Question

Acceleration of a particle is increasing linearly with time $t$ as $bt$. The particle starts from the origin with an initial velocity $v_0$. The distance travelled by the particle in time $t$ will be

• A. $v_{0}t+\frac{b{t}^3}{6}$
• B. $v_{0}t+\frac{b{t}^2}{3}$
• C. $v_{0}t+\frac{b{t}^3}{3}$
• D. $v_{0}t+\frac{b{t}^2}{2}$

Answer 1

The correct option is C $v_{0}t+\frac{b{t}^3}{3}$

$a=bt\frac{dv}{dt}=btdv=btdt$

Integrating both sides, we get
$v-v_o=\frac{b{t}^2}{2}v=v_o+\frac{b{t}^2}{2}\frac{dx}{dt}=v_o+\frac{b{t}^2}{2}dx=(v_o+\frac{b{t}^2}{2})dt$
Integrating both sides, we get
$\Delta x=v_{o}t+\frac{b{t}^3}{6}$