Power
- Define power as the rate of doing work, P = dW/dt
- Compute instantaneous power as the dot product P = F·v
- Distinguish average power from instantaneous power
Power is how fast energy is transferred
Two engines can do the same total work; the stronger one does it faster. Power measures that rate — the work done per unit time, P = dW/dt, measured in watts (1 W = 1 J/s). The average power over an interval is the total work divided by the elapsed time; the instantaneous power is the derivative, the rate right now.
Instantaneous power is force times velocity
Since dW = F·dx and v = dx/dt, dividing gives P = F·dx/dt = F·v. When a force acts along the direction of motion, P = Fv; if it acts at an angle θ, only the component along v counts, P = Fv cos θ. This is why a car engine delivering constant power produces less force at high speed — the same P must be shared over a larger v.
A motor lifts a 50 kg load straight up at a constant speed of 2 m/s. Using g = 10 m/s², what power does the motor deliver?
- 1.At constant speed the upward force equals the weight: F = mg = 50 × 10 = 500 N.
- 2.The force is along the (upward) velocity, so P = Fv.
- 3.Substitute: P = 500 × 2 = 1000 W.
- 4.Equivalently, the motor raises the load potential energy at mgh/t = mg·v = 1000 W.
A 20 N force pushes an object that moves at a constant 3 m/s in the direction of the force. What power does the force deliver?
A crane does 12000 J of work lifting a load over 6 s. What is its average power?
Use P = W/Δt when you know a total amount of work over an interval, and P = Fv (or F·v) when you want the power at a specific instant or speed. A force perpendicular to the velocity — like the centripetal force on a satellite — does zero work and delivers zero power.
Answer the 2 checkpoints as you read.
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