Kinetic vs Potential Energy: Formulas and Examples

Energy is one of the most important concepts in all of physics. It comes in many forms, but two of the most fundamental are kinetic energy (the energy of motion) and potential energy (stored energy due to position or configuration). Understanding these two types of energy and how they convert between each other is essential for studying mechanics, engineering, and the natural world.

This guide covers the formulas for kinetic and potential energy, the conservation of energy principle, elastic potential energy, the work-energy theorem, and detailed worked examples involving falling objects, roller coasters, and pendulums.

What Is Kinetic Energy?

Kinetic energy is the energy an object possesses due to its motion. Any object that is moving has kinetic energy. The faster it moves or the more massive it is, the more kinetic energy it has. The formula is:

Where:

• is the kinetic energy in joules (J)
• is the mass of the object in kilograms (kg)
• is the velocity of the object in metres per second (m/s)

Notice that kinetic energy depends on the square of velocity. This means that doubling the speed quadruples the kinetic energy. This is why high-speed collisions are so much more destructive than low-speed ones, and why stopping distance increases dramatically at higher speeds.

Units of Kinetic Energy

The SI unit of energy is the joule (J). One joule equals one kilogram metre squared per second squared:

Other common energy units include kilojoules (kJ), calories (cal), kilocalories (kcal), and electronvolts (eV) in atomic physics.

What Is Potential Energy?

Potential energy is stored energy based on an object's position or configuration. Unlike kinetic energy, an object does not need to be moving to have potential energy. The most common type is gravitational potential energy.

Gravitational Potential Energy

An object raised above a reference point (usually the ground) has gravitational potential energy because gravity can do work on it as it falls. The formula is:

Where:

• is the gravitational potential energy in joules (J)
• is the mass in kilograms (kg)
• is the acceleration due to gravity ( on Earth)
• is the height above the reference point in metres (m)

Gravitational potential energy is relative. It depends on the chosen reference level. If you place the reference at ground level, an object on a table 1 m high has . If you place the reference at the table level, the same object has zero PE.

Elastic Potential Energy

When a spring is compressed or stretched, it stores elastic potential energy. This energy is released when the spring returns to its natural length. The formula, derived from Hooke's law, is:

Where:

• is the spring constant in newtons per metre (N/m), measuring the stiffness of the spring
• is the displacement from the natural (equilibrium) length in metres (m)

Like kinetic energy, elastic potential energy depends on the square of the displacement. Compressing a spring twice as far stores four times the energy.

Conservation of Energy

One of the most powerful principles in physics is the conservation of energy: energy cannot be created or destroyed, only transformed from one form to another. In a closed system with no external forces (like friction or air resistance), the total mechanical energy remains constant:

Or equivalently:

This principle allows us to solve problems without knowing the forces at every point along the path. We only need to compare the energy at two different positions.

The Work-Energy Theorem

The work-energy theorem connects force, displacement, and kinetic energy. It states that the net work done on an object equals the change in its kinetic energy:

Where work is defined as:

Here is the force, is the displacement, and is the angle between the force and direction of motion. When force is applied in the direction of motion (), and work equals .

Worked Example 1: Falling Object

Problem: A 2 kg ball is dropped from a height of 10 m. Ignoring air resistance, find (a) its potential energy at the top, (b) its speed just before hitting the ground, and (c) its kinetic energy at impact.

(a) Potential energy at the top:

At the top, the ball is stationary, so . The total mechanical energy is 196.2 J.

(b) Speed at the bottom:

At ground level, , so all potential energy has converted to kinetic energy. Using conservation of energy:

The mass cancels from both sides:

(c) Kinetic energy at impact:

As expected, the kinetic energy at impact (196 J) equals the initial potential energy (196.2 J), confirming energy conservation. The slight difference is due to rounding.

Worked Example 2: Roller Coaster

Problem: A roller coaster car (mass 500 kg) starts from rest at the top of a 30 m hill. What is its speed at the bottom of the hill? What is its speed at the top of a second hill that is 20 m high? Ignore friction.

Speed at the bottom of the first hill:

At the top: (starts from rest), .

At the bottom ():

That is approximately 87 km/h.

Speed at the top of the second hill (20 m):

At the second hill, some kinetic energy has converted back to potential energy:

Key insight: The mass cancelled out completely. The speed at any point depends only on the height difference, not on the mass of the car. A heavier car reaches the same speed as a lighter one (ignoring friction).

Worked Example 3: Simple Pendulum

Problem: A pendulum consists of a 0.5 kg bob on a 1.2 m string. The bob is released from a height of 0.3 m above its lowest point. Find the speed of the bob at the lowest point.

At the release point, the bob is stationary ( ). At the lowest point, all potential energy has converted to kinetic energy:

Again, the mass cancels. The speed at the lowest point depends only on the height from which the bob is released. A pendulum continuously converts between kinetic and potential energy: maximum PE and zero KE at the endpoints, maximum KE and zero PE at the lowest point.

Worked Example 4: Spring and Elastic Potential Energy

Problem: A spring with spring constant is compressed by 0.15 m. A 0.1 kg ball is placed against it and released. What speed does the ball reach?

The elastic potential energy stored in the spring converts to kinetic energy of the ball:

The ball leaves the spring at approximately 6.71 m/s. This principle is used in spring-loaded mechanisms, from dart guns to car crash absorbers.

Energy Transformations in Everyday Life

Energy constantly transforms between kinetic and potential forms, as well as other types:

• Hydroelectric dams: Water stored at height (gravitational PE) flows downhill, turning turbines (KE), which generate electricity.
• Bungee jumping: Gravitational PE converts to KE during the fall, then to elastic PE as the cord stretches, and the cycle repeats.
• Bouncing ball: Gravitational PE at the top converts to KE during the fall, then to elastic PE as the ball deforms on impact, then back to KE and PE as it bounces upward. Each bounce is lower because some energy converts to heat and sound.
• Archery: The archer does work on the bowstring, storing elastic PE. When released, this converts to KE of the arrow.
• Regenerative braking: In electric vehicles, the motor runs in reverse during braking, converting KE back into electrical energy stored in the battery.

When Is Energy Not Conserved?

In real-world situations, friction, air resistance, and other non-conservative forces are present. These forces convert mechanical energy into thermal energy (heat), sound, or other forms. The total energy of the universe is still conserved, but the mechanical energy of the system is not.

In the presence of friction, the energy equation becomes:

Where is the energy lost to friction (always positive, as friction removes energy from the system).

Summary of Key Formulas

Frequently Asked Questions

Can an object have both kinetic and potential energy at the same time?

Yes. A ball thrown upward at the midpoint of its flight has both kinetic energy (it is still moving) and gravitational potential energy (it is above the ground). The total mechanical energy is the sum of both. Only at the very top of the arc does the ball have zero kinetic energy (for an instant), and only at ground level does it have zero gravitational PE.

Why does mass cancel in free-fall energy problems?

When you set , the mass appears on both sides and cancels. This means all objects fall at the same rate (ignoring air resistance), a principle famously demonstrated by Galileo. The speed at any height depends only on the height fallen, not on the mass.

What is the difference between kinetic energy and momentum?

Both involve mass and velocity, but they are different quantities. Momentum is (linear in velocity), while kinetic energy is (quadratic in velocity). Momentum is a vector (it has direction), while kinetic energy is a scalar (it is always positive). They are conserved under different conditions: momentum is conserved in all collisions, while kinetic energy is only conserved in perfectly elastic collisions.

Is potential energy always gravitational?

No. Gravitational PE is the most familiar type, but potential energy also exists in other forms. Elastic PE is stored in stretched or compressed springs. Chemical PE is stored in molecular bonds (released during combustion or digestion). Electrical PE exists in charged particles within electric fields. Nuclear PE is stored within atomic nuclei and released during fission or fusion.

How does friction affect energy conservation?

Friction converts mechanical energy (KE + PE) into thermal energy (heat). The total energy is still conserved (first law of thermodynamics), but the mechanical energy of the system decreases. This is why a ball bouncing on the floor reaches a lower height with each bounce: some energy is lost to heat and sound with every impact.