Conservation of mechanical energy

Mechanical energy = kinetic energy + potential energy.

Follow a falling box. The 2 kg box at 10 m: U = 200 J, K = 0. Total 200 J.

(using g = 10 m s⁻²). Every joule of potential energy lost becomes a joule of kinetic energy. When only gravity acts, mechanical energy is conserved: it stays the same.

A pendulum shows it. Release the bob at P. At the lowest point Q it is all kinetic energy; it swings up to R, almost as high as P, where it is all potential again. Slowly it swings a little lower each time and stops, because friction at the support and air resistance take away energy.

Down a slide. At the top, mgh; at the bottom, ½mv². So mgh = ½mv², which gives v = √(2gh). The m cancels: mass and the shape of the slide don't matter (if we ignore friction). A heavy child and a light child reach the bottom at the same speed.

Escape ramps save lives. On steep highways, a truck whose brakes fail can turn into a sand-filled escape ramp. A 10000 kg truck at 20 m s⁻¹ has 2,000,000 J of kinetic energy. Climbing a 30° ramp against a 50000 N drag from the sand, it stops after 20 m. With energy, we can find this without working through every second of the motion.

Falling water does work. In the Himalayas, the *gharat or panchakki* (water mill) turns the potential energy of falling water into kinetic energy that turns a millstone and grinds grain. Big dams do the same thing to make electricity.

In nature, the total energy of a system with no outside forces always stays the same. Energy only changes form.