Work, Energy and Power

Work done by a constant force over a displacement $d$ at angle $\theta$ to it:

Starting from $v^2 = u^2 + 2as$ and multiplying both sides by $\tfrac{1}{2}m$:

the work-energy theorem: net work equals the change in kinetic energy.

Lifting a mass steadily through height $h$ stores that same work as gravitational potential energy, $\Delta PE = mgh$, and power is simply how quickly work gets done:

“Mechanical energy is conserved” is taught as a blanket rule, then quietly restricted to “in the absence of friction and other non-conservative forces” in a footnote most students skip. Run the simulation above with any friction above zero: potential energy converts into kinetic energy exactly as before, but the two no longer sum to a constant. The missing energy hasn’t vanished, it’s the heat generated at the sliding surface, and it only shows up if you track it separately from “mechanical” energy.

A related trap: “friction always removes energy, so it never does positive work.” That’s true for kinetic friction opposing sliding, but static friction can do positive work, for example on a crate accelerating forward on a truck bed purely because static friction is dragging it along. Whether friction adds or removes energy depends on whether it acts with the motion or against it, not on some fixed rule about friction itself.