Conservation of energy, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Energy cannot be created or destroyed; it can only be transferred or transformed. That single principle solves an enormous range of physics problems, often far more simply than forces and kinematics, and it is central to AP Physics 1 and every physics course.

The principle

The law of conservation of energy says that the total energy of an isolated system stays constant. Energy can move between forms, such as kinetic, gravitational potential, elastic potential and internal (thermal) energy, but the total does not change. When energy seems to disappear, it has usually been transferred to internal energy through friction or air resistance, or carried away as sound.

The energy forms you will use

• Kinetic energy: K = ½mv².
• Gravitational potential energy near Earth's surface: U_g = mgh, measured from a chosen reference height.
• Elastic potential energy in a spring: U_s = ½kx².
• Internal (thermal) energy: increased by friction, equal to the friction force times the distance slid.

Systems and external work

Choosing the system carefully is the key skill. If the system includes the object and the Earth, gravity is internal and its energy appears as potential energy. Any force from outside the system that does work changes the system's total energy:

K_i + U_i + W_external = K_f + U_f + ΔE_internal

If no external work is done and there is no friction, mechanical energy (K + U) is conserved: K_i + U_i = K_f + U_f.

Why energy methods are so useful

Energy is a scalar, so there are no directions to resolve. The path does not matter for gravitational potential energy, only the change in height. That means a ball rolling down a curved, twisting frictionless track reaches the same speed at the bottom as one dropped straight down from the same height. Problems that would be very hard with forces become a few lines with energy. The trade-off is that energy methods give speeds and heights, not times; for timing questions you still need kinematics or Newton's laws.

Worked examples

Example 1: a falling object

A 0.50 kg ball is dropped from 20 m. Find its speed just before it lands, ignoring air resistance. (g = 9.8 m/s²)

• mgh = ½mv², so v = √(2gh) = √(2 × 9.8 × 20).
• v ≈ 19.8 m/s. The mass cancels.

Example 2: a roller coaster

A roller coaster car starts from rest at a height of 30 m and goes over a hill 12 m high. Find its speed at the top of the hill, ignoring friction.

• Loss in height: 18 m. ½v² = g × 18.
• v = √(2 × 9.8 × 18) ≈ 18.8 m/s.

Example 3: a spring launcher

A spring with k = 400 N/m is compressed 0.10 m and launches a 0.020 kg ball straight up. How high does the ball rise?

• Elastic energy: ½ × 400 × 0.10² = 2.0 J = mgh.
• h = 2.0/(0.020 × 9.8) ≈ 10.2 m, ignoring air resistance.

Example 4: a pendulum

A pendulum bob is released from 0.45 m above its lowest point. Find its speed at the bottom.

• v = √(2gh) = √(2 × 9.8 × 0.45) ≈ 2.97 m/s. The tension in the string does no work, because it is always perpendicular to the motion.

Example 5: with friction

A 2.0 kg box slides down a 5.0 m long ramp from a height of 3.0 m, starting from rest. Friction is 6.0 N. Find its speed at the bottom.

• Energy in: mgh = 2.0 × 9.8 × 3.0 = 58.8 J. Energy to internal (friction × distance): 6.0 × 5.0 = 30 J.
• Kinetic energy at the bottom: 58.8 − 30 = 28.8 J, so v = √(2 × 28.8/2.0) ≈ 5.4 m/s.

Example 6: work by an external force

A person pushes a 10 kg sled from rest with a 50 N horizontal force over 8.0 m on frictionless ice. Find the final speed.

• External work = 50 × 8.0 = 400 J, all becoming kinetic energy.
• ½ × 10 × v² = 400, so v ≈ 8.9 m/s.

Energy bar charts

AP Physics often asks for energy bar charts (sometimes called LOL diagrams). Draw bars for each form of energy at the start and the end, plus a bar for external work or energy transferred out of the system. The bars must balance: the total before plus work done on the system equals the total after. Sketching the bars before calculating helps you choose the right equation and catches missing energy stores.

Power and efficiency

Power is the rate of energy transfer: P = ΔE/Δt, in watts. For a force moving an object at constant speed, P = Fv. Efficiency is the useful energy output divided by the total energy input. No real machine is 100% efficient, because some energy is always transferred to internal energy of the surroundings.

Example 7: power of a lift

A lift raises a 600 kg load 20 m in 15 s at constant speed. Find the useful power output.

• Energy = mgh = 600 × 9.8 × 20 = 117,600 J.
• P = 117,600/15 ≈ 7.8 kW.

Common mistakes

• Forgetting to account for energy transferred by friction or air resistance.
• Including gravitational potential energy and also the work done by gravity, which counts it twice.
• Measuring heights from different reference levels in the same problem.
• Using the length of a slope instead of the vertical height in mgh.
• Forgetting to square v or take the square root when finding speed.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is conservation of energy on AP Physics 1?

Yes. Work, energy and power form a full unit of AP Physics 1, and energy reasoning is expected throughout the exam, including in free-response questions.

Is conservation of energy in GCSE and A level?

Yes. GCSE Physics uses energy stores and transfers, and A level Physics applies conservation of energy quantitatively in mechanics, electricity and thermal physics.

Key takeaways

• Energy is never created or destroyed, only transferred.
• Define your system; external work changes its energy.
• Without friction: K_i + U_i = K_f + U_f.
• With friction, include energy transferred to internal energy.
• Power is the rate of energy transfer; efficiency compares useful output to input.

Related guides

• Kinetic energy
• AP Physics 1 energy
• Hooke's law