Conservation of energy

The three mechanical energy stores

Kinetic energy — the energy of a moving object:

$$E_k = \tfrac{1}{2}mv^2$$

• $E_k$ — kinetic energy (J)
• $m$ — mass (kg)
• $v$ — speed (m s⁻¹)
• It depends on $v^2$, so doubling the speed quadruples the kinetic energy.

Gravitational potential energy — the energy of a raised object:

$$E_p = mgh$$

• $h$ — the vertical height gained (m), not the distance travelled along a slope
• $g = 9.8$ m s⁻²
• Only changes in height matter; you can measure $h$ from any convenient level as long as you are consistent.

Elastic potential energy — the energy stored in a stretched or compressed spring:

$$E_p = \tfrac{1}{2}kx^2$$

• $k$ — spring constant (N m⁻¹)
• $x$ — extension or compression from the natural length (m)

The law of conservation of energy

• Energy cannot be created or destroyed, only transformed from one form into another.
• The total energy of a closed system is constant.
• In an ideal system with no friction or air resistance, the mechanical total is constant:

$$E_k + E_p = \text{constant}$$

• This gives a powerful shortcut: for a falling or swinging object, energy at the start = energy at the end, and you never need to know the path taken or the time elapsed.

Using energy instead of the equations of motion

The energy method is often faster, and it works where the equations of motion do not.

• Object falling from height $h$ (starting from rest):

$$mgh = \tfrac{1}{2}mv^2 \;\Rightarrow\; v = \sqrt{2gh}$$

• The mass cancels, so all objects reach the same speed from the same height, exactly as free fall predicts.
• Use energy rather than the equations of motion when:
  • the path is curved (a slide, a pendulum, a rollercoaster),
  • the acceleration is not constant,
  • the question mentions friction losses or heat.
• Use the equations of motion when you need a time, which energy equations never contain.

Where the "lost" energy goes

• In a real system, some mechanical energy is always transformed into heat (friction, air resistance), sound, and permanent deformation.
• That energy is not destroyed — it has simply left the useful mechanical store.
• To account for it:

$$E_{\text{start}} = E_{\text{end}} + E_{\text{lost to heat and sound}}$$

• A slide with friction: $mgh = \tfrac{1}{2}mv^2 + \text{heat}$, so the child arrives slower than the frictionless calculation predicts, and the slide surface warms up slightly.