Work, Energy, and the Work-Energy Theorem

Energy bookkeeping often beats force analysis: you can predict a rollercoaster’s speed at the bottom of any hill without knowing anything about the track’s twists — because work only cares about force along the motion, and gravity’s work only cares about height.

Work is the mechanism by which forces transfer energy to or from an object.

Work done by a constant force

$$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$$

where $\theta$ is the angle between $\vec{F}$ and the displacement $\vec{d}$. Work is measured in joules ($J = N\,m$).

Work-energy theorem

$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

The net work equals the change in kinetic energy.

Work done by a variable force

$$W = \int_{x_i}^{x_f} F(x)\,dx$$

Power is the rate of doing work:

$$P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$$

Physics link: The work-energy theorem is not a new law — it is derived directly from Newton's second law via integration. It reframes $F=ma$ in terms of energy, which is often easier to use.

Common pitfall: A force does zero work when it is perpendicular to the motion — no matter how large it is. String tension in circular motion and the normal force on a sliding block are hard at work holding geometry, yet transfer no energy.