Conservation of Energy

Kinetic plus potential stays constant, until friction takes its cut.

15 minUpdated 21 October 2026mechanical energygravitational PEelastic PEE = mc²conservation of energy

Lesson at a glanceShow the key points

If internal forces are conservative and external forces do no work, mechanical energy E = K + U is constant.
Near Earth, raising a body by h increases gravitational PE by mgh; the zero level is your choice.
A spring stretched or compressed by x stores elastic PE ½kx² (zero at natural length).
With friction or external work: W(non-conservative) + W(external) = Ef − Ei.
Energy has many forms (thermal, chemical, light, sound, mass: E = mc²); in total it is never created or destroyed.

Of all the ideas in physics, this may be the most useful: energy changes form, but the total never changes. Use it and many hard motion problems need just one line.

Conservation of mechanical energy

Combine the last two lessons. If the internal forces are conservative and the external forces do no work, then $K_f - K_i = -(U_f - U_i)$ , which rearranges to

Conservation of mechanical energy

K_i + U_i = K_f + U_f \qquad (E = K + U \text{ is constant})

If non-conservative forces (like friction) or external forces do work, the mechanical energy changes by exactly that work:

When energy is not conserved(8.7, 8.8)

W_{nc} + W_{ext} = E_f - E_i

Gravitational potential energy

Take “block + Earth” as the system. The Earth is so massive that it hardly moves, so we can use a frame attached to it. When the block rises by h, gravity does $-mgh$ on it, so the system’s potential energy rises by $mgh$ . By custom we call this the potential energy “of the block”:

U = mgh \qquad (h \ll R_{\text{Earth}}, \ \text{measured above a zero level of your choice})

Interactive

A skater on a track: where does the energy go?

Drag the skater to a starting point and release. Without friction, kinetic plus potential energy stays fixed, so she always climbs back to her starting height. Add friction and the missing energy turns up as heat.

0.0

kinetic

17.9

potential

0.0

thermal

17.9

total

× 100 J

Track

Friction

\mu

0.000

Worked example

Down a frictionless slide

A child slides from rest down a smooth curved slide whose top is 5.0 m above its bottom. How fast is she going at the bottom? (g = 9.8 m/s²)

Interactive

Pendulum: trading speed for height

The string pulls perpendicular to the bob's motion, so it does no work. Only gravity does work, and mechanical energy is conserved: speed at angle θ follows from the height gained, L(1 − cos θ).

h = L(1-\cos\theta)

0.40m

v_\theta

2.83m/s

highest angle reached90.0°

Speed at the bottom

v_0

4.0m/s

String length

L

0.80m

Angle to check

\theta

60°

Worked example

A pendulum swinging up

A pendulum bob passes its lowest point at 4.0 m/s on a 0.80 m string. How fast is it moving when the string makes 60° with the vertical? (g = 10 m/s²)

Elastic potential energy of a spring

Stretch a massless spring slowly by x. The wall’s force on it does no work (its end doesn’t move); the block’s pull does $\int_0^x kx\,dx = \tfrac12kx^2$ . The spring’s kinetic energy stays zero, so all of that work is stored as potential energy. Compressing by x stores the same amount. Choosing U = 0 at natural length:

Elastic (strain) potential energy

U_{\text{spring}} = \tfrac12 k x^2

Worked example

A block hits a spring

A 0.50 kg block slides at 2.0 m/s on a smooth table into a spring (k = 200 N/m) fixed to a wall. How far does the spring compress?

All the forms of energy

Mechanical energy is not the whole story. A body at rest still has internal (thermal) energy from the random motion and interaction of its molecules. Light and sound carry energy; chemical reactions release it. And Einstein showed that mass itself is a form of energy:

Mass–energy equivalence

E = mc^2, \qquad c = 3\times10^8\ \text{m/s}

about 8.2 × 10⁻¹⁴ J is needed to create one electron

What to remember

1Conservative internal forces, no external work: $K + U$ stays constant.
2Otherwise $W_{nc} + W_{ext} = \Delta E$ .
3 $U_{grav} = mgh$ (any zero level); $U_{spring} = \tfrac12kx^2$ .
4Total energy, counting heat, light, chemical energy and mass (E = mc²), is always conserved.

Check yourself

Two balls are released from the same height: one dropped straight down, one rolled down a frictionless curved ramp. At the bottom…

Compressing a spring by 2 cm stores 0.1 J. Compressing it by 4 cm stores…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchConservation of energyKhan Academy PlayEnergy Skate Park: BasicsPhET · CU Boulder PlayEnergy Skate ParkPhET · CU Boulder ReadConservation of energyOpenStax University Physics §8.3 ReadPotential energy diagrams and stabilityOpenStax University Physics §8.4 ReadMechanical energyThe Physics Classroom ReadConservation of energyHyperPhysics ReadConservation of energyFeynman Lectures, Vol. I, Ch. 4The famous ‘Dennis the Menace’s blocks’ story

PreviousConservative Forces & Potential Energy Next chapter Centre of Mass, Momentum & Collisions

Conservation of Energy

Kinetic plus potential stays constant, until friction takes its cut.

15 minUpdated 21 October 2026mechanical energygravitational PEelastic PEE = mc²conservation of energy

Lesson at a glanceShow the key points

If internal forces are conservative and external forces do no work, mechanical energy E = K + U is constant.
Near Earth, raising a body by h increases gravitational PE by mgh; the zero level is your choice.
A spring stretched or compressed by x stores elastic PE ½kx² (zero at natural length).
With friction or external work: W(non-conservative) + W(external) = Ef − Ei.
Energy has many forms (thermal, chemical, light, sound, mass: E = mc²); in total it is never created or destroyed.

Of all the ideas in physics, this may be the most useful: energy changes form, but the total never changes. Use it and many hard motion problems need just one line.

Conservation of mechanical energy

Combine the last two lessons. If the internal forces are conservative and the external forces do no work, then $K_f - K_i = -(U_f - U_i)$ , which rearranges to

Conservation of mechanical energy

K_i + U_i = K_f + U_f \qquad (E = K + U \text{ is constant})

If non-conservative forces (like friction) or external forces do work, the mechanical energy changes by exactly that work:

When energy is not conserved(8.7, 8.8)

W_{nc} + W_{ext} = E_f - E_i

Gravitational potential energy

U = mgh \qquad (h \ll R_{\text{Earth}}, \ \text{measured above a zero level of your choice})

Interactive

A skater on a track: where does the energy go?

0.0

kinetic

17.9

potential

0.0

thermal

17.9

total

× 100 J

Track

Friction

\mu

0.000

Worked example

Down a frictionless slide

A child slides from rest down a smooth curved slide whose top is 5.0 m above its bottom. How fast is she going at the bottom? (g = 9.8 m/s²)

Interactive

Pendulum: trading speed for height

The string pulls perpendicular to the bob's motion, so it does no work. Only gravity does work, and mechanical energy is conserved: speed at angle θ follows from the height gained, L(1 − cos θ).

h = L(1-\cos\theta)

0.40m

v_\theta

2.83m/s

highest angle reached90.0°

Speed at the bottom

v_0

4.0m/s

String length

L

0.80m

Angle to check

\theta

60°

Worked example

A pendulum swinging up

A pendulum bob passes its lowest point at 4.0 m/s on a 0.80 m string. How fast is it moving when the string makes 60° with the vertical? (g = 10 m/s²)

Elastic potential energy of a spring

Elastic (strain) potential energy

U_{\text{spring}} = \tfrac12 k x^2

Worked example

A block hits a spring

A 0.50 kg block slides at 2.0 m/s on a smooth table into a spring (k = 200 N/m) fixed to a wall. How far does the spring compress?

All the forms of energy

Mass–energy equivalence

E = mc^2, \qquad c = 3\times10^8\ \text{m/s}

about 8.2 × 10⁻¹⁴ J is needed to create one electron

What to remember

1Conservative internal forces, no external work: $K + U$ stays constant.
2Otherwise $W_{nc} + W_{ext} = \Delta E$ .
3 $U_{grav} = mgh$ (any zero level); $U_{spring} = \tfrac12kx^2$ .
4Total energy, counting heat, light, chemical energy and mass (E = mc²), is always conserved.

Check yourself

Two balls are released from the same height: one dropped straight down, one rolled down a frictionless curved ramp. At the bottom…

Compressing a spring by 2 cm stores 0.1 J. Compressing it by 4 cm stores…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousConservative Forces & Potential Energy Next chapter Centre of Mass, Momentum & Collisions