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

AIPMTAIPMT 1995Motion in a Straight Line

Solution:

Acceleration $ \propto bt.$ i.e., $\frac{d^2 x}{dt^2} = a\, \propto bt $
Integrating, $\frac{dx}{dt} =\frac{bt^2}{2}+C$
Initially, $ t = 0, dx/dt=v_0 $
Therefore, $\frac{dx}{dt} =\frac{bt^2}{2}+v_0$
Integrating again, $x = \frac{bt^3}{6}+v_0 t + C$
When $t = 0, x = 0$
$ \Rightarrow C = 0.$
i.e., distance travelled by the particle in time t
$ = v_0 t + \frac{bt^3}{6}$.