First slide

Rectilinear Motion

Question

Acceleration of a particle at any time t is $\bar{a} = (2 t \hat{i} + 3 t^{2} \hat{j}) m / s^{2}$ . If initially particle is at rest find the velocity of the particle at time t=2 sec

Easy

A

$4 \hat{i} + 8 \hat{j}$
B

$8 \hat{i} + 4 \hat{j}$
C

$2 \hat{i} + 4 \hat{j}$
D

$4 \hat{i} + 2 \hat{j}$

Solution

Here acceleration is a function of time. We will have to go for integration for finding velocity at time ‘t’ . Thus
$\begin{array}{l} d \bar{v} = \bar{a} d t \\ \Rightarrow \int_{0}^{v} d v = \int_{0}^{2} a d t \\ \Rightarrow \bar{v} = \int_{0}^{2} (2 t \hat{i} + 3 t^{2} \hat{j}) d t \end{array}$
$\begin{array}{l} = {(\frac{2 t^{2}}{2} \hat{i} + \frac{3 t^{3}}{3} \hat{j})}_{0}^{2} \\ = {(t^{2} \hat{i} + t^{3} \hat{j})}_{0}^{2} \\ = (4 \hat{i} + 8 \hat{j}) m / s \end{array}$

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