Question

how do the kinetic energy and potential energy relate to the total

Answer 1

In classical mechanics the total (mechanical) energy E of a system is the sum of its kinetic energy K and its potential energy U:

- Kinetic energy: K = 1/2 m v^2 (for a point mass; more general expressions exist for rotations, fields, etc.).
- Potential energy: U(x) depends on configuration (examples: near Earth U = mgh, gravitational U = −Gm1m2/r, spring U = 1/2 k x^2).

Total mechanical energy:
E = K + U.

Key consequences and relations
- Conservation (conservative forces only): If only conservative forces act, work by those forces changes K exactly opposite to U so that E is constant in time:
ΔE = ΔK + ΔU = 0 ⇒ K + U = constant.
Equivalently ΔK = −ΔU.
- Work by non-conservative forces: If non-conservative forces (friction, air drag) do work W_nc, then
ΔE = W_nc (often negative, representing energy dissipated as heat).
- Motion constraint/turning points: For a given constant E, K = E − U. Since K ≥ 0, motion is allowed only where U ≤ E. Points where U = E are turning points (v = 0).
- Force from potential: For one-dimensional motion, the force is F = −dU/dx. The slope of U(x) determines acceleration and how K and U trade energy.
- Reference arbitrariness: U is defined up to an additive constant; only differences ΔU matter physically.

Example (pendulum): At the highest point U is maximum and K = 0; as it falls U decreases and K increases so that E stays constant (neglecting friction).

That’s the basic relationship: total mechanical energy is the sum of kinetic and potential energy, and energy is exchanged between those forms while the total stays fixed if only conservative forces act.