Summary

This chapter developed WBCHSE Unit 4's account of work, energy and power in the following order. Work done by a constant force is $W = \vec F\cdot\vec d = Fd\cos\theta$, a scalar; for a variable force it generalises to the integral $W = \int F(x)\,dx$, the area under the force-displacement graph, illustrated by deriving the work needed to stretch an ideal spring, $\tfrac12kx_0^2$. Kinetic energy, $K=\tfrac12mv^2$, connects to work through the work-energy theorem, $W_{\text{net}}=\Delta K$, derived here directly from Newton's second law using the chain rule $dv/dt = v\,dv/dx$. Power, the rate of doing work, is $P_{\text{avg}}=W/t$ on average and $P=\vec F\cdot\vec v$ instantaneously, with the watt and the practical energy unit kilowatt-hour both introduced. Conservative forces (gravity, the spring force) do path-independent work and zero net work around any closed path, unlike non-conservative forces (friction, drag), which dissipate mechanical energy irreversibly as heat and sound -- this distinction is exactly what allows a potential energy to be defined at all, giving $U=mgh$ for gravity and $U(x)=\tfrac12kx^2$ for a spring. Combining kinetic and potential energy as $E=K+U$ gives the law of conservation of mechanical energy, $E=\text{constant}$, whenever only conservative forces act -- applied here to motion in a vertical circle, where combining Newton's second law at the top ($T=0$ at the critical case) with energy conservation across the diameter $2r$ gives the two key results $v_{\text{top,min}}=\sqrt{gr}$ and $v_{\text{bottom,min}}=\sqrt{5gr}$. Finally, collisions were treated in both one and two dimensions: momentum is always conserved, but kinetic energy only in a perfectly elastic collision, for which the standard velocity formulas were derived and their equal-mass, very-light-body and very-heavy-body limits worked out; perfectly inelastic collisions, where the bodies stick together, represent the maximum possible kinetic-energy loss consistent with momentum conservation; and the **coefficient of restitu …