A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of $\mathit{a}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{80}\mathbf{ }{\mathbf{\text{m/s}}}^{\mathbf{\text{2}}}$. A worker assists the cart by pushing on the crate with a force that is eastward and has magnitude that depends on time according to $\mathit{F}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\left(5.40 \text{N/s}\right)\mathit{t}$. What is the instantaneous power supplied by this force at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{s}}$?

Acceleration of car is$a=2.80 {\text{m/s}}^{\text{2}}$

Time dependent force is$F\left(t\right)=\left(5.40 \text{N/s}\right)t$

Power is defined as work done per unit time. It is given as

$\mathit{P}\mathbf{=}\frac{\mathit{W}}{\mathit{t}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\text{or}}\mathbf{ }\mathbf{ }\mathit{F}\mathit{v}$ ...(i)

Where, $W$is the work done, role="math" localid="1660032624791" $t$is the time , $F$is the force and$v$ is the velocity

The final velocity of the crate is expressed as follows

$v=u+at$

Where $u$is the initial velocity

In the question, it is given that the cart start from rest so that the initial velocity $u=0$

So,

$v=at$ ...(ii)

By using equation (i) and equation (ii)

role="math" localid="1660032854672" $\begin{array}{rcl}P& =& Fat\\ & =& \left(5.40 \text{N/s}\right)t\times at\\ & =& a\times \left(5.40 \text{N/s}\right)t^2\end{array}$

Substitute the values in above equation

$\begin{array}{rcl}P& =& 2.80 {\text{m/s}}^{\text{2}}\times 5.40 \text{N/s}\times {\left(\text{5.00} \text{s}\right)}^2\\ & =& 378 \text{W}\end{array}$

Hence the instantaneous power supplied by this force at$t=5.00 \text{s}$is$378 \text{W}$