Energy Transformations and Calculations

Most real physical systems involve several energy transformations, not just one.

A roller coaster, bouncing ball, spring launcher, hydroelectric dam, bungee jumper, or vehicle may transfer energy through several stores before the process is complete.

The key skill is learning to track the energy:

Where is the energy initially? → Where does it go? → How much remains?

Energy is never created or destroyed. It is transferred between stores and, in real systems, some energy is often dissipated into the surroundings.

The Mechanical Energy Stores

Three important mechanical energy stores are:

Kinetic Energy

Energy associated with motion.

Eₖ = ½mv²

where:

• Eₖ = kinetic energy (J)
• m = mass (kg)
• v = speed (m/s)

Gravitational Potential Energy

Energy associated with position in a gravitational field.

ΔEₚ = mgΔh

where:

• ΔEₚ = change in gravitational potential energy (J)
• m = mass (kg)
• g = gravitational field strength (N/kg)
• Δh = vertical height change (m)

Elastic Potential Energy

Energy stored when an elastic object is stretched or compressed.

Eₑ = ½kx²

where:

• Eₑ = elastic potential energy (J)
• k = spring constant (N/m)
• x = extension or compression (m)

These equations can be combined to analyze complex systems.

---

Energy Transformations

An energy transformation occurs when energy changes from one store to another.

Examples include:

gravitational potential → kinetic

kinetic → gravitational potential

elastic potential → kinetic

kinetic → elastic potential

gravitational potential → elastic potential

kinetic → thermal

A physical system may involve several transformations in sequence.

---

Energy Flow Diagrams

An energy flow diagram shows how energy moves through a system.

For a falling object:

gravitational potential energy → kinetic energy

For a spring launcher:

elastic potential energy → kinetic energy

For a ball thrown upward:

kinetic energy → gravitational potential energy

These simple diagrams become particularly useful when several transformations occur.

---

Multi-Step Energy Flow

Consider a cart released from the top of a hill that rolls downward and compresses a spring.

The energy flow is:

gravitational potential → kinetic → elastic potential

If friction is present:

gravitational potential → kinetic → elastic potential + thermal energy

The thermal energy represents mechanical energy transferred to the surroundings.

---

Useful and Dissipated Energy

Real systems are rarely perfectly efficient.

Some energy is usually transferred into less useful forms, especially:

• thermal energy
• sound
• vibration

This is often described as energy dissipation.

For example:

chemical → kinetic + thermal + sound

Energy has not disappeared.

It has simply become more spread out in the surroundings and is often less useful for performing the intended task.

---

Conservation of Energy

The fundamental principle is:

total energy before = total energy after

For an ideal mechanical system:

initial mechanical energy = final mechanical energy

For a real system:

initial energy = useful final energy + dissipated energy

This provides the foundation for solving multi-step energy problems.

---

A Systematic Problem-Solving Method

Complex energy problems become much easier when solved systematically.

Identify the System

Determine which objects are involved.

For example:

• ball + Earth
• cart + track + Earth
• cart + spring
• roller coaster + Earth

Identify the Initial Energy Stores

Ask:

• Is anything moving?
• Is anything elevated?
• Is anything stretched or compressed?

Identify the Final Energy Stores

Ask the same questions at the final position.

Identify Energy Losses

Look for:

• friction
• air resistance
• sound
• deformation

Draw an Energy Flow

For example:

GPE → KE → elastic PE + thermal

Write the Energy Equation

Then substitute numerical values.

This method prevents many calculation mistakes.

---

Worked Example: Hill to Kinetic Energy

A 500 kg roller-coaster car begins from rest 30 m above the bottom of a hill.

Ignore friction.

Use:

g = 9.8 N/kg

Initial GPE:

Eₚ = mgh

Eₚ = 500 × 9.8 × 30

Eₚ = 147,000 J

Initial KE:

0 J

At the bottom:

GPE → KE

Therefore:

Eₖ = 147,000 J

Now:

Eₖ = ½mv²

147,000 = 0.5 × 500 × v²

147,000 = 250v²

v² = 588

v ≈ 24.2 m/s

The coaster reaches approximately 24.2 m/s.

Worked Example: Roller Coaster with Friction

Now suppose the same coaster loses 27,000 J of mechanical energy because of friction and air resistance.

Initial mechanical energy:

147,000 J

Energy dissipated:

27,000 J

Remaining mechanical energy:

147,000 − 27,000 = 120,000 J

At the bottom:

Eₖ = 120,000 J

Therefore:

120,000 = 0.5 × 500 × v²

120,000 = 250v²

v² = 480

v ≈ 21.9 m/s

Without friction:

24.2 m/s

With friction:

21.9 m/s

The coaster is slower because some mechanical energy has been transferred into the surroundings.

---

Energy Flow for the Roller Coaster

The idealized system is:

GPE → KE

The real system is:

GPE → KE + thermal + sound

Notice that total energy is conserved in both cases.

What changes is the amount of energy remaining in the mechanical energy stores.

---

Worked Example: Hill to Spring

A 4 kg cart starts from rest at a height of 5 m.

It rolls down a frictionless track and compresses a spring with:

k = 800 N/m

Find the maximum compression.

Initial energy:

GPE = mgh

GPE = 4 × 9.8 × 5

GPE = 196 J

At maximum compression:

• cart is momentarily stationary
• KE = 0
• energy is stored in the spring

Therefore:

GPE → elastic PE

196 = ½kx²

196 = 0.5 × 800 × x²

196 = 400x²

x² = 0.49

x = 0.70 m

The spring compresses by 0.70 m.

---

Tracking the Intermediate Stage

The previous problem actually contains three stages.

Stage A — Top of Hill

GPE: 196 J

KE: 0 J

Elastic PE: 0 J

Stage B — Bottom Before Spring

GPE: 0 J

KE: 196 J

Elastic PE: 0 J

Stage C — Maximum Compression

GPE: 0 J

KE: 0 J

Elastic PE: 196 J

So:

GPE → KE → elastic PE

The energy changes form twice, but the total mechanical energy remains 196 J.

---

Worked Example: Hill, Friction, and Spring

Now suppose 36 J is dissipated by friction before the cart reaches maximum compression.

Initial energy:

196 J

Dissipated:

36 J

Energy remaining:

196 − 36 = 160 J

Therefore:

160 = ½ × 800 × x²

160 = 400x²

x² = 0.40

x ≈ 0.632 m

The spring compresses approximately:

0.63 m

Less energy reaches the spring because some energy has been dissipated.

---

Spring Launcher to Height

A spring launcher can involve:

elastic PE → KE → GPE

Suppose a spring stores 50 J and launches a 1 kg object vertically upward.

Ignoring losses, the object eventually reaches maximum height.

At maximum height:

KE = 0

Therefore:

elastic PE → GPE

50 = mgΔh

50 = 1 × 9.8 × Δh

Δh = 50/9.8

Δh ≈ 5.1 m

The object rises approximately 5.1 m.

---

Spring Launcher with Energy Loss

Suppose the spring still stores 50 J, but 10 J is dissipated through friction and air resistance.

Energy available for GPE:

50 − 10 = 40 J

Therefore:

40 = 1 × 9.8 × Δh

Δh ≈ 4.08 m

The object reaches only about 4.1 m.

The energy flow is:

elastic PE → kinetic → GPE + thermal + sound

---

Bouncing Ball

A bouncing ball involves several energy transformations.

Before falling:

gravitational potential energy

During the fall:

GPE → KE

During impact:

KE → elastic PE + thermal + sound

During rebound:

elastic PE → KE

As the ball rises:

KE → GPE

Therefore:

GPE → KE → elastic PE → KE → GPE

Why a Ball Does Not Return to Its Original Height

If all mechanical energy were recovered, the ball would return to its original height.

In reality, energy is dissipated through:

• sound
• thermal energy
• internal deformation
• air resistance

Therefore:

final GPE < initial GPE

and:

rebound height < original height

---

Worked Example: Bouncing Ball

A 0.50 kg ball is dropped from 4 m.

Use:

g = 9.8 N/kg

Initial GPE:

Eₚ = 0.50 × 9.8 × 4

Eₚ = 19.6 J

Suppose 5.6 J is dissipated during the fall and bounce.

Energy remaining:

19.6 − 5.6 = 14.0 J

At maximum rebound height:

KE = 0

Therefore:

14.0 = 0.50 × 9.8 × h

14.0 = 4.9h

h ≈ 2.86 m

The ball rebounds to approximately 2.9 m.

---

Pendulum Energy Transformations

A pendulum repeatedly transforms energy between:

GPE ↔ KE

At the highest points:

• GPE maximum
• KE minimum

At the lowest point:

• GPE minimum
• KE maximum

In a real pendulum:

GPE ↔ KE + thermal + sound

As energy is dissipated, the pendulum's maximum height gradually decreases.

Energy Transfer in a Trampoline

A trampoline involves several stores.

A person standing above the trampoline has GPE.

During the fall:

GPE → KE

As the trampoline stretches:

KE → elastic PE

As the trampoline returns to shape:

elastic PE → KE

As the person rises:

KE → GPE

Therefore:

GPE → KE → elastic PE → KE → GPE

Some energy is also dissipated as:

• thermal energy
• sound

---

Bungee Jumping

A bungee jump is another excellent multi-step system.

Initially:

GPE

As the jumper falls:

GPE → KE

As the cord begins stretching:

KE + GPE → elastic PE

At maximum extension:

• speed is momentarily zero
• KE = 0
• substantial energy is stored elastically

During rebound:

elastic PE → KE → GPE

Energy Flow Diagrams for Complex Systems

Energy flow diagrams can be written simply using arrows.

Roller Coaster

GPE → KE → GPE → KE

With friction:

GPE → KE + thermal + sound

Spring Launcher

elastic PE → KE

Spring Launcher Upward

elastic PE → KE → GPE

Bouncing Ball

GPE → KE → elastic PE → KE → GPE

Hydroelectric Dam

GPE of water → KE of water → mechanical energy of turbine → electrical energy

Vehicle

chemical energy → kinetic energy + thermal energy + sound

Regenerative Braking

kinetic energy → electrical energy → chemical energy in battery

These diagrams make complicated processes easier to analyze.

---

Sankey Diagrams

A Sankey diagram is another way of representing energy transfers.

The width of each arrow represents the amount of energy transferred.

For example, suppose a machine receives:

100 J input

and produces:

70 J useful output

with:

30 J dissipated

The energy accounting is:

100 J → 70 J useful + 30 J dissipated

The arrows in a Sankey diagram would show a larger useful-energy branch and a smaller dissipated-energy branch.

The total widths leaving the system must equal the input width because energy is conserved.

---

Efficiency

Efficiency describes how much input energy becomes useful output energy.

efficiency = useful energy output / total energy input

As a percentage:

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

Efficiency cannot exceed 100%.

---

Worked Example: Efficiency

A motor receives:

500 J

of electrical energy.

It transfers:

400 J

into useful mechanical energy.

Efficiency:

efficiency = 400/500 × 100

efficiency = 80%

Energy dissipated:

500 − 400 = 100 J

Therefore:

500 J electrical → 400 J mechanical + 100 J thermal/sound

---

Worked Example: Finding Useful Energy

A machine is 75% efficient and receives 2000 J of energy.

Useful energy:

useful output = efficiency × input

Convert:

75% = 0.75

Therefore:

useful output = 0.75 × 2000

useful output = 1500 J

Energy dissipated:

2000 − 1500 = 500 J

---

Worked Example: Finding Input Energy

A machine produces 600 J of useful mechanical energy and is 80% efficient.

0.80 = 600/input

Therefore:

input = 600/0.80

input = 750 J

Dissipated energy:

750 − 600 = 150 J

---

Cars and Energy Transformations

A moving vehicle involves many energy transfers.

Initially, fuel or a battery provides stored energy.

For a fuel-powered vehicle:

chemical energy → kinetic energy + thermal energy + sound

When the vehicle climbs a hill:

kinetic/chemical energy → gravitational potential energy

When braking:

kinetic energy → thermal energy

The brakes become hot because mechanical energy has been transferred into thermal energy.

Regenerative Braking

Some electric and hybrid vehicles recover part of the vehicle's kinetic energy during braking.

Instead of transferring nearly all of it into thermal energy in the brakes:

kinetic energy → electrical energy → stored energy in battery

The process is not perfectly efficient, so some energy is still dissipated.

This demonstrates how engineering can improve energy efficiency by reducing unwanted energy transfers.

---

Hydroelectric Power

Water stored behind a dam possesses gravitational potential energy.

The sequence is approximately:

GPE of water → KE of water → rotational KE of turbine → electrical energy

Some energy is dissipated through:

• turbulence
• friction
• sound
• electrical resistance

Worked Example: Hydroelectric System

Suppose 1000 kg of water falls through 20 m.

Use:

g = 9.8 N/kg

GPE lost:

ΔEₚ = mgΔh

ΔEₚ = 1000 × 9.8 × 20

ΔEₚ = 196,000 J

Suppose the system is 85% efficient.

Electrical energy produced:

0.85 × 196,000

= 166,600 J

Energy dissipated:

196,000 − 166,600

= 29,400 J

---

Multi-Step Problem: Spring, Motion, and Height

A spring with:

k = 600 N/m

is compressed by:

x = 0.30 m

It launches a 0.50 kg cart up a ramp.

The cart loses 5 J because of friction.

How high does it rise vertically?

Step 1: Calculate Elastic Energy

Eₑ = ½kx²

Eₑ = 0.5 × 600 × 0.30²

Eₑ = 27 J

Step 2: Subtract Dissipated Energy

27 − 5 = 22 J

Step 3: Convert Remaining Energy to GPE

22 = mgΔh

22 = 0.50 × 9.8 × Δh

22 = 4.9Δh

Δh ≈ 4.49 m

The cart rises approximately 4.5 m vertically.

The complete energy flow is:

elastic PE → KE → GPE + thermal

---

Multi-Step Problem: Falling onto a Spring

A 2 kg object falls through 3 m before compressing a spring.

The spring constant is:

k = 500 N/m

Ignore friction and, for this simplified example, take the 3 m as the total vertical GPE decrease being converted to elastic energy.

Initial GPE available:

Eₚ = mgh

Eₚ = 2 × 9.8 × 3

Eₚ = 58.8 J

At maximum compression:

KE = 0

Therefore:

58.8 = ½ × 500 × x²

58.8 = 250x²

x² = 0.2352

x ≈ 0.485 m

The spring compresses approximately 0.49 m.

In a more detailed vertical-spring problem, the additional downward movement while the spring compresses would also change gravitational potential energy and must be included.

---

Evaluating Real Systems

Real-world energy analysis involves more than calculating numbers.

We should also ask:

• Which energy transfers are useful?
• Which are unwanted?
• Where is energy dissipated?
• Can the system be made more efficient?
• What assumptions are being made?
• Are friction and air resistance significant?
• Does the mathematical model accurately represent the real system?

This turns energy calculations into an engineering tool.

---

Reducing Energy Dissipation

Engineers often try to reduce unwanted energy transfers.

Examples include:

Lubrication

Reduces friction between moving surfaces.

Streamlining

Reduces air resistance.

Thermal Insulation

Reduces unwanted thermal energy transfer.

Low-Resistance Bearings

Reduce friction in rotating systems.

Regenerative Braking

Recovers some kinetic energy rather than dissipating it entirely as heat.

Efficient Motors

Reduce unwanted thermal energy transfers.

However, unwanted energy transfers can rarely be eliminated completely.

---

When Dissipation Is Useful

Not all energy dissipation is unwanted.

Vehicle brakes deliberately transform:

kinetic energy → thermal energy

This is useful because it slows the vehicle.

Shock absorbers deliberately dissipate mechanical energy to reduce bouncing.

Protective materials may transform kinetic energy through deformation and thermal processes during impacts.

Therefore, whether an energy transfer is "useful" depends on the purpose of the system.

---

Choosing the Correct Equation

Use:

Eₖ = ½mv²

when motion or speed is involved.

Use:

ΔEₚ = mgΔh

when vertical height changes.

Use:

Eₑ = ½kx²

when a spring or elastic system is stretched or compressed.

Use:

efficiency = useful output/input

when useful and wasted energy are being compared.

Most challenging problems require more than one equation.

---

Worked Example: Three Equations in One Problem

A 1.0 kg cart starts from rest 5 m above the bottom of a track.

It descends and loses 9 J through friction before compressing a spring with:

k = 400 N/m

Find the maximum spring compression.

Initial GPE

Eₚ = mgh

Eₚ = 1 × 9.8 × 5

Eₚ = 49 J

Subtract Dissipation

49 − 9 = 40 J

Transfer to Spring

At maximum compression:

KE = 0

Therefore:

40 = ½kx²

40 = 0.5 × 400 × x²

40 = 200x²

x² = 0.20

x ≈ 0.447 m

Therefore:

x ≈ 0.45 m

Energy flow:

GPE → KE → elastic PE + thermal

---

Energy Accounting Table

An energy table can help organize complex problems.

Check:

49 J = 40 J + 9 J

The energy accounting balances.

---

Common Mistakes

Treating Energy as Disappearing

Incorrect:

"9 J of energy was lost."

Better:

"9 J was transferred from mechanical energy stores to thermal energy in the surroundings."

---

Ignoring Intermediate Transformations

A spring launching an object upward is not simply:

elastic → gravitational

The actual sequence includes:

elastic → kinetic → gravitational

Even if the calculation can compare the initial and final states directly.

---

Assuming Mechanical Energy Is Always Conserved

If friction or air resistance is significant:

mechanical energy decreases

but:

total energy remains conserved

---

Forgetting to Subtract Dissipated Energy

If:

Initial energy = 100 J

Dissipated energy = 20 J

then useful mechanical energy remaining is:

80 J

---

Using the Wrong Energy Equation

Speed → kinetic energy.

Height → gravitational potential energy.

Spring deformation → elastic potential energy.

---

Forgetting Squared Quantities

Remember:

Eₖ = ½mv²

and:

Eₑ = ½kx²

Both v and x are squared.

---

Using Centimetres for Spring Extension

Convert to metres.

20 cm = 0.20 m

---

Using Total Ramp Length for GPE

Use vertical height change, not distance along the ramp.

---

Assuming Efficiency Can Exceed 100%

An efficiency above 100% would imply more energy leaves the system than enters it.

That violates conservation of energy.

---

Confusing Useful and Dissipated Energy

Dissipated energy is not destroyed.

It has been transferred into less useful stores, commonly thermal energy in the surroundings.

---

Check Your Understanding

1. Name the three main mechanical energy stores studied in this unit.
2. Write the equations for kinetic, gravitational potential, and elastic potential energy.
3. Explain what is meant by an energy transformation.
4. Construct an energy flow diagram for a falling ball.
5. Construct an energy flow diagram for a spring launching a cart.
6. Construct an energy flow diagram for a bouncing ball.
7. Describe the energy transformations occurring during a bungee jump.
8. A 2 kg object falls through 5 m. Calculate the GPE lost using g = 9.8 N/kg.
9. Ignoring air resistance, calculate its KE after falling.
10. Calculate its speed.
11. A 3 kg cart starts 4 m above the bottom of a frictionless track. Calculate its kinetic energy at the bottom.
12. A spring with k = 500 N/m is compressed by 0.20 m. Calculate the stored elastic energy.
13. If all this energy is transferred to a 0.50 kg cart, calculate its speed.
14. A spring stores 40 J and launches a 1 kg object vertically upward. Calculate its maximum height, ignoring losses.
15. Repeat Question 14 if 8 J is dissipated.
16. A ball initially has 30 J of GPE and loses 6 J during its fall and bounce. How much mechanical energy remains?
17. Explain why the ball does not return to its original height.
18. A roller coaster starts with 250 kJ of GPE. During its descent, 40 kJ is dissipated. How much kinetic energy can it have at the bottom?
19. Explain where the 40 kJ went.
20. A machine receives 800 J and produces 600 J of useful energy. Calculate its efficiency.
21. How much energy was dissipated in Question 20?
22. A system is 70% efficient and receives 1500 J. Calculate its useful output.
23. A 5 kg cart descends 6 m and then compresses a spring with k = 1000 N/m. Calculate the maximum compression, ignoring friction.
24. Repeat Question 23 if 50 J is dissipated before maximum compression.
25. Describe the energy transformations in a hydroelectric power station.
26. Explain how regenerative braking changes the energy flow in a vehicle.
27. Why does a real pendulum gradually stop?
28. Explain the difference between conservation of mechanical energy and conservation of total energy.
29. Give an example where energy dissipation is useful.
30. Explain why tracking energy through several stages can be more useful than examining only the beginning and end of a process.

---

Key Terms

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

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

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

Kinetic energy — Energy associated with motion.

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

Elastic potential energy — Energy stored by elastic deformation.

Conservation of energy — Principle that energy cannot be created or destroyed.

Conservation of mechanical energy — Conservation of the total kinetic and potential energy of a system when no mechanical energy is transferred elsewhere.

Energy dissipation — Transfer and spreading of energy into less useful stores, commonly thermal energy in the surroundings.

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

Air resistance — Resistive force acting against motion through air.

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

Useful energy — Energy transferred in the form required for the intended purpose.

Dissipated energy — Energy transferred into less useful stores and spread through the surroundings.

Energy flow diagram — Representation using arrows to show transfers and transformations of energy.

Sankey diagram — Diagram in which arrow widths represent quantities of energy transferred.

System — Object or collection of objects chosen for analysis.

Energy accounting — Tracking energy entering, leaving, and changing stores within a system.

---

Key Takeaways

• Physical systems often involve several energy transformations.
• The three main mechanical energy stores are kinetic, gravitational potential, and elastic potential energy.
• 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.
• Energy flow diagrams help track transformations through complex systems.
• A falling object transforms gravitational potential energy into kinetic energy.
• A spring launcher transforms elastic potential energy into kinetic energy.
• A bouncing ball can involve gravitational, kinetic, and elastic potential energy.
• A system can pass through several intermediate energy stores even when calculations compare only its initial and final states.
• Total energy is always conserved.
• Mechanical energy is conserved only when energy is not transferred out of mechanical stores.
• Friction and air resistance commonly transfer mechanical energy into thermal energy.
• Dissipated energy has not disappeared.
• Energy accounting should balance: energy in = energy out + change in stored energy.
• Efficiency measures the fraction of input energy transferred usefully.
• Real systems are less than 100% efficient.
• Engineers improve efficiency by reducing unwanted energy transfers.
• Some dissipation, such as braking or damping, can be useful.
• Multi-step problems are easiest when the initial state, final state, intermediate transformations, and dissipated energy are identified before calculations begin.

A useful approach for almost every complex energy problem is:

Identify stores → draw energy flow → calculate initial energy → account for dissipation → calculate final energy → solve for the unknown

The most important principle remains:

Energy does not disappear. Track where it goes.