Last Updated: September 5, 2026
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred between systems.
Quick Summary
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transferred between systems or transformed from one form into another. In a closed or isolated system, the total amount of energy remains constant.
For many introductory physics problems, conservation of energy is applied through mechanical energy:
[
E = KE + PE
]
When only conservative forces do work:
[
KE_i + PE_i = KE_f + PE_f
]
The most important formulas include:
- Kinetic energy: (KE = \frac{1}{2}mv^2)
- Gravitational potential energy: (GPE = mgh)
- Elastic potential energy: (EPE = \frac{1}{2}kx^2)
- Mechanical energy: (E = KE + PE)
- Conservation of mechanical energy: (KE_i + PE_i = KE_f + PE_f)
- Work-energy theorem: (W_{\text{net}} = \Delta KE)
Key Takeaways
- The law of conservation of energy says that total energy is conserved.
- Energy can change form or transfer between systems, but it is not simply destroyed.
- Kinetic energy depends on an object’s mass and speed.
- Gravitational potential energy depends on mass, gravitational acceleration, and height.
- Elastic potential energy is stored in stretched or compressed elastic objects.
- Mechanical energy is the sum of kinetic and potential energy.
- Mechanical energy remains constant when non-conservative forces do no work.
- Friction can reduce mechanical energy while total energy remains conserved because energy is transferred into thermal energy and other forms.
- The work-energy theorem states that net work equals the change in kinetic energy.
- Conservation of energy is useful for free-fall, roller-coaster, spring, braking, collision, and engineering problems.
Who Is This Conservation of Energy Guide For?
This conservation of energy guide is designed for GCSE, IGCSE, A-Level, IB, and AP Physics students, as well as beginners studying introductory college physics and mechanical engineering. It is useful when you need to understand the law of conservation of energy, choose the correct energy formula, solve physics problems, or explain energy transformations in an exam.
Use this guide for problems involving kinetic energy, gravitational potential energy, elastic potential energy, mechanical energy, falling objects, roller coasters, springs, braking systems, and other mechanical energy transformations.
For exam questions, focus on identifying the initial and final energy states, selecting the correct conservation equation, and checking whether friction or another non-conservative force transfers energy out of the mechanical system.
Who Is This Conservation of Energy Guide For?
This conservation of energy guide is designed for GCSE, IGCSE, A-Level, IB Physics, AP Physics, and introductory college physics students learning mechanics and energy.
It is also useful for engineering and STEM learners studying mechanical systems, vehicle dynamics, energy transfers, machines, and real-world applications of physics.
Use this guide when you need to:
- State the law of conservation of energy in a physics exam.
- Explain what the conservation of energy principle means.
- Identify the correct conservation of energy formula.
- Calculate kinetic or potential energy.
- Solve mechanical energy conservation problems.
- Determine the speed of a falling or moving object.
- Analyze roller-coaster and braking problems.
- Understand when kinetic energy is conserved.
- Apply the work-energy theorem.
- Distinguish total energy conservation from mechanical energy conservation.
For students reviewing the broader mechanics framework, the Newton’s Laws of Motion Guide provides the force-based approach that complements energy methods.
What Is the Law of Conservation of Energy?
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transferred from one system to another or transformed from one form into another.
This is one of the fundamental conservation laws in physics. Energy may appear as kinetic energy, gravitational potential energy, elastic energy, thermal energy, chemical energy, electrical energy, nuclear energy, light, or other forms.
The important idea is that the total energy of a properly defined closed or isolated system remains constant.
A simple example is a ball falling from a height. At the beginning, the ball has gravitational potential energy. As it falls, gravitational potential energy decreases while kinetic energy increases. If air resistance is ignored, the total mechanical energy remains constant.
OpenStax describes conservation of energy as a principle in which total energy remains constant while energy can change form or transfer between systems.
State the Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred from one system to another.
Law of Conservation of Energy in One Sentence
A concise sentence suitable for a physics exam is:
Energy cannot be created or destroyed; it can only be transferred or transformed, so the total energy of a closed system remains constant.
This directly answers questions such as:
- What is the law of conservation of energy?
- State the law of conservation of energy.
- State the principle of conservation of energy.
- What does the law of conservation of energy state?
- Describe the law of conservation of energy.
- What is the law of conservation of energy in physics?
What Does the Law of Conservation of Energy Mean?
It means that the total amount of energy is preserved even when energy changes from one form to another.
For example, when an object falls:
[
GPE \rightarrow KE
]
The object loses gravitational potential energy while gaining kinetic energy.
For a stretched spring launching an object:
[
EPE \rightarrow KE
]
For a braking vehicle:
[
KE \rightarrow \text{thermal energy + sound + deformation}
]
In each case, saying that energy is “lost” usually means that energy has been transferred into a form that is less useful for the process being studied. The energy itself has not disappeared.
Conservation of Energy Formula
What Is the Conservation of Energy Formula?
The basic conservation of mechanical energy formula is (KE_i + PE_i = KE_f + PE_f) when only conservative forces do work.
The general mechanical-energy relationship is:
[
E = KE + PE
]
For an initial and final state:
[
\boxed{KE_i + PE_i = KE_f + PE_f}
]
This form is appropriate when mechanical energy is conserved.
For example, if gravitational potential energy changes into kinetic energy:
mgh_f + \frac{1}{2}mv_f^2
]
This is one of the most useful conservation of energy equations in introductory mechanics. OpenStax gives the same mechanical-energy relationship for systems in which non-conservative work is absent.
Law of Energy Conservation Formula
The phrase law of energy conservation formula can refer to the mathematical expression used for a particular system.
For mechanical systems:
[
\boxed{KE_i + PE_i = KE_f + PE_f}
]
If non-conservative forces such as friction do work, mechanical energy is not necessarily constant. A more general relationship is:
[
E_f = E_i + W_{\text{nc}}
]
where (W_{\text{nc}}) represents work done by non-conservative forces under the chosen system convention.
Therefore, do not automatically use:
[
KE_i + PE_i = KE_f + PE_f
]
when friction, drag, or another non-conservative force is doing significant work.
Mechanical Energy
Mechanical energy is the sum of an object’s kinetic energy and potential energy.
[
\boxed{E_{\text{mech}} = KE + PE}
]
Mechanical energy can include different forms of potential energy depending on the system:
[
E_{\text{mech}} = KE + GPE + EPE
]
For a system involving gravitational and elastic potential energy:
\frac{1}{2}mv^2 + mgh + \frac{1}{2}kx^2
]
Mechanical energy is conserved when the relevant non-conservative forces do no work.
Law of Conservation of Mechanical Energy
The law of conservation of mechanical energy states that the total kinetic plus potential energy remains constant when only conservative forces do work.
Therefore:
[
\boxed{KE_i + PE_i = KE_f + PE_f}
]
Gravity and ideal spring forces are conservative forces. Friction and air resistance are non-conservative forces.
This distinction is important because total energy conservation is more general than mechanical energy conservation. Mechanical energy can decrease while the total energy of the complete system remains constant because some mechanical energy is transferred into thermal, sound, or other forms. OpenStax makes this distinction explicitly when discussing non-conservative work.
Kinetic Energy
Kinetic energy is the energy associated with an object’s motion.
The formula is:
[
\boxed{KE = \frac{1}{2}mv^2}
]
where:
- (m) = mass in kilograms
- (v) = speed in meters per second
- (KE) = kinetic energy in joules
Because velocity is squared, speed has a particularly strong effect on kinetic energy.
If speed doubles:
[
KE’ = \frac{1}{2}m(2v)^2
]
[
KE’ = 4KE
]
Therefore, doubling speed produces four times the kinetic energy for the same mass.
When Is Kinetic Energy Conserved?
Kinetic energy is conserved only in situations where the total kinetic energy before and after the process remains the same, such as an ideal perfectly elastic collision.
Kinetic energy is not conserved in every interaction.
For example, in an inelastic collision, some kinetic energy can be transformed into:
- Thermal energy
- Sound
- Deformation
- Internal energy
However, total energy remains conserved when the complete system is considered.
This distinction is especially important when solving collision and mechanics questions.
For related energy comparisons, see the Kinetic Energy vs Potential Energy Guide.
Potential Energy
Potential energy is stored energy associated with an object’s position, configuration, or interaction with other objects.
Two important forms in introductory mechanics are gravitational potential energy and elastic potential energy.
Gravitational Potential Energy
Gravitational potential energy is energy associated with an object’s position in a gravitational field.
Near Earth’s surface:
[
\boxed{GPE = mgh}
]
where:
- (m) = mass in kg
- (g) = gravitational acceleration, approximately (9.8\text{ m/s}^2)
- (h) = height relative to a chosen reference level
- GPE = gravitational potential energy in joules
The zero-height reference can be chosen conveniently. What matters for most calculations is the change in gravitational potential energy:
[
\Delta GPE = mg\Delta h
]
Does Gravitational Potential Energy Depend on the Reference Level?
Yes, its numerical value depends on the chosen zero reference level, but physical predictions depend on differences in potential energy.
For example, you can choose the floor, ground, or bottom of a ramp as (h=0), provided you use the same reference consistently.
Elastic Potential Energy
Elastic potential energy is energy stored when an elastic object is stretched or compressed.
For an ideal spring:
[
\boxed{EPE = \frac{1}{2}kx^2}
]
where:
- (k) = spring constant in N/m
- (x) = extension or compression in m
- EPE = elastic potential energy in J
The formula assumes ideal spring behavior.
When a compressed spring launches an object, elastic potential energy can be transformed into kinetic energy:
[
EPE \rightarrow KE
]
The Work-Energy Theorem
What Is the Work-Energy Theorem?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy.
The equation is:
[
\boxed{W_{\text{net}} = \Delta KE}
]
Therefore:
\frac{1}{2}mv_f^2
\frac{1}{2}mv_i^2
]
where (v_i) is initial speed and (v_f) is final speed.
Positive net work increases kinetic energy, while negative net work decreases kinetic energy.
OpenStax gives the same relationship for the work-energy theorem: net work on an object equals its change in kinetic energy.
How Is Work Related to Conservation of Energy?
Work is one way energy is transferred into or out of a system.
For conservative forces:
[
W_c = -\Delta PE
]
and the work-energy theorem gives:
[
W_c = \Delta KE
]
Therefore:
[
\Delta KE + \Delta PE = 0
]
which leads to:
[
KE + PE = \text{constant}
]
This is the connection between the work-energy theorem and conservation of mechanical energy.
Conservation of Energy vs Conservation of Mechanical Energy
These two ideas are related but are not identical.
| Concept | Meaning |
|---|---|
| Conservation of energy | Total energy is conserved |
| Conservation of mechanical energy | (KE + PE) remains constant under the required conditions |
| Kinetic energy conservation | Total KE remains unchanged in a particular process |
| Work-energy theorem | Net work equals change in KE |
For example, when a car brakes, its kinetic energy decreases. Mechanical energy decreases if the thermal energy produced by braking is outside the mechanical-energy accounting.
But the total energy is still conserved because the lost mechanical energy has been transferred mainly into thermal energy, along with sound and other forms.
Worked Examples
Example 1: Free-Falling Object
A (4) kg ball is dropped from rest from a height of (20) m. Ignore air resistance and use:
[
g = 10\text{ m/s}^2
]
Find its speed just before reaching the ground.
Step 1: Identify the Initial Energy
The ball starts from rest, so:
[
KE_i = 0
]
Its initial gravitational potential energy is:
[
GPE_i = mgh
]
[
GPE_i = (4)(10)(20)
]
[
GPE_i = 800\text{ J}
]
Step 2: Apply Conservation of Energy
At the ground, take:
[
GPE_f = 0
]
Therefore:
[
800 = KE_f
]
[
800 = \frac{1}{2}(4)v^2
]
Step 3: Solve for Speed
[
800 = 2v^2
]
[
v^2 = 400
]
[
\boxed{v = 20\text{ m/s}}
]
The gravitational potential energy has been transformed into kinetic energy.
Example 2: Roller-Coaster Energy
A (500) kg roller-coaster car starts from rest at a height of (25) m. Find its speed at a point (10) m above the ground. Ignore friction and use:
[
g = 10\text{ m/s}^2
]
Step 1: Initial Mechanical Energy
Because the car starts from rest:
[
KE_i = 0
]
So:
[
E_i = mgh_i
]
[
E_i = (500)(10)(25)
]
[
E_i = 125,000\text{ J}
]
Step 2: Final Potential Energy
[
GPE_f = mgh_f
]
[
GPE_f = (500)(10)(10)
]
[
GPE_f = 50,000\text{ J}
]
Step 3: Find Final Kinetic Energy
[
KE_f = 125,000 – 50,000
]
[
KE_f = 75,000\text{ J}
]
Step 4: Calculate Speed
[
75,000 = \frac{1}{2}(500)v^2
]
[
v^2 = 300
]
[
\boxed{v \approx 17.3\text{ m/s}}
]
Example 3: Braking Distance Using Work-Energy
A (1000) kg vehicle travels at (25) m/s. The brakes apply a constant resistive force of (5000) N. Find the stopping distance.
Step 1: Calculate Initial Kinetic Energy
[
KE_i = \frac{1}{2}mv^2
]
[
KE_i = \frac{1}{2}(1000)(25)^2
]
[
KE_i = 312,500\text{ J}
]
Step 2: Apply the Work-Energy Theorem
At rest:
[
KE_f = 0
]
The work done by the braking force is:
[
W = -Fd
]
Therefore:
[
-Fd = KE_f – KE_i
]
[
-5000d = -312,500
]
Step 3: Solve
[
d = \frac{312,500}{5000}
]
[
\boxed{d = 62.5\text{ m}}
]
The negative work done by the brakes removes the vehicle’s kinetic energy.
Examples of the Law of Conservation of Energy
Examples of conservation of energy include a falling object changing gravitational potential energy into kinetic energy, a roller coaster converting potential energy into kinetic energy, a spring launching an object, and a vehicle converting kinetic energy into thermal energy during braking.
Example 1: Falling Object
[
GPE \rightarrow KE
]
As the object falls, height decreases and speed increases.
Example 2: Roller Coaster
[
GPE \leftrightarrow KE
]
At high points, the car has more gravitational potential energy. At low points, it has more kinetic energy, assuming negligible friction.
Example 3: Spring
[
EPE \rightarrow KE
]
A compressed spring stores elastic potential energy that can become kinetic energy when released.
Example 4: Braking Vehicle
[
KE \rightarrow \text{thermal energy + sound + deformation}
]
The vehicle’s kinetic energy decreases, but the total energy is conserved.
Example 5: Hydroelectric Power
Water stored at a height has gravitational potential energy. As it falls, that energy becomes kinetic energy and is then converted into electrical energy through turbines and generators.
Real-World Applications of Energy Conservation
The conservation of energy principle is used across physics, mechanical engineering, automotive engineering, aerospace, electrical power systems, and renewable-energy technology.
| System | Initial Energy | Main Transformation |
|---|---|---|
| Roller coaster | Gravitational PE | Kinetic energy |
| Car braking | Kinetic energy | Thermal energy and sound |
| Spring launcher | Elastic PE | Kinetic energy |
| Hydroelectric plant | Gravitational PE | Electrical energy |
| Solar cell | Radiant energy | Electrical energy |
| Battery-powered motor | Chemical energy | Electrical → mechanical energy |
| Nuclear reactor | Nuclear energy | Thermal energy → electrical energy |
These applications show why conservation of energy is more than an exam formula. It is a general framework for tracking energy transfers and transformations in physical and engineering systems.
Conservation of Energy and Matter
Is Conservation of Energy the Same as Conservation of Matter?
No. Conservation of energy and conservation of matter are different principles, although modern physics connects mass and energy through relativity.
In classical physics, conservation of energy tracks energy transfers and transformations, while conservation of mass tracks matter in appropriate systems.
For most GCSE, IGCSE, A-Level, IB, and introductory mechanics problems, treat these as distinct conservation concepts unless the question specifically involves relativistic mass-energy relationships.
Common Mistakes
Mistake 1: Saying Energy Is Destroyed
Energy is not destroyed. It can be transferred or transformed into another form.
Mistake 2: Assuming Mechanical Energy Is Always Conserved
Mechanical energy is conserved only when the relevant non-conservative forces do no work.
Mistake 3: Assuming Kinetic Energy Is Always Conserved
Kinetic energy is not conserved in every collision or process.
Mistake 4: Forgetting the Initial Kinetic Energy
If an object already has a starting speed, include:
[
KE_i = \frac{1}{2}mv_i^2
]
Do not automatically assume (v_i=0).
Mistake 5: Using the Wrong Height
For gravitational potential energy, use the height relative to your selected reference level and remain consistent.
Mistake 6: Confusing Total Energy With Mechanical Energy
Total energy can remain constant even when mechanical energy decreases because mechanical energy may be transformed into thermal, sound, deformation, or other forms.
Frequently Asked Questions
What is the law of conservation of energy?
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or transformed, so the total energy of a closed system remains constant.
What does the law of conservation of energy state?
It states that the total energy of a closed or isolated system remains constant, although energy can change form or transfer between systems.
What is the principle of conservation of energy?
The principle of conservation of energy says that energy is a conserved quantity: it can be transformed or transferred but is not simply created or destroyed.
What is the formula for conservation of mechanical energy?
The standard formula is (KE_i + PE_i = KE_f + PE_f), provided the relevant non-conservative forces do no work.
What is the law of energy conservation formula?
For mechanical systems, a common conservation-of-energy formula is (KE_i + PE_i = KE_f + PE_f). For gravitational motion, this can be written as (\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f).
When is kinetic energy conserved?
Kinetic energy is conserved when the total kinetic energy before and after a process is unchanged, such as in an ideal perfectly elastic collision. Total energy, however, is conserved much more generally.
What does delta K mean in physics?
Delta K, written (\Delta K), means the change in kinetic energy.
[
\Delta K = K_f – K_i
]
Since:
[
K = \frac{1}{2}mv^2
]
we have:
\frac{1}{2}mv_f^2
\frac{1}{2}mv_i^2
]
The work-energy theorem states:
[
W_{\text{net}} = \Delta K
]
What is the law of conservation of energy in physics?
In physics, the law of conservation of energy is the principle that total energy remains constant in an appropriately defined closed or isolated system, even when energy changes form or moves between parts of the system.
What are examples of the law of conservation of energy?
Common examples include falling objects, roller coasters, springs, braking vehicles, hydroelectric systems, batteries, and many other systems in which energy changes form or transfers between systems.
Related Physics Resources
For further study, use these related guides:
- Kinetic Energy vs Potential Energy Guide — compare kinetic and potential energy formulas and transformations.
- Newton’s Laws of Motion Guide — understand the force-based approach to mechanics.
- SUVAT Equations Guide — solve constant-acceleration motion problems.
- Projectile Motion Guide — study two-dimensional motion.
- Conservation of Momentum Guide — compare energy conservation with momentum conservation.
For an authoritative external reference, see OpenStax: Conservation of Energy, which explains mechanical energy, conservative and non-conservative forces, and energy conservation.
For the work-energy theorem, see OpenStax: Work-Energy Theorem.
Conclusion
The law of conservation of energy is one of the fundamental principles of physics. It states that energy cannot be created or destroyed; instead, it can be transferred or transformed from one form into another.
For mechanical problems, remember these key equations:
[
\boxed{KE = \frac{1}{2}mv^2}
]
[
\boxed{GPE = mgh}
]
[
\boxed{EPE = \frac{1}{2}kx^2}
]
[
\boxed{E_{\text{mech}} = KE + PE}
]
[
\boxed{KE_i + PE_i = KE_f + PE_f}
]
and:
[
\boxed{W_{\text{net}} = \Delta KE}
]
The most important step is to identify what type of energy is present, what system is being analyzed, whether non-conservative forces do work, and what energy transfer occurs between the initial and final states.
Once those decisions are clear, conservation of energy becomes one of the simplest and most powerful methods for solving physics problems.