Kinetic and Potential Energy Basics

Kinetic energy is the energy an object has because of its motion, E_k = ½mv², and gravitational potential energy is the energy it has because of its height in a gravitational field, E_p = mgh. In brief: kinetic energy depends on the square of the speed, so doubling the speed quadruples the energy. When an object falls, potential energy is converted to kinetic energy; when it rises, the reverse happens. Both are measured in joules.

A roller coaster at the top of a hill has a lot of potential energy, and at the bottom it has a lot of kinetic energy. This guide defines the two forms of energy, shows the formulas and how they are used, explains the conversion between them, compares them in a table, works through a falling ball example and corrects common mistakes.

What is kinetic energy?

Every moving object has kinetic energy (KE). It is given by E_k = ½mv², where m is the mass (kg) and v is the speed (m/s). It is always positive or zero, and it does not depend on direction. Because of the squared speed, a car at 30 m/s has nine times the kinetic energy of the same car at 10 m/s, which is one reason why high-speed crashes are so dangerous and why braking distance grows quickly with speed.

What is gravitational potential energy?

An object lifted above the ground has gravitational potential energy (GPE) because the gravitational field can do work on it when it falls. Near the Earth’s surface, the change in GPE is ΔE_p = mgΔh, with g = 9.8 N/kg (m/s²). Heights are measured from a chosen reference level, which can be the floor or the ground; only changes in potential energy have physical meaning. The formula holds when g can be taken as constant, so for small changes in height compared with the radius of the Earth.

How do they convert into one another?

When an object falls, gravity does work on it, and gravitational potential energy decreases while kinetic energy increases by the same amount (if there is no air resistance). When it is thrown upward, the kinetic energy decreases as potential energy increases. In a swinging pendulum or on a roller coaster, the energy changes back and forth between the two forms, and the total of the two (the mechanical energy) stays constant if friction is negligible. In real systems, some energy is transferred to thermal energy by friction and air resistance.

Kinetic and potential energy compared

Worked example: a falling ball

A 0.50 kg ball is dropped from a height of 20 m. Ignore air resistance. Use g = 9.8 m/s².

1. Initial potential energy: E_p = mgh = 0.50 × 9.8 × 20 = 98 J. The kinetic energy is zero.
2. At a height of 5.0 m: E_p = 0.50 × 9.8 × 5.0 = 24.5 J. The potential energy lost is 98 − 24.5 = 73.5 J, so the kinetic energy is 73.5 J.
3. Speed at 5.0 m: ½ × 0.50 × v² = 73.5, so v² = 294 and v = 17.1 m/s.
4. Just before landing: all 98 J is kinetic: v = √(2 × 98 ÷ 0.50) = √392 = 19.8 m/s. Check with v = √(2gh) = √(2 × 9.8 × 20) = 19.8 m/s, which does not depend on the mass.

How were the formulas found?

Gottfried Leibniz proposed in the 1680s that the quantity mv² (vis viva) is conserved in collisions, and Émilie du Châtelet argued in the 1740s, from experiments dropping balls into clay, that the depth of the dent is proportional to v². The term “kinetic energy” was coined by Lord Kelvin in the 1850s, and “potential energy” by William Rankine in 1853.

Which mistakes are common?

Mistake 1: Forgetting to square the speed

Kinetic energy depends on v². Doubling v gives four times the energy.

Mistake 2: Leaving out the factor ½

E_k = ½mv², while E_p = mgh has no ½.

Mistake 3: Thinking that potential energy is an absolute number

It depends on the reference level. The change in energy is what matters.

Mistake 4: Using the path length instead of the vertical height

For gravitational potential energy, only the vertical change in height counts.