Conservation of Mechanical Energy

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

The main mechanical energy stores considered here are:

• kinetic energy — energy associated with motion
• gravitational potential energy — energy associated with position in a gravitational field
• elastic potential energy — energy stored when an elastic object is stretched or compressed

In an ideal system where no mechanical energy is transferred to the surroundings:

total mechanical energy remains constant

This is the principle of conservation of mechanical energy.

Mechanical energy can change from one form to another, but the total remains the same.

Mechanical Energy

The total mechanical energy of a system can include:

Emechanical = Eₖ + Eₚ + Eₑ

where:

• Eₖ = kinetic energy
• Eₚ = gravitational potential energy
• Eₑ = elastic potential energy

The equations we have already studied are:

Eₖ = ½mv²

ΔEₚ = mgΔh

Eₑ = ½kx²

These three equations can be combined using conservation of mechanical energy.

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Conservation of Mechanical Energy

Consider a system at two different times.

If there are no energy losses:

initial mechanical energy = final mechanical energy

Therefore:

Eₖ(initial) + Eₚ(initial) + Eₑ(initial) = Eₖ(final) + Eₚ(final) + Eₑ(final)

Not every problem contains all three energy stores.

For example, a falling object might involve only:

gravitational potential energy ↔ kinetic energy

A spring launcher might involve:

elastic potential energy ↔ kinetic energy

A spring launching an object upward might involve:

elastic potential → kinetic → gravitational potential

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Conservation of Energy vs Conservation of Mechanical Energy

These ideas are related but not identical.

The conservation of energy states that energy cannot be created or destroyed.

Energy may be:

• transferred
• transformed
• stored in different ways

The total energy of a closed system remains constant.

Mechanical energy, however, can decrease when energy is transferred from mechanical stores into other forms.

For example:

kinetic energy → thermal energy

because of friction.

Total energy is still conserved, but mechanical energy is not.

This distinction is extremely important.

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Energy Transformations

Mechanical energy can move between different stores.

Falling Object

gravitational potential → kinetic

Object Thrown Upward

kinetic → gravitational potential

Spring Launcher

elastic potential → kinetic

Object Compressing a Spring

kinetic → elastic potential

Roller Coaster

gravitational potential ↔ kinetic

Pendulum

gravitational potential ↔ kinetic

These transformations can occur repeatedly.

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Falling Objects

Consider a ball held above the ground.

At the top:

• gravitational potential energy is high
• kinetic energy is zero if the ball is stationary

As the ball falls:

• gravitational potential energy decreases
• kinetic energy increases

Ignoring air resistance:

GPE lost = KE gained

Therefore:

mgΔh = ½mv²

Worked Example: Falling Ball

A 2 kg ball falls from rest through a vertical distance of 5 m.

Use:

g = 9.8 N/kg

Initial kinetic energy:

Eₖ = 0 J

Gravitational potential energy lost:

ΔEₚ = mgΔh

ΔEₚ = 2 × 9.8 × 5

ΔEₚ = 98 J

Ignoring air resistance:

KE gained = 98 J

Therefore, just before reaching the lower position:

Eₖ = 98 J

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Finding Speed Using Energy

We can use conservation of energy to calculate the ball's speed.

We know:

mgΔh = ½mv²

For a fall of 5 m:

2 × 9.8 × 5 = 0.5 × 2 × v²

98 = v²

Therefore:

v = √98

v ≈ 9.9 m/s

The ball reaches approximately 9.9 m/s.

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Why Mass Cancels

Starting with:

mgΔh = ½mv²

m appears on both sides.

Cancel m:

gΔh = ½v²

Therefore:

v = √(2gΔh)

This means that, ignoring air resistance, objects dropped from the same height reach the same speed regardless of their mass.

A heavier object begins with more gravitational potential energy, but it also requires proportionally more kinetic energy to reach the same speed.

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Energy at Different Points During a Fall

Suppose an object starts with 100 J of GPE and 0 J of KE.

Ignoring air resistance:

GPE decreases while KE increases.

But:

total mechanical energy = 100 J throughout

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Objects Moving Upward

The transformation works in the opposite direction when an object moves upward.

Suppose a ball is thrown vertically upward.

Initially:

• kinetic energy is high
• gravitational potential energy is relatively low

As it rises:

kinetic → gravitational potential

The ball slows down.

At maximum height:

v = 0

so:

Eₖ = 0

The initial kinetic energy has been transferred into gravitational potential energy, assuming air resistance is negligible.

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Worked Example: Ball Thrown Upward

A 0.50 kg ball is thrown upward at 10 m/s.

Initial kinetic energy:

Eₖ = ½mv²

Eₖ = 0.5 × 0.50 × 10²

Eₖ = 25 J

At maximum height:

Eₖ = 0

Ignoring air resistance:

GPE gained = 25 J

Use:

ΔEₚ = mgΔh

25 = 0.50 × 9.8 × Δh

25 = 4.9Δh

Therefore:

Δh = 25/4.9

Δh ≈ 5.1 m

The ball rises approximately 5.1 m.

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Roller Coasters

Roller coasters are excellent examples of mechanical energy transformations.

At the top of a hill:

• GPE is high
• KE may be relatively low

As the coaster descends:

GPE → KE

The coaster speeds up.

At the bottom:

• GPE is relatively low
• KE is high

As the coaster climbs another hill:

KE → GPE

The coaster slows down.

Worked Example: Roller Coaster

A 500 kg roller-coaster car starts from rest at the top of a 20 m hill.

Ignore friction.

Use:

g = 9.8 N/kg

Initial GPE relative to the bottom:

Eₚ = mgh

Eₚ = 500 × 9.8 × 20

Eₚ = 98,000 J

Initial KE:

Eₖ = 0 J

Total mechanical energy:

98,000 J

At the bottom:

Eₚ = 0 J

Therefore:

Eₖ = 98,000 J

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Finding the Roller Coaster's Speed

At the bottom:

Eₖ = ½mv²

Therefore:

98,000 = 0.5 × 500 × v²

98,000 = 250v²

v² = 392

v ≈ 19.8 m/s

The coaster reaches approximately:

19.8 m/s

if friction and air resistance are ignored.

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Energy at an Intermediate Height

Suppose the same coaster reaches another point 8 m above the bottom.

Total mechanical energy remains:

98,000 J

GPE at 8 m:

Eₚ = 500 × 9.8 × 8

Eₚ = 39,200 J

Therefore:

Eₖ = 98,000 − 39,200

Eₖ = 58,800 J

Now calculate speed:

58,800 = 0.5 × 500 × v²

58,800 = 250v²

v² = 235.2

v ≈ 15.3 m/s

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Why Later Roller-Coaster Hills Are Usually Lower

In an ideal frictionless system, a roller coaster released from rest could rise again to its original height.

In reality, some mechanical energy is continually transferred to:

• thermal energy through friction
• thermal energy through air resistance
• sound

Therefore, less mechanical energy remains available.

Without an additional energy input:

later hill < original hill

in most real roller-coaster designs.

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Pendulums

A pendulum continually transfers energy between gravitational potential energy and kinetic energy.

At the highest point:

• GPE is maximum
• KE is zero momentarily

As the pendulum swings downward:

GPE → KE

At the lowest point:

• GPE is minimum
• KE is maximum
• speed is maximum

As it swings upward:

KE → GPE

Pendulum Energy Positions

Consider three positions:

Position A — Highest Point

GPE: maximum

KE: zero

Position B — Lowest Point

GPE: minimum

KE: maximum

Position C — Opposite Highest Point

GPE: maximum

KE: zero

In an ideal system:

total mechanical energy remains constant

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Why a Real Pendulum Eventually Stops

A real pendulum does not swing forever.

Mechanical energy is gradually transferred by:

• air resistance
• friction at the pivot

into:

• thermal energy
• sound

Therefore, the pendulum reaches a slightly lower height on each swing.

Eventually, it stops.

Energy has not disappeared.

It has been transferred to the surroundings.

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Elastic Potential Energy

Springs introduce another mechanical energy store:

elastic potential energy

For an ideal spring:

Eₑ = ½kx²

A compressed or stretched spring can transfer its stored elastic energy into kinetic energy.

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Spring Launcher

Consider a compressed spring launching a cart.

Before release:

• elastic potential energy is high
• kinetic energy is zero

During release:

elastic potential → kinetic

When the spring reaches its natural length:

• elastic potential energy may be zero
• kinetic energy is maximum

assuming no energy losses.

Worked Example: Spring Launcher

A spring has:

k = 400 N/m

and is compressed by:

x = 0.20 m

Calculate the elastic potential energy.

Eₑ = ½kx²

Eₑ = 0.5 × 400 × 0.20²

Eₑ = 8 J

The spring launches a 0.25 kg cart.

Ignoring friction:

elastic energy lost = kinetic energy gained

Therefore:

Eₖ = 8 J

Use:

Eₖ = ½mv²

8 = 0.5 × 0.25 × v²

8 = 0.125v²

v² = 64

v = 8 m/s

The cart leaves the spring at 8 m/s.

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Spring Launching an Object Upward

A spring can transfer energy through several mechanical stores.

Initially:

elastic potential energy

After release:

kinetic energy

As the object rises:

gravitational potential energy

The overall transformation is:

elastic potential → kinetic → gravitational potential

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Worked Example: Spring to Height

A spring stores 30 J of elastic potential energy and launches a 0.60 kg object vertically upward.

Ignoring losses, at maximum height:

Eₖ = 0

Therefore:

elastic energy = gained GPE

30 = mgΔh

Using:

g = 9.8 N/kg

30 = 0.60 × 9.8 × Δh

30 = 5.88Δh

Δh ≈ 5.10 m

The object rises approximately 5.1 m above its starting point.

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Multiple Energy Stores

Some problems involve all three mechanical energy stores.

For example, imagine a cart:

1. begins at the top of a hill
2. rolls downward
3. compresses a spring

Initially:

gravitational potential energy

During motion:

kinetic energy

At maximum spring compression:

elastic potential energy

The transformation is:

GPE → KE → elastic PE

Ignoring losses:

initial GPE = final elastic PE

Therefore:

mgh = ½kx²

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Worked Example: Hill to Spring

A 2 kg cart begins from rest 5 m above a spring.

At the bottom, it compresses a spring with:

k = 1000 N/m

Ignore friction.

Initial GPE:

Eₚ = mgh

Eₚ = 2 × 9.8 × 5

Eₚ = 98 J

At maximum compression, the cart momentarily stops.

Therefore:

Eₖ = 0

All available mechanical energy is stored elastically:

98 = ½ × 1000 × x²

98 = 500x²

x² = 0.196

x ≈ 0.443 m

The spring compresses approximately 0.44 m.

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A General Problem-Solving Method

Conservation-of-energy problems become much easier if you use a consistent process.

Identify the Initial State

Ask:

• Is the object moving?
• Is it elevated?
• Is a spring stretched or compressed?

Determine which energy stores are present.

Identify the Final State

Again determine:

• kinetic energy
• gravitational potential energy
• elastic potential energy

Write Conservation of Energy

For an ideal system:

initial mechanical energy = final mechanical energy

Remove Zero Terms

For example:

If the object starts from rest:

initial KE = 0

If the spring is initially undeformed:

initial elastic PE = 0

If the reference height is chosen as zero:

GPE = 0

Substitute Equations

Use:

Eₖ = ½mv²

ΔEₚ = mgΔh

Eₑ = ½kx²

Solve

Rearrange for the unknown quantity.

Check

Ask:

• Are the units correct?
• Is the answer physically reasonable?
• Did I include energy losses if the question mentions them?

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Energy Bar Charts

Energy bar charts can help visualize conservation.

Suppose a falling object initially has:

100 J GPE

At the top:

GPE: 100 J
KE: 0 J

Halfway:

GPE: 50 J
KE: 50 J

At the bottom:

GPE: 0 J
KE: 100 J

The sizes of the individual stores change, but their total remains:

100 J

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Friction

Friction transfers mechanical energy into thermal energy.

Consider a sliding object.

Initially it has kinetic energy.

As friction slows it:

kinetic energy → thermal energy

The object's mechanical energy decreases.

However:

total energy is still conserved

The "missing" mechanical energy has been transferred to the thermal energy stores of the object and surroundings.

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Work Done Against Friction

If friction acts with approximately constant force:

energy transferred by friction = Fd

where:

• F = frictional force in N
• d = distance traveled in m

For a system involving friction:

initial mechanical energy = final mechanical energy + energy transferred thermally

A useful form is:

Emechanical(initial) = Emechanical(final) + Fd

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Worked Example: Friction

A cart initially has:

500 J of mechanical energy

As it travels, friction transfers:

120 J

to thermal energy.

Remaining mechanical energy:

500 − 120 = 380 J

The cart now has:

380 J of mechanical energy

But total energy remains:

380 J mechanical + 120 J thermal = 500 J

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Worked Example: Roller Coaster with Friction

A roller-coaster car begins with:

100,000 J of GPE

During the descent, 15,000 J is transferred to thermal energy through friction and air resistance.

Mechanical energy at the bottom:

100,000 − 15,000 = 85,000 J

If GPE at the bottom is zero:

Eₖ = 85,000 J

The coaster therefore moves more slowly than it would in an ideal frictionless system.

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Air Resistance

Air resistance, or drag, is another force that transfers mechanical energy.

For a falling object:

Without air resistance:

GPE → KE

With air resistance:

GPE → KE + thermal energy

Therefore, the object gains less kinetic energy than predicted by a frictionless calculation.

Mechanical Energy Is Not Always Conserved

Mechanical energy is conserved when only conservative forces, such as gravity and ideal spring forces, transfer energy within the system.

Mechanical energy is not conserved when significant energy is transferred by forces such as:

• friction
• air resistance
• other forms of drag

However, total energy remains conserved.

This distinction should always be stated clearly.

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Efficiency

Real machines and systems are not perfectly efficient.

Some energy is transferred into less useful forms.

Efficiency can be calculated using:

efficiency = useful energy output / total energy input

As a percentage:

efficiency (%) = useful energy output / total energy input × 100

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Worked Example: Spring Launcher Efficiency

A spring initially stores:

50 J

The launched object receives:

40 J of kinetic energy

Efficiency:

efficiency = 40/50 × 100

efficiency = 80%

Energy transferred elsewhere:

50 − 40 = 10 J

That energy may have been transferred into:

• thermal energy
• sound
• vibration

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Real-World Energy Transfers

Mechanical energy transformations occur constantly.

Hydroelectric Dam

gravitational potential → kinetic → mechanical → electrical

Bow and Arrow

elastic potential → kinetic

Bungee Jump

gravitational potential → kinetic → elastic potential

Trampoline

gravitational potential → kinetic → elastic potential → kinetic → gravitational potential

Pendulum

gravitational potential ↔ kinetic

Roller Coaster

gravitational potential ↔ kinetic

Spring Launcher

elastic potential → kinetic

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Bungee Jumping

A bungee jump demonstrates all three mechanical energy stores.

At the platform:

• GPE is high
• KE is zero
• elastic PE is zero if the cord is slack

During the fall:

GPE → KE

When the cord stretches:

GPE + KE → elastic PE

Near maximum extension:

• speed becomes zero momentarily
• KE becomes zero
• elastic PE is large

The stretched cord then accelerates the jumper upward.

Pendulum with Friction

An ideal pendulum would continue swinging to the same height forever.

A real pendulum reaches slightly lower heights on successive swings.

Why?

Mechanical energy is transferred into:

• thermal energy at the pivot
• thermal energy through air resistance
• sound

Eventually:

mechanical energy → mostly thermal energy

and the pendulum stops.

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Conservation of Energy Does Not Mean Constant Speed

A common misconception is that conservation of energy means an object's speed remains constant.

It does not.

For a roller coaster:

GPE decreases → KE increases → speed increases

Total mechanical energy can remain constant even while speed changes dramatically.

What remains constant is the sum of the mechanical energy stores, not each individual store.

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Conservation Does Not Mean Each Energy Store Is Constant

Suppose:

Initial:

GPE = 100 J

KE = 0 J

Later:

GPE = 30 J

KE = 70 J

The individual stores changed.

But:

100 + 0 = 30 + 70

Therefore:

100 J = 100 J

Mechanical energy has been conserved.

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Reference Heights

Gravitational potential energy depends on the chosen reference height.

Often we choose:

h = 0

at the lowest point of the motion.

The choice of reference does not change the physics as long as it is used consistently.

What matters in energy calculations is usually the change in height.

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Common Mistakes

Saying Energy Is "Used Up"

Energy is not destroyed.

It is transferred or transformed.

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Saying Friction Destroys Energy

Friction transfers mechanical energy into thermal energy.

Total energy remains conserved.

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Assuming Mechanical Energy Is Always Conserved

Mechanical energy is conserved only when energy is not being transferred from mechanical stores into other forms.

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Confusing Mechanical Energy with Total Energy

Mechanical energy includes:

• kinetic
• gravitational potential
• elastic potential

Thermal energy is not mechanical energy.

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Forgetting Zero-Energy Conditions

If an object starts from rest:

Eₖ = 0

If a spring is undeformed:

Eₑ = 0

Using a chosen reference level:

Eₚ = 0

Recognizing these conditions can simplify calculations greatly.

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Forgetting That Speed Is Squared

Kinetic energy:

Eₖ = ½mv²

not:

Eₖ = ½mv

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Forgetting That Spring Deformation Is Squared

Elastic energy:

Eₑ = ½kx²

not:

Eₑ = ½kx

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Using Path Length Instead of Vertical Height

For gravitational potential energy, use vertical height change.

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Ignoring Friction When the Question Includes It

If friction transfers energy, include that transfer in the energy balance.

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Check Your Understanding

1. Define mechanical energy.
2. State the principle of conservation of mechanical energy.
3. Name the three mechanical energy stores considered in this topic.
4. State the equations for kinetic, gravitational potential, and elastic potential energy.
5. Explain the difference between conservation of total energy and conservation of mechanical energy.
6. A ball loses 200 J of GPE while falling. Ignoring air resistance, how much KE does it gain?
7. A 2 kg object falls 10 m from rest. Calculate the GPE lost using g = 9.8 N/kg.
8. Use conservation of energy to determine its kinetic energy after falling 10 m.
9. Calculate its speed.
10. Explain why mass cancels when calculating the speed of an object falling from rest through a known height.
11. A roller coaster begins from rest 25 m above the bottom. Calculate its speed at the bottom, ignoring friction.
12. A coaster begins with 120,000 J of mechanical energy. At one point it has 45,000 J of GPE. How much KE does it have?
13. Describe the energy transformations during one complete swing of a pendulum.
14. Where does a pendulum have maximum kinetic energy?
15. Where does it have maximum gravitational potential energy?
16. Why does a real pendulum eventually stop?
17. A spring stores 18 J. It launches a 0.25 kg object. Calculate the object's speed, ignoring losses.
18. A spring with k = 500 N/m is compressed by 0.20 m. Calculate its elastic potential energy.
19. If all this energy is transferred to a 0.40 kg cart, calculate the cart's speed.
20. Describe the energy transformations during a bungee jump.
21. A system begins with 800 J of mechanical energy. Friction transfers 150 J into thermal energy. How much mechanical energy remains?
22. Explain why total energy is still conserved in Question 21.
23. A roller coaster begins with 200 kJ of GPE and loses 30 kJ to friction. How much kinetic energy could it have at the bottom?
24. A 3 kg object is launched vertically upward with 150 J of kinetic energy. Ignoring air resistance, calculate its maximum height above the starting point.
25. A 2 kg cart rolls from a height of 4 m and compresses a spring with k = 800 N/m. Calculate the maximum spring compression, ignoring friction.
26. Explain why conservation of mechanical energy does not mean speed remains constant.
27. Explain why later roller-coaster hills are generally lower than the first major hill.
28. A spring launcher receives 60 J of stored elastic energy but produces only 48 J of projectile kinetic energy. Calculate its efficiency.
29. Where did the remaining energy in Question 28 go?
30. Give one real-world example involving transformations among all three mechanical energy stores.

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Key Terms

Mechanical energy — Energy associated with motion and position, including kinetic and potential energy stores.

Conservation of mechanical energy — Principle that total mechanical energy remains constant when no mechanical energy is transferred to other forms.

Conservation of energy — Principle that energy cannot be created or destroyed but can be transferred between stores and systems.

Kinetic energy — Energy associated with motion.

Gravitational potential energy — Energy associated with position in a gravitational field.

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

Energy transformation — Change from one form or store of energy to another.

Energy transfer — Movement of energy between objects, systems, or stores.

Friction — Force opposing relative motion between surfaces that can transfer mechanical energy into thermal energy.

Air resistance — Drag force acting on an object moving through air.

Drag — Resistive force acting on an object moving through a fluid.

Thermal energy — Energy associated with the microscopic motion and interactions of particles.

Efficiency — Fraction of input energy transferred into the desired useful output.

Conservative force — Force for which energy can be stored as potential energy and recovered, such as gravity or an ideal spring force.

Energy dissipation — Spreading of energy into less concentrated, often less useful stores such as thermal energy in the surroundings.

Reference height — Chosen level from which gravitational height is measured.

Closed system — System that does not exchange energy with its surroundings.

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Key Takeaways

• Mechanical energy includes kinetic, gravitational potential, and elastic potential energy.
• In an ideal system with no mechanical energy losses, total mechanical energy remains constant.
• Mechanical energy can be transformed between different stores without being destroyed.
• Falling objects transform gravitational potential energy into kinetic energy.
• Objects moving upward transform kinetic energy into gravitational potential energy.
• Springs can transform elastic potential energy into kinetic energy.
• Moving objects can compress springs and transfer kinetic energy into elastic potential energy.
• Roller coasters continuously exchange gravitational potential and kinetic energy.
• Pendulums continuously exchange gravitational potential and kinetic energy.
• Spring launchers transfer elastic potential energy into kinetic energy.
• Systems such as bungee jumps can involve all three mechanical energy stores.
• Conservation of mechanical energy can be used to calculate unknown speeds, heights, spring compressions, and energies.
• A useful general relationship is initial mechanical energy = final mechanical energy for an ideal system.
• Kinetic energy is calculated using Eₖ = ½mv².
• Changes in gravitational potential energy are calculated using ΔEₚ = mgΔh.
• Elastic potential energy is calculated using Eₑ = ½kx².
• Friction and air resistance transfer mechanical energy into thermal energy and sometimes sound.
• When friction is present, mechanical energy may decrease, but total energy is still conserved.
• Real roller coasters, pendulums, and spring systems therefore do not perfectly conserve mechanical energy.
• Energy is never "used up" or destroyed; it is transferred to other stores.
• Conservation of energy does not mean that speed, height, kinetic energy, or potential energy remains constant.
• It is the total energy accounting that must balance.

The central idea can be summarized as:

GPE ↔ KE ↔ Elastic PE

For an ideal system:

Eₖ + Eₚ + Eₑ = constant

For a real system with friction:

initial mechanical energy = final mechanical energy + energy transferred to the surroundings

So when solving energy problems, the key question is not:

"Where did the energy go?"

but:

"Which energy store did it transfer into?"