An object of mass 1000 g experiences a time dependent force $\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N$. The power generated by the force at time $t$ is:

<p><strong>Given Information:</strong></p> <p><p>Force: $\vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{N}$</p></p> <p><p>Mass of the object, $m = 1000 \, \text{g} = 1 \, \text{kg}$</p></p> <p><strong>Determine Acceleration:</strong></p> <p><p>Using Newton's second law, $\overrightarrow{F} = m \overrightarrow{a}$.</p></p> <p><p>Thus, the acceleration, $\overrightarrow{a} = 2t \hat{i} + 3t^2 \hat{j}$.</p></p> <p><strong>Velocity Calculation:</strong></p> <p><p>To find velocity, integrate acceleration with respect to time:</p> <p>$ \frac{d\overrightarrow{v}}{dt} = 2t \hat{i} + 3t^2 \hat{j} $</p></p> <p><p>Integrate to find $\overrightarrow{v}$:</p> <p>$ \overrightarrow{v} = \int (2t \hat{i} + 3t^2 \hat{j}) \, dt = t^2 \hat{i} + t^3 \hat{j} + \vec{C} $</p> <p>(Assuming initial velocity is zero, $\vec{C} = 0$, hence $\overrightarrow{v} = t^2 \hat{i} + t^3 \hat{j}$).</p></p> <p><strong>Calculate Power:</strong></p> <p><p>Power is defined as the dot product of force and velocity vectors:</p> <p>$ P = \overrightarrow{F} \cdot \overrightarrow{v} = (2t \hat{i} + 3t^2 \hat{j}) \cdot (t^2 \hat{i} + t^3 \hat{j}) $</p></p> <p><p>Computing the dot product gives:</p> <p>$ P = (2t \cdot t^2) + (3t^2 \cdot t^3) = 2t^3 + 3t^5 $</p></p> <p>The power generated at time $t$ is $P = (2t^3 + 3t^5) \, \text{W}$.</p>