Understanding Δ

The Greek letter Δ (delta) means change in.

Therefore:

Δh = final height − initial height

and:

ΔEₚ = final gravitational potential energy − initial gravitational potential energy

If an object moves upward:

Δh is positive

so:

ΔEₚ is positive

If an object moves downward:

Δh is negative

so:

ΔEₚ is negative

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Raising an Object

Suppose an object moves from:

2 m → 7 m

Then:

Δh = 7 − 2

Δh = +5 m

Its gravitational potential energy increases.

If the object instead moves from:

7 m → 2 m

then:

Δh = 2 − 7

Δh = −5 m

Its gravitational potential energy decreases.

The important quantity is the change in height, not simply the final height.

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Worked Example: Raising a Box

A 5 kg box is lifted vertically by 3 m.

Use:

g = 9.8 N/kg

ΔEₚ = mgΔh

ΔEₚ = 5 × 9.8 × 3

ΔEₚ = 147 J

The gravitational potential energy increases by:

147 J

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Worked Example: Using g = 10 N/kg

A 12 kg object is raised 4 m.

Use:

g = 10 N/kg

ΔEₚ = mgΔh

ΔEₚ = 12 × 10 × 4

ΔEₚ = 480 J

The object's gravitational potential energy increases by 480 J.

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How Mass Affects Gravitational Potential Energy

Consider two objects raised through the same height in the same gravitational field.

Object A

Mass = 2 kg
Height change = 5 m

Using g = 10 N/kg:

ΔEₚ = 2 × 10 × 5 = 100 J

Object B

Mass = 4 kg
Height change = 5 m

ΔEₚ = 4 × 10 × 5 = 200 J

Doubling the mass doubles the change in gravitational potential energy.

Therefore:

mass × 2 → ΔEₚ × 2

Gravitational potential energy is directly proportional to mass.

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How Height Affects Gravitational Potential Energy

Now keep mass constant.

A 3 kg object is raised by 2 m:

ΔEₚ = 3 × 10 × 2 = 60 J

Raise the same object by 4 m:

ΔEₚ = 3 × 10 × 4 = 120 J

Doubling the height change doubles the change in gravitational potential energy.

Therefore:

height × 2 → ΔEₚ × 2

This is different from kinetic energy, where doubling speed produces four times the kinetic energy.

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Comparing Kinetic and Gravitational Potential Energy

For kinetic energy:

Eₖ = 0.5mv²

Speed is squared.

Therefore:

speed × 2 → Eₖ × 4

For gravitational potential energy near Earth's surface:

ΔEₚ = mgΔh

Height is not squared.

Therefore:

height × 2 → ΔEₚ × 2

This distinction is important.

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Gravitational Field Strength

The symbol g represents gravitational field strength.

Near Earth's surface:

g ≈ 9.8 N/kg

This means that every kilogram of mass experiences approximately 9.8 N of gravitational force.

Recall the equation for weight:

W = mg

Therefore:

ΔEₚ = mgΔh

can also be written as:

ΔEₚ = WΔh

So the change in gravitational potential energy equals:

weight × vertical height change

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Worked Example Using Weight

A crate has a weight of 300 N and is lifted vertically 2.5 m.

ΔEₚ = WΔh

ΔEₚ = 300 × 2.5

ΔEₚ = 750 J

The crate gains 750 J of gravitational potential energy.

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Why g Matters

Gravitational field strength is not the same everywhere.

Different planets and moons have different values of g.

For the same object raised through the same height:

stronger gravitational field → greater change in gravitational potential energy

weaker gravitational field → smaller change in gravitational potential energy

This is because more work must be done against a stronger gravitational force.

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Mass and Weight Are Different

Mass describes the amount of matter in an object.

Unit:

kilogram (kg)

Weight is the gravitational force acting on an object.

Unit:

newton (N)

The relationship is:

W = mg

Mass remains the same when an object travels to another planet, but its weight can change because gravitational field strength changes.

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Worked Example: Different Gravitational Fields

A 10 kg object is raised 5 m.

On Earth

Use:

g = 9.8 N/kg

ΔEₚ = 10 × 9.8 × 5

ΔEₚ = 490 J

In a gravitational field of 2 N/kg

ΔEₚ = 10 × 2 × 5

ΔEₚ = 100 J

The same object gains less gravitational potential energy because the gravitational field is weaker.

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Lowering an Object

When an object is lowered, its gravitational potential energy decreases.

Suppose a 6 kg object moves downward by 4 m.

Using:

g = 10 N/kg

The height change is:

Δh = −4 m

Therefore:

ΔEₚ = 6 × 10 × (−4)

ΔEₚ = −240 J

The negative sign means the gravitational potential energy has decreased by 240 J.

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What Happens to the Lost Energy?

When gravitational potential energy decreases, the energy does not disappear.

It is transferred elsewhere.

For a freely falling object:

gravitational potential energy → kinetic energy

If air resistance is significant:

gravitational potential energy → kinetic energy + thermal energy

Energy is conserved.

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Falling Objects

Imagine a ball held above the ground.

Before release:

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

As it falls:

• gravitational potential energy decreases
• kinetic energy increases

Immediately before impact:

• gravitational potential energy relative to the ground is lower
• kinetic energy is greater

Energy Conservation During a Fall

Ignoring air resistance:

decrease in gravitational potential energy = increase in kinetic energy

Therefore:

−ΔEₚ = ΔEₖ

For example, if an object loses:

500 J of gravitational potential energy

it gains:

500 J of kinetic energy

assuming no energy is transferred to the surroundings.

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Worked Example: Falling Object

A 2 kg object falls through 10 m.

Use:

g = 9.8 N/kg

Change in gravitational potential energy:

ΔEₚ = 2 × 9.8 × (−10)

ΔEₚ = −196 J

The object loses 196 J of gravitational potential energy.

Ignoring air resistance, it gains 196 J of kinetic energy.

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Connecting GPE and Kinetic Energy

Recall:

Eₖ = 0.5mv²

and:

ΔEₚ = mgΔh

For an object falling from rest with negligible air resistance:

lost gravitational potential energy = gained kinetic energy

Therefore:

mgΔh = 0.5mv²

This relationship can be used to calculate the speed of a falling object without directly using acceleration equations.

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Worked Example: Finding Speed from Energy

A ball falls 5 m from rest.

Ignore air resistance and use:

g = 9.8 N/kg

Energy conservation gives:

mgΔh = 0.5mv²

Mass appears on both sides, so it cancels:

gΔh = 0.5v²

Substitute:

9.8 × 5 = 0.5v²

49 = 0.5v²

v² = 98

v ≈ 9.9 m/s

Notice that the mass was not required.

Ignoring air resistance, objects falling from the same height reach the same speed regardless of mass.

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Choosing a Reference Height

Gravitational potential energy is usually measured relative to a chosen reference level.

For example, we might choose:

• the floor
• ground level
• the bottom of a hill
• the lowest point of a roller coaster

as:

h = 0

What usually matters physically is the change in gravitational potential energy, not the absolute value.

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Example: Shelf Heights

A 4 kg book moves from a shelf 1 m above the floor to a shelf 3 m above the floor.

The change in height is:

Δh = 3 − 1

Δh = 2 m

Using g = 9.8 N/kg:

ΔEₚ = 4 × 9.8 × 2

ΔEₚ = 78.4 J

A common mistake would be to use 3 m rather than the 2 m change in height.

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Vertical Height Matters

The equation uses vertical height change, not the total distance traveled.

Imagine carrying a box up a long ramp.

Suppose:

Ramp distance = 10 m

Vertical height gained = 3 m

For gravitational potential energy, use:

Δh = 3 m

not:

10 m

Different Paths to the Same Height

Suppose identical objects are moved from the same starting height to the same final height.

Object A is lifted vertically.

Object B moves up a ramp.

Object C follows a curved path.

If all three have the same:

• mass
• starting height
• final height

then they have the same change in gravitational potential energy.

The path does not matter for ΔEₚ.

Only the vertical height difference matters.

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Rearranging the Equation

Starting with:

ΔEₚ = mgΔh

we can rearrange the equation to find other quantities.

Finding Mass

m = ΔEₚ/(gΔh)

Finding Height Change

Δh = ΔEₚ/(mg)

Finding Gravitational Field Strength

g = ΔEₚ/(mΔh)

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Worked Example: Finding Height

A 20 kg object gains 1960 J of gravitational potential energy.

How high was it raised?

Use:

Δh = ΔEₚ/(mg)

Δh = 1960/(20 × 9.8)

Δh = 1960/196

Δh = 10 m

The object was raised 10 m.

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Worked Example: Finding Mass

An object gains 1500 J of gravitational potential energy when raised 5 m.

Use:

g = 10 N/kg

m = ΔEₚ/(gΔh)

m = 1500/(10 × 5)

m = 1500/50

m = 30 kg

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Roller Coasters

Roller coasters provide an excellent example of energy transfer.

At the top of a hill:

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

As the coaster moves downhill:

gravitational potential energy → kinetic energy

The coaster speeds up.

At the bottom:

• gravitational potential energy is lower
• kinetic energy is greater

Why the First Roller-Coaster Hill Is Often High

The first major hill gives the coaster a large gravitational potential energy store.

As the coaster descends, this energy can be transferred into kinetic energy.

However, some energy is continuously transferred to the surroundings through:

• friction
• air resistance
• sound

Therefore, without additional energy input, later hills usually cannot be higher than the initial hill.

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Hydroelectric Dams

Water stored high behind a dam has gravitational potential energy.

When the water is released:

gravitational potential energy → kinetic energy of water

The moving water turns turbines:

kinetic energy → rotational kinetic energy

The generator then transfers energy electrically.

A simplified pathway is:

gravitational potential → kinetic → mechanical → electrical

Why Dam Height Matters

From:

ΔEₚ = mgΔh

greater height means more gravitational potential energy for the same mass of water.

Therefore, a larger vertical drop can provide more energy per kilogram of water.

This height difference is often called the head in hydroelectric systems.

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Pumped Hydroelectric Storage

Gravitational potential energy can also be used to store energy.

When excess electricity is available:

electricity → pumps → water moved uphill

The water gains gravitational potential energy.

Later:

water moves downhill → turbines → electricity

A simplified sequence is:

electrical energy → gravitational potential energy → electrical energy

This is called pumped hydroelectric storage.

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Cranes and Lifting

A crane transfers energy when it raises a load.

Suppose a crane lifts a 2000 kg load through 15 m.

Using:

g = 9.8 N/kg

ΔEₚ = 2000 × 9.8 × 15

ΔEₚ = 294,000 J

or:

294 kJ

At least this much energy must be transferred to the gravitational potential energy store of the load.

Real cranes require more energy because they are not 100% efficient.

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Elevators

An elevator also demonstrates gravitational potential energy.

When an elevator rises:

electrical energy → gravitational potential energy + other energy transfers

When it descends:

gravitational potential energy decreases

Some modern elevator systems can recover part of this energy using regenerative systems.

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Human Body and GPE

When you climb stairs, your body gains gravitational potential energy.

Suppose a 60 kg person climbs vertically 4 m.

Using:

g = 9.8 N/kg

ΔEₚ = 60 × 9.8 × 4

ΔEₚ = 2352 J

or approximately:

2.35 kJ

The person's body must transfer at least this much energy mechanically to increase gravitational potential energy.

Because the body is not perfectly efficient, the chemical energy used is greater.

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GPE and Mountains

Climbing a mountain increases gravitational potential energy.

For a 70 kg climber gaining 1000 m of vertical height:

ΔEₚ = 70 × 9.8 × 1000

ΔEₚ = 686,000 J

ΔEₚ = 686 kJ

This represents the increase in gravitational potential energy of the climber alone.

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Gravitational Potential Energy Beyond Earth's Surface

The equation:

ΔEₚ = mgΔh

works well near Earth's surface when gravitational field strength is approximately constant.

Over very large distances, however, g changes significantly.

For satellites, planets, and spacecraft, a more general gravitational potential energy equation is required.

For ordinary near-Earth calculations, however:

ΔEₚ = mgΔh

is an excellent approximation.

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Graph of GPE Against Height

For constant mass and gravitational field strength:

ΔEₚ ∝ Δh

Therefore, a graph of gravitational potential energy against height is a straight line.

If height doubles:

GPE doubles

If height triples:

GPE triples

This differs from kinetic energy against speed, which produces a curved relationship because speed is squared.

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Graph of GPE Against Mass

For constant height and gravitational field strength:

ΔEₚ ∝ m

The graph is again a straight line.

Doubling the mass doubles the gravitational potential energy change.

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Real-World Comparison

Imagine two roller-coaster cars at the same height.

Car A

Mass = 500 kg

Car B

Mass = 1000 kg

Car B has twice the mass.

Therefore, at the same height:

Car B has twice the gravitational potential energy relative to the same reference level.

If both descend the same vertical distance, Car B loses twice as much gravitational potential energy.

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Common Mistakes

Using the Total Height Instead of the Change in Height

If an object moves from 4 m to 10 m:

Incorrect:

Δh = 10 m

Correct:

Δh = 10 − 4 = 6 m

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Using Distance Instead of Vertical Height

If an object moves 20 m along a ramp but rises only 5 m:

Use:

Δh = 5 m

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Forgetting Mass Must Be in Kilograms

Convert grams to kilograms first.

500 g = 0.500 kg

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Confusing Mass and Weight

Mass is measured in:

kg

Weight is measured in:

N

Remember:

W = mg

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Confusing g with Mass

In:

ΔEₚ = mgΔh

g represents gravitational field strength, not grams.

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Forgetting the Sign of Δh

Moving upward:

Δh > 0

Moving downward:

Δh < 0

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Assuming Gravitational Potential Energy Disappears During a Fall

Energy is transferred.

Typically:

gravitational potential → kinetic

with some energy possibly transferred thermally because of air resistance.

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Using Horizontal Distance

Moving horizontally at the same height does not change gravitational potential energy.

If:

Δh = 0

then:

ΔEₚ = 0

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Check Your Understanding

1. Define gravitational potential energy.
2. State the equation for the change in gravitational potential energy.
3. What does Δ mean?
4. State the units for mass, gravitational field strength, height, and energy.
5. Calculate the GPE gained by a 5 kg object raised 6 m. Use g = 9.8 N/kg.
6. Calculate the GPE gained by a 20 kg object raised 3 m. Use g = 10 N/kg.
7. A 500 g ball is raised 4 m. Calculate its increase in GPE using g = 9.8 N/kg.
8. Explain what happens to GPE when mass doubles.
9. Explain what happens to GPE when height doubles.
10. An object moves from 3 m above the ground to 11 m above the ground. What is Δh?
11. A 10 kg object moves from 12 m to 5 m. Calculate ΔEₚ using g = 9.8 N/kg.
12. What does a negative value of ΔEₚ mean?
13. An object gains 980 J of GPE when raised 5 m. Calculate its mass.
14. A 25 kg object gains 4900 J of GPE. Calculate the height through which it was raised.
15. A ramp is 12 m long but raises an object vertically by 3 m. Which value should be used for Δh?
16. Explain why different paths to the same final height produce the same change in GPE.
17. Describe the energy transfers as a ball falls.
18. Explain how a roller coaster demonstrates transfers between GPE and kinetic energy.
19. Explain how a hydroelectric dam uses gravitational potential energy.
20. A 1000 kg roller-coaster car descends 20 m. Calculate the GPE lost using g = 9.8 N/kg.
21. Ignoring friction, how much kinetic energy would the coaster gain?
22. Explain why later roller-coaster hills are usually lower than the first major hill.
23. Explain why water stored higher behind a dam has greater potential to generate energy.
24. Describe how pumped hydroelectric storage stores energy.
25. Compare the effects of doubling height on GPE with doubling speed on kinetic energy.

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Key Terms

Gravitational potential energy (GPE) — Energy associated with an object's position in a gravitational field.

Energy store — A way of describing where energy is held within a physical system.

Joule (J) — SI unit of energy.

Mass — Measure of the amount of matter in an object, measured in kilograms.

Weight — Gravitational force acting on an object.

Gravitational field — Region in which an object with mass experiences a gravitational force.

Gravitational field strength (g) — Gravitational force per unit mass, measured in N/kg.

Height — Vertical position relative to a chosen reference level.

Change in height (Δh) — Difference between final and initial vertical position.

Delta (Δ) — Symbol meaning "change in."

Reference level — Chosen position from which height or gravitational potential energy is measured.

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

Kinetic energy — Energy associated with motion.

Conservation of energy — Principle that energy cannot be created or destroyed, although it can be transferred between stores.

Head — Vertical height difference available to water in a hydroelectric system.

Pumped hydroelectric storage — Energy-storage system that uses electricity to pump water uphill and later recovers energy as the water flows downhill.

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Key Takeaways

• Gravitational potential energy is associated with position in a gravitational field.
• Raising an object increases its gravitational potential energy.
• Lowering an object decreases its gravitational potential energy.
• Near Earth's surface, the change in gravitational potential energy is calculated using ΔEₚ = mgΔh.
• Mass must normally be measured in kilograms.
• Height must be the vertical change in height, not the total distance traveled.
• Gravitational field strength near Earth's surface is approximately 9.8 N/kg.
• Some school problems approximate g as 10 N/kg.
• Doubling mass doubles the change in gravitational potential energy.
• Doubling height doubles the change in gravitational potential energy.
• A stronger gravitational field produces a greater change in gravitational potential energy for the same mass and height.
• Moving upward produces a positive ΔEₚ.
• Moving downward produces a negative ΔEₚ.
• When an object falls, gravitational potential energy can be transferred into kinetic energy.
• Air resistance can transfer some energy to thermal stores in the object and surroundings.
• Energy is conserved during these transfers.
• Roller coasters continually transfer energy between gravitational potential and kinetic energy stores.
• Water stored behind a dam possesses gravitational potential energy that can ultimately be transferred into electrical energy.
• Pumped hydroelectric systems use gravitational potential energy to store energy.
• The equation ΔEₚ = mgΔh is most appropriate near Earth's surface, where g can be treated as approximately constant.

The central relationship is:

ΔEₚ = mgΔh

So:

mass × 2 → ΔEₚ × 2

height × 2 → ΔEₚ × 2

gravitational field strength × 2 → ΔEₚ × 2

Unlike kinetic energy, none of these quantities is squared. Gravitational potential energy near Earth's surface therefore changes linearly with mass, gravitational field strength, and vertical height.