Potential energy and conservation

Lift a book and the work you did doesn't vanish — it's stored, ready to come back the instant you let go. That storage account, plus one rule about the total, solves problems that would take pages of kinematics.

Potential energy

Lifting a mass m by height h against gravity takes work W = mgh. That work is banked as gravitational potential energy:

$PE = mgh$

Only changes in height matter, so you choose where h = 0 is — the floor, the tabletop, sea level — whatever makes the problem cleanest.

Conservation

When gravity is the only force doing work, the sum stays fixed:

$\tfrac{1}{2}mv^2 + mgh = \text{constant}$

Speed and height become two forms of the same budget — climbing buys height with speed, falling converts it back:

The shortcut in action

A ball rolls off a 5 m ledge. How fast does it hit the ground?

Kinematics route: find the fall time from ½gt², get the velocity components, combine them. Energy route: mgh becomes ½mv², so

$v = \sqrt{2gh} = \sqrt{2 \times 9.81 \times 5} \approx 9.9 \text{ m/s}$

Notice what the energy route didn't need: the time, the path, even the mass (it cancels). Toss the ball sideways off the ledge, slide it down a frictionless ramp, drop it straight — same landing speed every time, because only the height changed the budget. That path-independence is the superpower; kinematics has to re-solve every geometry, conservation doesn't. (The direction of landing differs — energy is a scalar and doesn't track direction. That's the price of the shortcut.)

When the account leaks

Friction and drag do negative work and bleed mechanical energy away as heat and sound. Then the honest ledger reads: initial KE + PE = final KE + PE + energy lost. The total — including the heat — is still conserved; that's the full conservation of energy, one of the deepest principles in physics. Mechanical conservation is the friction-free special case, and knowing when it applies is half the skill.

Why this matters

You now have two ways to attack any mechanics problem: forces (F = ma, full detail) and energy (totals, no clock). The last module adds the third and final currency — momentum — the one that survives even collisions.