First slide

Different question variety formed

Question

The 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

Moderate

A

$v_{o} t + \frac{1}{3} {bt}^{2}$
B

$v_{o} t + \frac{1}{3} b t^{3}$
C

$v_{o} t + \frac{1}{6} {bt}^{3}$
D

$v_{o} t + \frac{1}{2} {bt}^{2}$

Solution

$\frac{dv}{dt} = bt \Rightarrow dv = bt dt \Rightarrow v = \frac{{bt}^{2}}{2} + K_{1}$
At t = 0, v = $v_{o} \Rightarrow K_{1} = v_{o}$
We get, $v = \frac{1}{2} {bt}^{2} + v_{o}$
Again $\frac{dx}{dt} = \frac{1}{2} {bt}^{2} + v_{o} \Rightarrow x = \frac{1}{2} \frac{{bt}^{3}}{3} + v_{o} t + K_{2}$
At t = 0, x = 0 $\Rightarrow K_{2} = 0$
$∴ x = \frac{1}{6} {bt}^{3} + v_{o} t$

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