frac{\left( \alpha t^{2} + \ \beta \right)}{2}$
Explanation
(b) Given, $a = \alpha t + \beta$
$\frac{dv}{dt} = \ \alpha t + \beta$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \int_{0}^{t}{dv} = \int_{0}^{t}{\alpha t}dt + \int_{0}^{t}\beta dt$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v = \frac{\alpha t^{2}}{2} + \beta t$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.