Q.Arrive at an expression for power and velocity. Give some examples for the same.

Step 1. Start from the definition of work done by a force $\vec F$ over a displacement $d\vec r$: $W=\displaystyle\int \vec F\cdot d\vec r$.

Step 2. Differentiate both sides with respect to time. On the left, $\dfrac{dW}{dt}$ is, by definition, the instantaneous power $P$. On the right, using $\dfrac{d\vec r}{dt}=\vec v$ (the instantaneous velocity), $\dfrac{d}{dt}\displaystyle\int\vec F\cdot d\vec r = \vec F\cdot\dfrac{d\vec r}{dt}=\vec F\cdot \vec v$.

Step 3. Equating the two sides: $\boxed{P=\vec F\cdot\vec v}$ -- the instantaneous power delivered by a force is the dot product of the force and the velocity of the point it acts on.

Step 4. Example 1 -- vehicle engine. A car's engine must overcome resistive forces (air drag, rolling friction) and, if accelerating, also supply $ma$; the total driving force multiplied by the current speed gives the instantaneous power the engine must deliver, $P=(F_{\text{resistive}}+ma)v$. …

Derive P = F.v by differentiating the work integral with respect to time and using dr/dt = v, then …

• Stating P = F.v without deriving it from the work integral, when a derivation is explicitly asked for. …