Conservation of Energy

You’ll be able to

State conservation of mechanical energy for systems acted on only by conservative forces
Solve motion problems using KE_i + U_i = KE_f + U_f
Account for energy removed by friction using the work of nonconservative forces

Mechanical energy is conserved when only conservative forces act

Add kinetic and potential energy to get the total mechanical energy E = KE + U. If the only forces doing work are conservative, E stays constant: as an object speeds up, KE rises exactly as much as U falls. This turns hard dynamics problems into simple bookkeeping — equate the total energy at two instants and solve, never needing the force or the time in between.

Conservation of mechanical energy

KE_i + U_i = KE_f + U_f

Valid when only conservative forces (gravity, springs) do work. Pick a reference level for U and apply it consistently at both instants.

Friction breaks the balance

A nonconservative force such as friction or drag does path-dependent work and removes mechanical energy as heat, so E is not conserved when it acts. The generalized statement is that the work of the nonconservative forces equals the change in mechanical energy: W_nc = E_f − E_i. For sliding friction the lost energy is the friction force times the path length, |W_friction| = f·d.

Energy with a nonconservative force

KE_i + U_i + W_nc = KE_f + U_f

W_nc is the (usually negative) work of nonconservative forces. For friction over a distance d, W_nc = −f·d, draining mechanical energy into heat.

Worked example

A 2 kg block is released from rest at the top of a frictionless ramp of height 1.8 m. Using g = 10 m/s², find its speed at the bottom.

1.Only gravity does work, so mechanical energy is conserved: KE_i + U_i = KE_f + U_f.
2.Take the bottom as U = 0. Starting from rest: 0 + mgh = ½mv_f² + 0.
3.The mass cancels: gh = ½v_f² → v_f² = 2gh = 2(10)(1.8) = 36.
4.Therefore v_f = 6 m/s — independent of the ramp shape, since only the height change matters.

Answer: v_f = √(2gh) = √36 = 6 m/s

Checkpoint

A ball is dropped from rest at height h and falls freely. Using energy conservation, its speed just before it hits the ground is:

Checkpoint

A 3 kg box slides across a level floor and is brought to rest by friction, which does −30 J of work. If no other force does work, what was the box initial kinetic energy?

Watch out

Only invoke "energy is conserved" when the forces doing work are conservative. The moment friction, drag, or an applied push enters, switch to KE_i + U_i + W_nc = KE_f + U_f and account for the energy those forces add or remove.

Answer the 2 checkpoints as you read.