Power = rate of doing work. P = $\frac{dW}{dt}$ = F.v (instantaneous). Average power = total work / total time.

Constant power motion: P = mav = constant. As v increases, a decreases. v = sqrt$\frac{2Pt}{m}$. The vehicle never has constant acceleration under constant power.

Terminal/maximum velocity: When driving force = friction: $v_{max}$ = $\frac{P}{f}$. On an incline: $v_{max}$ = $\frac{P}{f + mg*sin(theta}$).

Practical applications: Pumps (P = $\frac{mgh}{t}$), elevators (P = Mgv), engines. Always account for efficiency: $P_{useful}$ = eta * $P_{input}$.

Key insight: Power determines how FAST work can be done, not how much. A small engine can do the same total work as a large one, just takes longer.