Potential energy is energy an object has because of where it is, not because of how fast it’s moving. Gravitational potential energy (PE) is the energy stored when an object is lifted against gravity. Lift a book onto a shelf and you’ve put energy into it. Nothing is moving, but the energy is there, ready to turn into motion if the book falls.
PE = m g h
Mass m is in kilograms, height h is in meters, and g is 9.8 N/kg (the pull of gravity near Earth’s surface). The answer is in joules, the same unit as work and kinetic energy.
💡g = 9.8 N/kg is the same number as the 9.8 m/s² you used for free-fall acceleration in Unit 1. A newton per kilogram and a meter per second squared are the same unit.
BMath
Where mgh Comes From
In the lab you lifted masses with a spring scale. That’s exactly the argument. Lifting takes work (Activity 4-1-1), and the work you do is the energy you store.
Deriving the PE formula from lifting
Given
A mass m lifted straight up at a constant speed
Height gained: h
Solve
Constant speed → net force = 0
Your lifting force F = weight = m g
W_you = F × h = m g h
Gravity pulls down while the object moves up: W_gravity = −m g h
Net work = +mgh − mgh = 0 → ΔKE = 0 (speed didn’t change)
The energy you supplied is stored: ΔPE = +m g h
✓ PE = mgh is the energy stored by the work you did against gravity.
🔑Notice that the object’s speed didn’t change, so none of your work became KE. All of it went into PE. That’s why your spring-scale numbers matched mgh in Section 1.
CConcept
Where Is Zero? Only Changes Matter
Height has to be measured from somewhere. That starting level is the zero level, and you get to choose it. The floor is common, but a table top or the bottom of a hill works just as well. Here is a 2.0 kg book on a shelf 1.5 m above the floor, with different zero levels:
Zero level at
Book’s height h
Book’s PE
The floor
1.5 m
29.4 J
The table (0.75 m up)
0.75 m
14.7 J
The shelf itself
0 m
0 J
The ceiling (1.0 m above the shelf)
−1.0 m
−19.6 J
🔑The PE value changes with the zero level, but the energy released when the book falls doesn’t. Falling 0.75 m always releases 14.7 J, and falling to the floor always releases 29.4 J. Real problems only ever ask about changes in PE, so any consistent zero level works.
DExample
GPE Explorer
Set the zero level to the table and then lower the block below 2 m. Watch the PE go negative. Then switch back to the ground and see the same block, with the same energy released by a fall, described with different numbers.
Raise the block, add mass, and move the zero level. Watch which numbers change and which don’t.
Zero level at:
PE = 2 × 9.8 × (6.0) m
Potential energy118 J
Falling to the ground releases 118 J, wherever you put zero.
EWatch Out
Three Common Mix-Ups
⚠️“Height always means height above the floor.” It means height above whatever zero level you chose. If the object is below that level, its PE is negative. That isn’t an error.
⚠️“Heavier objects fall faster.” In a fall with no air resistance, mass cancels out of the energy equation and everything reaches the same speed. A feather seems to fall slowly only because air pushes back on it so much.
⚠️“The energy belongs to the object alone.” Strictly, gravitational PE is shared by the object and the Earth, since it comes from their separation. Saying “the book’s PE” is a handy shorthand, and it’s fine for everything in this unit.
FMath
Worked Examples
Example 1: find the PE
Given
Mass m = 2.5 kg
Height h = 1.8 m above the floor (zero level at the floor)
Solve
PE = m g h
PE = (2.5 kg)(9.8 N/kg)(1.8 m)
PE = 24.5 × 1.8
PE = 44.1 J
✓ The object has 44.1 J of gravitational potential energy.
Example 2: find the height from the PE
Given
A 0.50 kg object has PE = 24.5 J
Find its height above the zero level
Solve
PE = m g h so h = PE ÷ (m g)
h = 24.5 ÷ (0.50 × 9.8)
h = 24.5 ÷ 4.9
h = 5.0 m
✓ The object is 5.0 m above the zero level.
Example 3: a change in PE
Given
A 0.20 kg ball drops from 1.5 m to 0.5 m above the floor
Solve
ΔPE = m g (h_final − h_start)
ΔPE = (0.20)(9.8)(0.5 − 1.5)
ΔPE = 1.96 × (−1.0)
ΔPE = −1.96 J
✓ PE dropped by 1.96 J. That energy went somewhere else, mostly into motion.
GMath
From Stored Energy to Speed
When an object falls and nothing else takes energy away, all the PE it loses becomes KE. Set them equal and see what happens to the mass:
m g h = ½ m v² → v = √(2 g h)
Mass cancels, so the speed after a drop depends only on how far it dropped:
Height of drop
Speed at the bottom
Roughly
0.20 m
2.0 m/s
about 4 mph
1.0 m
4.4 m/s
about 10 mph
5.0 m
9.9 m/s
about 22 mph
10 m
14 m/s
about 31 mph
30 m
24 m/s
about 54 mph
Worked example: a pendulum at the bottom of its swing
Given
A pendulum bob is released from rest 0.15 m above the lowest point of its swing
Ignore friction and air
Solve
PE lost = KE gained: m g h = ½ m v²
v = √(2 g h)
v = √(2 × 9.8 × 0.15) = √2.94
v ≈ 1.7 m/s
✓ The bob passes the bottom at about 1.7 m/s, whatever its mass.
⚠️These are the best-case speeds. In a real drop, some energy goes to friction, to air resistance, and (for a rolling object) into spinning. Your measured speeds should come out lower than √(2gh), which is exactly the missing energy you chased in Section 4.
HExample
PE ↔ KE Swap Explorer
Watch the swap happen. As the object falls, the cyan PE shrinks and the orange KE grows, and together they always add up to the same total. Then add friction and watch a new slice appear. You’ll study that slice in detail in Activity 4-2-2.
An object is released from rest and falls through the drop. Slide along the drop and watch energy trade from one form to another. The total never changes. Try the friction slider to see what a real drop looks like.
Total energy mgH = 196 J (constant)
PE 98.0 JKE 98.0 JHeat 0.00 J
Speed here9.9 m/s
With no friction, PE lost equals KE gained. Change the mass and watch the speed stay put.
IConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1A 3.0 kg box sits on a shelf 2.0 m above the floor. Using the floor as zero, what is its gravitational PE? (g = 9.8 N/kg)
Q2The same object is lifted to twice the height. Its PE becomes…
Q3A 1 kg ball and a 5 kg ball are dropped from the same height with no air resistance. Which one hits the ground with a higher speed?
Q4A book sits on a table. You choose the table top as your zero level. The book falls to the floor 0.75 m below. Its PE at the floor is…
Score: 0 of 4
JConcept
Looking Ahead
You now have both halves of the energy swap: PE = mgh for height and KE = ½mv² for motion. Next you’ll put them together.