Conservation of Energy in Practice

The law of conservation of energy states that energy cannot be created or destroyed, only transferred from one store to another.

Falling objects (no air resistance)

When an object falls freely, GPE converts to KE:

$mgh = \frac{1}{2}mv^2$

The mass cancels:

$v = \sqrt{2gh}$

This gives the speed at any height, regardless of the object’s mass.

Pendulums

At the highest point: maximum GPE, zero KE (momentarily at rest).

At the lowest point: maximum KE, minimum GPE.

Energy continuously converts between GPE and KE. If no air resistance, the pendulum swings to the same height each time.

Roller coasters

The highest point of the track determines the maximum speed at the bottom:

$v_{\text{max}} = \sqrt{2gh_{\text{max}}}$

No hill after the first drop can be taller than the starting height (without an additional energy input), because there would not be enough energy.

With friction and air resistance

Some energy is dissipated as thermal energy, so:

$\text{useful KE} = \text{GPE lost} - \text{energy dissipated}$

The object reaches a lower speed than the frictionless calculation predicts.

Common errors and how to correct them

• Forgetting to account for energy dissipation in real systems.
• Using the wrong height (use vertical height change, not distance along a slope).

How to apply this in an exam

Set up the energy equation: initial energy = final energy + dissipated energy. Cancel mass if it appears on both sides. Solve for the unknown.