Question

The acceleration $a$ of 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. $\frac{\left(\alpha t^2\right)+\beta }{2}$

Answer 1

The correct option is B $\frac{\alpha t^2}{2}+\beta t$

According to given relation
acceleration a=αt+βa=αt+β
As a=dvdt⇒αt+β=dvdta=dvdt⇒αt+β=dvdt
Since particle starts from rest, its initial velocity is zero
At time t=0t=0,velocity =0=0
⇒∫v0dv=∫v0(αt+β)dt⇒∫0vdv=∫0v(αt+β)dt
or v=αt22+βt