Conservation of Energy

Mechanical energy is the energy associated with the motion and position of objects. A roller coaster racing down a hill, a swinging pendulum, a falling ball, and a compressed spring all involve mechanical energy.

One of the most important ideas in physics is that energy can move between different stores, but energy cannot be created or destroyed. This is the law of conservation of energy.

Learning Targets

• I can define mechanical energy as the energy associated with the motion and position of objects.
• I can identify situations in which energy is stored as kinetic, gravitational potential, or elastic potential energy.
• I can explain how mechanical energy can be transferred between different energy stores within a system.
• I can describe real-world examples of mechanical energy transformations, such as roller coasters, pendulums, and springs.
• I can predict qualitative changes in mechanical energy as objects move, speed up, slow down, rise, or fall.

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1. What Is Mechanical Energy?

Mechanical energy is energy associated with an object's motion or position.

Three important mechanical energy stores are:

• Kinetic energy – associated with motion
• Gravitational potential energy – associated with position in a gravitational field
• Elastic potential energy – associated with stretching or compressing an elastic object

A moving car has a kinetic energy store.

A book sitting on a high shelf has a gravitational potential energy store.

A compressed spring has an elastic potential energy store.

Mechanical energy can be transferred between these stores as objects move.

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2. Kinetic Energy

Any moving object has energy in its kinetic store.

Examples include:

• A moving car
• A rolling football
• A running person
• A falling rock
• A spinning turbine
• A roller coaster moving along a track

The amount of kinetic energy depends on the object's mass and speed.

The faster an object moves, the greater its kinetic energy.

The greater its mass, the greater its kinetic energy at the same speed.

Kinetic energy = ½ × mass × speed²

or:

KE = ½mv²

Kinetic energy is measured in joules (J).

Example

Two identical bicycles are travelling at different speeds.

Bicycle A travels at 5 m/s.

Bicycle B travels at 10 m/s.

Bicycle B has more kinetic energy because it is moving faster.

Notice that speed is squared in the kinetic energy equation. Doubling the speed therefore increases the kinetic energy by a factor of four.

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3. Gravitational Potential Energy

An object can store energy because of its position in a gravitational field.

This is called gravitational potential energy (GPE).

Examples include:

• A book on a shelf
• A roller coaster at the top of a hill
• Water stored behind a dam
• A person standing at the top of stairs
• A rock held above the ground

The higher an object is raised, the greater its gravitational potential energy relative to the chosen reference level.

Gravitational potential energy also depends on mass and gravitational field strength.

Gravitational potential energy = mass × gravitational field strength × height

or:

GPE = mgh

The unit is the joule (J).

Example

If you lift a backpack from the floor onto a table, its gravitational potential energy increases.

If the backpack falls from the table, its gravitational potential energy decreases while its kinetic energy increases.

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4. Elastic Potential Energy

Energy can also be stored when an elastic object is stretched or compressed.

This is called elastic potential energy.

Examples include:

• A stretched elastic band
• A compressed spring
• A stretched bow
• A trampoline pushed downward
• A diving board bending

The further a spring is stretched or compressed within its elastic limit, the more energy is stored in its elastic potential store.

When released, this energy can be transferred to a kinetic store.

For example:

Elastic potential store of spring → mechanical transfer → kinetic store of object

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5. The Conservation of Energy

The law of conservation of energy states:

Energy cannot be created or destroyed. It can only be transferred between stores.

Imagine a falling ball.

At the beginning, the ball has a large gravitational potential energy store.

As the ball falls:

• Gravitational potential energy decreases.
• Kinetic energy increases.
• The ball speeds up.

Ignoring air resistance, the decrease in gravitational potential energy corresponds to the increase in kinetic energy.

We can describe this using ordinary copyable text:

Gravitational potential energy → kinetic energy

The energy has not disappeared. It has moved from one store to another.

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6. Conservation of Mechanical Energy

In an ideal system where there is no friction or air resistance, the total mechanical energy remains constant.

For a simple system involving kinetic and gravitational potential energy:

Total mechanical energy = Kinetic energy + Gravitational potential energy

Suppose an object has:

GPE = 800 J

KE = 200 J

Its total mechanical energy is:

800 J + 200 J = 1000 J

Later, it might have:

GPE = 300 J

KE = 700 J

The total is still:

300 J + 700 J = 1000 J

The amounts stored as kinetic and gravitational potential energy have changed, but the total mechanical energy has remained constant.

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7. Falling Objects

A falling object provides one of the simplest examples of mechanical energy transfer.

Imagine dropping a ball.

At the top

The ball is high above the ground.

GPE: high

KE: low

While falling

The ball loses height and speeds up.

GPE: decreasing

KE: increasing

Just before hitting the ground

The ball is moving fastest.

GPE: low

KE: high

Therefore:

GPE decreases → KE increases

If air resistance is negligible, the total mechanical energy remains approximately constant.

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8. Objects Moving Upward

Now imagine throwing a ball upward.

The process reverses.

As the ball rises:

• Its speed decreases.
• Its kinetic energy decreases.
• Its height increases.
• Its gravitational potential energy increases.

Therefore:

KE decreases → GPE increases

At the highest point, the ball is momentarily stationary.

Its kinetic energy is at a minimum, while its gravitational potential energy is at a maximum relative to the release point.

Then it begins falling again.

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9. Roller Coasters

Roller coasters provide an excellent example of repeated transfers between gravitational potential and kinetic energy.

At the top of the first hill, the coaster has a large gravitational potential energy store.

As it moves downhill:

GPE decreases

KE increases

Speed increases

Near the bottom:

GPE is relatively low

KE is relatively high

Speed is high

As the coaster climbs another hill:

KE decreases

GPE increases

Speed decreases

This repeated transfer between gravitational potential and kinetic energy occurs throughout the ride.

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10. Predicting Roller-Coaster Motion

The relationship between energy and motion allows us to make predictions.

Going downhill

Height decreases → GPE decreases → KE increases → speed increases.

Going uphill

Height increases → GPE increases → KE decreases → speed decreases.

At the bottom of a hill

GPE is relatively low and KE is relatively high.

At the top of a hill

GPE is relatively high and KE is relatively low.

This means we can often predict how the speed of a roller coaster changes simply by looking at the shape of the track.

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11. Why Is the First Roller-Coaster Hill Usually the Highest?

A roller coaster usually receives a major energy input as it is pulled up its first large hill.

At the top, the coaster has a large gravitational potential energy store.

As it travels around the track, energy is repeatedly transferred between gravitational potential and kinetic stores.

However, friction and air resistance transfer some mechanical energy into thermal stores.

Without another significant energy input, the coaster therefore cannot naturally climb a later hill higher than the initial hill.

This is why roller-coaster designers must carefully consider the heights of hills throughout the ride.

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12. Pendulums

A swinging pendulum provides another clear example of conservation of mechanical energy.

Consider a pendulum released from one side.

Position A — Highest Point

The pendulum is momentarily stationary.

GPE: maximum

KE: minimum

Position B — Moving Down

The pendulum loses height and speeds up.

GPE: decreasing

KE: increasing

Position C — Lowest Point

The pendulum is moving fastest.

GPE: minimum

KE: maximum

Position D — Moving Up

The pendulum gains height and slows down.

GPE: increasing

KE: decreasing

Position E — Highest Point

The pendulum again becomes momentarily stationary.

GPE: maximum

KE: minimum

In an ideal system:

GPE → KE → GPE → KE → GPE

The total mechanical energy remains constant.

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13. Why Does a Real Pendulum Eventually Stop?

A real pendulum does not swing forever.

This does not mean energy has been destroyed.

Air resistance and friction at the pivot transfer some mechanical energy into thermal stores in the surroundings.

Some energy may also be transferred by sound waves.

As a result, the pendulum's mechanical energy gradually decreases.

The total energy of the larger system is still conserved.

This is an important distinction:

Total energy is always conserved.

But:

Mechanical energy may not remain constant if energy is transferred to non-mechanical stores.

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14. Springs

Springs demonstrate transfers between elastic potential energy and kinetic energy.

Imagine compressing a spring and placing a small cart against it.

Before release:

Elastic potential energy: high

Kinetic energy: zero

Release the spring.

As the spring expands:

Elastic potential energy decreases

Kinetic energy of cart increases

The cart speeds up.

The transfer can be represented as:

Elastic potential energy → kinetic energy

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15. Elastic Bands and Bows

The same principle applies to an elastic band or bow.

When you stretch a bow:

Chemical store in muscles → mechanical transfer → elastic potential store of bow

When the bowstring is released:

Elastic potential store → mechanical transfer → kinetic store of arrow

The arrow accelerates forward.

A stretched elastic band works in the same way.

The more it is stretched within its safe elastic range, the more elastic potential energy can be stored.

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16. Bouncing Ball

A bouncing ball involves several mechanical energy stores.

As the ball falls:

GPE → KE

When the ball strikes the ground, it deforms.

Some kinetic energy is temporarily transferred into an:

Elastic potential store

As the ball returns to its original shape:

Elastic potential energy → kinetic energy

The ball rises again:

KE → GPE

So the sequence is approximately:

GPE → KE → Elastic PE → KE → GPE

In reality, some energy is also dissipated through heating and sound.

Therefore, each bounce is usually lower than the previous one.

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17. Skateboarder on a Ramp

A skateboarder moving along a U-shaped ramp behaves similarly to a pendulum.

At the top:

GPE high → KE low

Moving downward:

GPE decreases → KE increases

At the bottom:

GPE low → KE high

Moving upward:

KE decreases → GPE increases

This means the skateboarder is generally moving fastest near the bottom of the ramp.

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18. Mechanical Energy in Real Systems

In ideal physics problems, we sometimes ignore friction and air resistance.

In reality, these forces are almost always present.

Consider a bicycle coasting along a flat road.

Initially, the bicycle has kinetic energy.

Friction and air resistance cause it to slow down.

Its kinetic energy decreases.

Where did the energy go?

Much of it is transferred into thermal stores of:

• The tyres
• The road
• The bicycle
• The surrounding air

Some energy is also transferred through sound waves.

The energy has not disappeared.

It has been transferred into other stores and pathways.

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19. Predicting Changes in Mechanical Energy

You can often predict energy changes without performing calculations.

For elastic systems:

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20. Worked Example — Ball Rolling Down a Hill

A ball begins at rest at the top of a hill.

At the top

Speed: zero

KE: zero

GPE: maximum

Halfway down

The ball has lost height and gained speed.

KE: increasing

GPE: decreasing

At the bottom

The ball is moving fastest.

KE: maximum

GPE: minimum

Ignoring friction:

Total mechanical energy remains constant.

The energy has simply shifted between stores.

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21. Worked Example — Throwing a Ball Upward

A ball leaves a person's hand travelling upward.

Immediately after release, it has substantial kinetic energy.

As it rises:

Speed decreases

KE decreases

GPE increases

At maximum height:

Speed = 0 m/s momentarily

KE = minimum

GPE = maximum

As the ball falls:

GPE decreases

KE increases

The process reverses.

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22. Conservation of Energy and Friction

Suppose a roller coaster begins with 10,000 J of mechanical energy.

Later, it has:

6,000 J gravitational potential energy

3,000 J kinetic energy

That accounts for:

6,000 J + 3,000 J = 9,000 J mechanical energy

The remaining:

10,000 J − 9,000 J = 1,000 J

may have been transferred to thermal stores and sound because of friction and air resistance.

Total energy is still:

10,000 J

Therefore:

Mechanical energy decreased, but total energy was conserved.

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23. Mechanical Energy vs Total Energy

These two ideas should not be confused.

Conservation of total energy

Energy cannot be created or destroyed.

This is always true.

Conservation of mechanical energy

Mechanical energy remains constant only when energy is not being transferred significantly into or out of the system through processes such as friction or air resistance.

In an ideal roller coaster:

GPE + KE = constant

In a real roller coaster:

Some mechanical energy is gradually transferred into thermal stores and through sound.

The total energy still remains conserved.

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24. Common Mistakes

Mistake 1 — Saying energy disappears

Incorrect:

"The roller coaster loses energy."

Better:

"Energy is transferred from mechanical stores into thermal stores because of friction."

Mistake 2 — Thinking a stationary object has no energy

A stationary object has no kinetic energy, but it may have gravitational or elastic potential energy.

A roller coaster stationary at the top of a hill can have a large gravitational potential energy store.

Mistake 3 — Thinking faster means more GPE

Speed affects kinetic energy.

Height affects gravitational potential energy.

Mistake 4 — Assuming mechanical energy is always conserved

Mechanical energy is conserved only under suitable conditions.

Friction and air resistance can transfer energy from mechanical stores into thermal stores.

Mistake 5 — Saying potential energy "turns into motion"

A more precise statement is:

Energy is transferred from a potential energy store to a kinetic energy store.

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25. Key Terms

Mechanical energy – energy associated with the motion and position of objects.

Kinetic energy – energy stored by a moving object.

Gravitational potential energy – energy associated with an object's position in a gravitational field.

Elastic potential energy – energy stored when an elastic object is stretched or compressed.

Conservation of energy – the principle that energy cannot be created or destroyed.

Mechanical energy conservation – mechanical energy remains constant when there are no significant transfers to other stores.

Energy transfer – movement of energy from one store to another.

Dissipation – energy spreading into less useful stores, often thermal stores of the surroundings.

Friction – a force that opposes motion and can transfer mechanical energy into thermal stores.

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26. Check Your Understanding

1. What is mechanical energy?

2. Name three mechanical energy stores discussed in these notes.

3. What happens to the kinetic energy of an object when its speed increases?

4. What happens to the gravitational potential energy of an object as it rises?

5. Describe the energy changes as a ball falls.

6. At what point in a pendulum's swing is its kinetic energy greatest?

7. At what point is its gravitational potential energy greatest?

8. Describe the energy transfer when a compressed spring launches a toy car.

9. Why does a bouncing ball eventually stop bouncing?

10. A roller coaster is travelling downhill. Predict what happens to its GPE, KE and speed.

11. Explain why a real roller coaster cannot rely on mechanical energy alone to climb a hill higher than its original starting hill.

12. A student says, "The pendulum eventually stops, so its energy has been destroyed." Explain why this statement is incorrect.

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Key Takeaways

• Mechanical energy is associated with the motion and position of objects.
• Important mechanical energy stores include kinetic, gravitational potential and elastic potential energy.
• Moving faster increases kinetic energy.
• Moving higher increases gravitational potential energy.
• Stretching or compressing an elastic object increases its elastic potential energy.
• Falling objects transfer energy from gravitational potential stores to kinetic stores.
• Rising objects transfer energy from kinetic stores to gravitational potential stores.
• Springs can transfer energy between elastic potential and kinetic stores.
• Roller coasters and pendulums repeatedly transfer energy between gravitational potential and kinetic stores.
• In an ideal system without friction or air resistance, mechanical energy remains constant.
• In real systems, some mechanical energy is transferred into thermal stores and through sound.
• Total energy is always conserved, even when mechanical energy decreases.

Did You Know?

The same principles can be explored with PhET's Energy Skate Park: Basics. The simulation lets learners watch the kinetic and gravitational potential energy change as a skater moves along different tracks, making conservation of mechanical energy visible in real time. OpenStax specifically uses this simulation when teaching mechanical energy and conservation.