03 / CHAPTER 02
Kinetic energy and the work theorem
Derive the work–energy relation from Newton’s law.
ANCHOR EQUATION
Kinetic energy
In symbols: K = ½mv²
The work theorem turns a force history into a speed change.
The work theorem comes from Newton’s law
Kinetic energy is not an independent magic quantity: it is the accumulated effect of net force while an object moves. The square on speed comes from integrating velocity with respect to displacement.
Replace time with position
Start with dW = F dx and F = ma. Since a = dv/dt and v = dx/dt, a dx = (dv/dt)(v dt) = v dv. Therefore dW = m v dv. Integrating from v₁ to v₂ gives W_net = ½m(v₂² − v₁²) = ΔK, with K = ½mv². The theorem works even when force varies, provided all net work is included.
A 2 kg cart speeds from 3 to 7 m/s. ΔK = ½(2)(49 − 9) = 40 J, so the net force must have done 40 J of work. A constant 10 N net force would require 4 m of displacement.
Energy does not identify the time history
Work–energy gives the final speed without telling how long the change took or the detailed path. Add momentum or the force law when time, impulse, or trajectory matters.
Work it: A 1000 kg car slows from 20 to 12 m/s. Find the work done by the net force and the average braking force if the car travels 40 m during braking.