Kinetic vs. Potential Energy: Formulas, Roller Coaster Examples and Physics Practice Quiz

Key facts

• Classical translational kinetic energy is KE = ½mv², where m is kilograms and v is meters per second.
• Near Earth's surface gravitational potential energy relative to a height reference is PE = mgh, where g is about 9.8 meters per second squared.
• A 2-kg object at 3 m/s has 9 joules of kinetic energy.
• A 5-kg object elevated 4 m above a chosen reference has 196 joules of gravitational potential energy using g=9.8 m/s².
• Double speed with the same mass and kinetic energy increases by a factor of four, not two.

Kinetic energy is energy associated with motion; potential energy is stored energy associated with an object’s position or configuration. For students studying movement near Earth’s surface, the two essential formulas are kinetic energy: KE = ½mv² and gravitational potential energy: PE = mgh, measured in joules (J). A roller coaster is a familiar example: as a car moves downhill, it generally loses gravitational potential energy and gains kinetic energy; as it climbs, the reverse happens.

Kinetic vs. potential energy at a glance

Feature	Kinetic energy	Gravitational potential energy
What it describes	Energy of motion	Energy related to height in a gravitational field
Basic formula	KE = ½mv²	PE = mgh
Key factors	Mass and speed squared	Mass, gravitational acceleration and height relative to a reference
SI unit	Joule (J)	Joule (J)
Example	A moving roller coaster car	A roller coaster car waiting high on a lift hill

This introductory comparison covers translational kinetic energy and gravitational potential energy. Potential energy also has other forms, including elastic energy stored in a spring and electrical potential energy. Not all potential energy is gravitational, and not all kinetic energy is simple straight-line motion.

What is kinetic energy? Formula and units

In classical physics, translational kinetic energy is:

KE = ½ × m × v²

• m = mass measured in kilograms (kg).
• v = speed measured in meters per second (m/s).
• KE = kinetic energy measured in joules (J).

At ordinary classroom speeds, the formula is appropriate. It is not the complete expression for objects moving at speeds close to the speed of light. See OpenStax University Physics: Kinetic Energy for the underlying derivation and limits.

Worked example 1: a moving 2-kg object

A 2-kg object moves at 3 m/s. Substitute the values:

KE = ½ × 2 × 3² = 1 × 9 = 9 J

If the object moves at 6 m/s with the same mass, KE becomes ½ × 2 × 6² = 36 J. Doubling speed increased the kinetic energy fourfold, because speed is squared. This is a common source of wrong answers on energy tests.

What is gravitational potential energy?

Near the surface of Earth, the change in gravitational potential energy can be calculated as:

ΔPE = m × g × Δh

For a chosen reference height of zero, the corresponding expression is PE = mgh. Here m is mass in kilograms, g is gravitational acceleration (approximately 9.8 m/s² near Earth’s surface) and h is the height difference in meters from the chosen reference.

The numerical value of gravitational PE depends on which height is defined as zero. For example, the floor can be a convenient reference in a classroom or the bottom of a track in a roller-coaster problem. The change in potential energy between two heights is what determines the associated energy transfer.

Worked example 2: a 5-kg object on a shelf

Suppose a 5-kg object is 4 m above the reference floor. Then:

PE = 5 × 9.8 × 4 = 196 J

This is its gravitational potential energy relative to the floor, not a statement that its total energy is 196 J under every possible reference or circumstance.

How energy changes on a roller coaster

The roller-coaster photograph accompanying this article shows a real ride on an elevated track. It illustrates height differences and movement but is not a measurement of the particular ride’s speed, mass or energy. The calculations below use a simplified hypothetical track to show the physics.

When a coaster car climbs a lift hill, an external motor generally does work to raise its gravitational potential energy. When it rolls down, gravity can convert some of that PE into kinetic energy as its speed increases. As it climbs a second hill, the car slows as kinetic energy is converted back to PE. Brakes and friction transfer mechanical energy into thermal energy and other forms.

Worked example 3: descending 5 meters without friction

Assume a coaster car starts at rest and drops 5 m vertically, with no relevant friction, air resistance or other energy transfer. The decrease in gravitational potential energy equals the increase in kinetic energy:

mgh = ½mv²

Mass cancels, giving v = √(2gh). Substitute g = 9.8 m/s² and h = 5 m:

v = √(2 × 9.8 × 5) = √98 ≈ 9.90 m/s

This is an idealized result. A real car may move more slowly because of rolling resistance, air drag and braking, and it may have had initial speed or receive additional power. For a detailed discussion of the conservation equations, read OpenStax Physics: Mechanical Energy and Conservation of Energy.

What is conservation of mechanical energy?

Mechanical energy is the sum of kinetic and potential energy in the model being studied:

Mechanical energy = KE + PE

If nonconservative forces do no net work on the system and the relevant conditions hold, then:

KE(initial) + PE(initial) = KE(final) + PE(final)

For example, if a simplified system has 100 J of total mechanical energy and 40 J is potential energy at one point, the kinetic energy there is 60 J. Mechanical energy may change in the presence of friction or external work. Total energy is nevertheless conserved when transferred thermal, sound or other forms are accounted for; energy has not vanished.

Potential energy vs. kinetic energy in daily life

• A parked car at rest on a hill: it has gravitational PE relative to lower positions but no translational KE while stationary.
• A moving bicycle: it has translational KE. It may also have gravitational PE depending on its reference height.
• A stretched rubber band: it stores elastic potential energy and can release energy as it snaps back.
• A ball thrown upward: its KE generally decreases as it rises; gravitational PE increases. On the way down, the reverse occurs, aside from air resistance.
• A roller coaster: motion and elevation continually change as the train follows the track, with external work and losses where relevant.

Why speed matters more than students expect

In the formula KE = ½mv², doubling an object’s mass at unchanged speed doubles kinetic energy. Doubling its speed at unchanged mass quadruples kinetic energy. Tripling speed multiplies KE by nine. This square relationship is central to the work needed to accelerate or stop an object, although a real braking event involves many other physical and safety factors.

The work–energy theorem connects the net work done on a particle with the change in its kinetic energy: W(net) = ΔKE. It is explained in OpenStax Physics: Work and the Work–Energy Theorem.

Why real roller coasters do not conserve mechanical energy perfectly

Real coasters experience wheel and bearing friction, air drag, braking and possibly powered sections. If you model just the car and Earth, some energy leaves the mechanical form through nonconservative work. A car that starts at rest at one height will generally not reach the same height again unaided after repeated frictional losses. A lift motor or launch system can add mechanical energy.

For ideal classroom problems, it is reasonable to neglect losses when the question explicitly asks for a frictionless model. For real engineering or safety analysis, it is not reasonable to ignore those effects without evidence.

How do kinetic energy and EV regenerative braking relate?

When a vehicle slows, its translational kinetic energy decreases. Conventional friction brakes convert much of the relevant mechanical energy to heat. In an electric vehicle, regenerative braking can convert a portion of the motion’s energy into electrical energy for the traction battery, subject to motor, battery, temperature and traction limits. Regeneration is not 100% efficient and does not remove the need for mechanical brakes.

For more about electric vehicles, read our guide to regenerative braking and brake pads. For the force and acceleration foundations, revisit our Newton’s three laws of motion lesson.

Practice quiz: six energy questions with worked answers

1. A 2-kg object moves at 4 m/s. Find its kinetic energy.
2. A 3-kg object is 5 m above a reference level. Find its gravitational PE using g = 9.8 m/s².
3. An object’s speed doubles without changing mass. By what factor does its kinetic energy change?
4. A system starts from rest and falls 5 m without resistance. What is its ideal speed at the bottom?
5. A frictionless system has 100 J of mechanical energy. At one point its PE is 40 J. What is KE?
6. A 4-kg cart moves at 10 m/s. What is its kinetic energy?

Answer key and calculations

1. 16 J: KE = ½ × 2 × 4² = 16.
2. 147 J: PE = 3 × 9.8 × 5 = 147.
3. Four times: (2v)² = 4v².
4. About 9.90 m/s: v = √(2 × 9.8 × 5) = √98.
5. 60 J: KE = 100 − 40 = 60.
6. 200 J: KE = ½ × 4 × 10² = 200.

Every calculation above uses SI units and ordinary nonrelativistic mechanics. A kilogram measures mass, not force; use meters, seconds and g in m/s² to obtain joules. Do not use mph directly in KE = ½mv² unless you convert it into m/s first.

Common exam mistakes

• Forgetting to square speed: the v² in ½mv² changes the scaling dramatically.
• Mixing kilograms and pounds: use consistent SI units.
• Mixing joules with kilowatt-hours: 1 kWh = 3.6 million J. Both are energy units but very different sizes.
• Assuming all real mechanical energy stays mechanical: friction and sound can transfer energy to other forms.
• Not defining zero height: potential energy depends on the reference height, while differences between positions remain meaningful.
• Assuming stationary means no energy: an elevated stationary object can have gravitational PE.

Primary physics references

Consult OpenStax: Work and the Work–Energy Theorem, OpenStax: Conservation of Mechanical Energy and OpenStax: Gravitational Potential Energy for further learning and more problems.

Student physics guide reviewed November 9, 2026. Featured image: real roller coaster photograph by PsychologyAssociation, April 2026, Wikimedia Commons CC0 1.0. The image is not a measurement of the ride’s energy or forces; examples use explicitly simplified numbers.