Conservation of Energy

You’ll be able to

State the conservation of mechanical energy for frictionless systems
Convert between potential and kinetic energy in falls, ramps, and swings
Account for energy lost to friction as thermal energy

Mechanical energy is conserved without friction

When only gravity and springs act (no friction, no air resistance), the total mechanical energy — kinetic plus potential — stays constant. Energy is not created or destroyed; it merely converts from one form to another. A falling ball trades PE for KE; a rising ball trades KE for PE. This lets you connect the start and end of a motion directly: KE_i + PE_i = KE_f + PE_f.

Falling: mass cancels out

Drop an object from height h. All its potential energy mgh becomes kinetic energy ½mv² at the bottom: mgh = ½mv². The mass appears on both sides and cancels, leaving v = √(2gh). Every object, heavy or light, gains the same speed falling the same height (ignoring air resistance) — the energy version of "all objects fall at the same rate."

Conservation of mechanical energy

KE_i + PE_i = KE_f + PE_f

Valid when no friction or drag removes energy. With friction, add the dissipated thermal energy to the final side.

Energy with friction

KE_i + PE_i = KE_f + PE_f + E_thermal

Friction converts mechanical energy to heat. Total energy is still conserved — it just leaves the mechanical account.

Worked example

A ball is released from rest at the top of a frictionless ramp 5 m above the ground (g = 10 m/s²). Find its speed at the bottom.

1.At the top: all energy is potential, PE = mgh; at the bottom: all energy is kinetic, ½mv².
2.Set them equal: mgh = ½mv². The mass cancels from both sides.
3.So gh = ½v² → v² = 2gh = 2 × 10 × 5 = 100.
4.Take the square root: v = 10 m/s.

Answer: v = 10 m/s at the bottom, independent of the ball’s mass

Tip

For "released from rest, find the speed" problems, jump straight to energy conservation. Set the initial PE equal to the final KE — you avoid the forces and the ramp angle entirely, since only the height matters.

Checkpoint

An object is dropped from rest at a height of 20 m (g = 10 m/s², ignore air resistance). What is its speed just before it hits the ground?

Checkpoint

A cart at the top of a frictionless track is at rest and has 200 J of gravitational potential energy. What is its kinetic energy at the bottom, where its height is zero?

On the exam

If a problem mentions friction or air resistance, mechanical energy is not conserved — the "missing" energy became heat. Write KE_i + PE_i = KE_f + PE_f + E_thermal and treat the thermal term as the energy removed.

Answer the 2 checkpoints as you read.