Question

Acceleration of a particle starting from rest varies with time $(t)$ according to relation $a=\alpha t+\beta$ $(\alpha$ and $\beta$ are positive constant). The velocity $v$ of particle as a function of time is given as.

• A. $v=\frac{\alpha t^{2}+\beta}{2}$
• B. $v=\alpha t^{2}+\beta t$
• C. $v=\frac{\alpha t^{2}}{2}+\beta t$
• D. $v=\alpha t^{2}+2 \beta t$

Answer 1

Answer: (C)

Since acceleration is variable so to find $v$ we use integration
$v=\int a d t$
$v=\int\limits_{0}^{t}(\alpha t+\beta) d t $
$u=0$
$=\alpha\left[\frac{t^{2}}{2}\right]_{0}^{t}+[\beta t]_{0}^{t}$
$=\frac{\alpha t^{2}}{2}+\beta t$