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An object of mass $3\,kg$ is at rest. If a force $\vec F\, = \,\left( {6{t^2}\,\hat i\,\, + \,4t\,\hat j} \right)\,N$ is applied on the object, then the velocity of the object at $t = 3\,s$ will be
${18\,\hat i\,\, + \,3\,\hat j}$
${18\,\hat i\,\, + \,6\,\hat j}$
${3\,\hat i\,\, + \,18\,\hat j}$
${18\,\hat i\,\, + \,4\,\hat j}$
Solution
$\frac{d \vec {v}}{d t}=\vec {a}=\frac{\vec {F}}{m}=\frac{6 t^{2} \hat{i}+4 t \hat{j}}{3} m / s^{2}$
as acceleration depending on time so its not a
case of constant acceleration hence $\overrightarrow{v}=\int_{0}^{i} \vec{a} \mathrm{d} t$
$\therefore \quad \overrightarrow{\mathrm{v}}=\int_{0}^{3}\left(\frac{6 \mathrm{t}^{2}}{3} \hat{\mathrm{i}}+\frac{4 \mathrm{t}}{3} \hat{\mathrm{j}}\right) \mathrm{dt}$
$=\left[\frac{6 t^{3}}{9} \hat{i}+\frac{4 t^{2}}{6} \hat{j}\right]_{0}^{3}=18 \hat{i}+16 \hat{j}$
