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Step-by-Step Solution
Step 1: Identify the Given Quantities
The mass of the object is given as $3\text{ kg}$, and the force acting on the object is
$ \overrightarrow{F} = 6t^2 \,\widehat{i} + 4t \,\widehat{j}.$
Initially, the object is at rest.
Step 2: Relate Force and Acceleration
From Newton's second law, $\overrightarrow{F} = m\,\overrightarrow{a}$. Hence, the acceleration is
$$
\overrightarrow{a} = \frac{\overrightarrow{F}}{m} = \frac{6t^2 \,\widehat{i} + 4t \,\widehat{j}}{3}
= 2t^2 \,\widehat{i} + \frac{4}{3}t \,\widehat{j}.
$$
Step 3: Express Acceleration as Derivative of Velocity
Acceleration is the time derivative of velocity:
$$
\overrightarrow{a} = \frac{d\overrightarrow{v}}{dt} = 2t^2 \,\widehat{i} + \frac{4}{3}t \,\widehat{j}.
$$
Step 4: Integrate Acceleration to Find Velocity
To find the velocity, integrate the acceleration with respect to time. Since the object starts from rest, the initial velocity at $t = 0$ is zero:
$$
\overrightarrow{v} = \int_0^3 \left( 2t^2 \,\widehat{i} + \frac{4}{3}t \,\widehat{j}\right) dt.
$$
Step 5: Perform the Integration
Compute the integral:
$$
\overrightarrow{v}
= \left[\frac{2}{3}t^3 \,\widehat{i} + \frac{4}{6}t^2 \,\widehat{j} \right]_0^3
= \left(\frac{2}{3} \cdot 3^3\right)\widehat{i} + \left(\frac{4}{6} \cdot 3^2\right)\widehat{j}
= 18 \,\widehat{i} + 6 \,\widehat{j}.
$$
Step 6: State the Final Velocity
Thus, the velocity of the object at $t = 3 \text{ s}$ is
$$
\overrightarrow{v} = 18\,\widehat{i} + 6\,\widehat{j}.
$$
