Work Energy and Power

Step 1: Write the equation for powerPower $( P )$ is force $( F )$ multiplied by velocity $( v )$ :$P = F \cdot v$Step 2: Replace force with mass and accelerationForce is mass $( m )$ times acceleration $\left( \;\frac{d v}{d t} \right)$ :$P = m \;\frac{d v}{d t} \cdot v$Step 3: Rearrange to separate variablesMove terms so you have all $v$ terms on one side and $t$ terms on the other:$v \;\frac{d v}{d t} = \;\frac{P}{m}$Step 4: Integrate both sidesIntegrate with respect to $v$ and $t$ :$\int v \, dv = \;\frac{P}{m} \int dt$After integrating:$\;\frac{v^2}{2} = \;\frac{P}{m} t$ Step 5: Solve for velocity $( v )$$v = \sqrt{ \;\frac{2 P t}{m} }$Step 6: Find the relationship between displacement and timeVelocity is the rate of change of displacement, so:$\;\frac{d s}{d t} = \sqrt{ \;\frac{2 P t}{m} }$Step 7: Integrate to get displacement $( s )$Integrate both sides with respect to $t$ :$\int ds = \sqrt{ \;\frac{2 P}{m} } \int t^{1/2} \, dt$The right side integrates to $t^{3/2}$ :$s \propto t^{3/2}$Step 8: AnswerThe value of $x$ is $\;\frac{3}{2}$.