Gravitational Potential Energy¶
The Formula¶
What It Means¶
Gravitational potential energy is stored work. When you lift a rock, you spend effort fighting gravity. That effort doesn't vanish — it gets stored in the rock's position. Let the rock go, and gravity converts that stored energy back into motion (kinetic energy). The higher you lift it, the more energy is stored. The heavier it is, the more energy is stored.
This formula says exactly how much: mass times gravitational acceleration times height. Simple, but getting here was anything but.
Why It Works — The Story Behind the Formula¶
The 50-Year Fight Over "What Is Energy?"¶
In the late 1600s, physics had a problem. Two giants of science were at war over a seemingly simple question: what is the "quantity of motion" in the universe?
René Descartes said it was \(mv\) — mass times velocity. He called it "quantity of motion" and argued it was conserved. When things collide, the total \(mv\) stays the same. Clean and simple.
Gottfried Wilhelm Leibniz said Descartes was wrong. In 1686, he published Brevis demonstratio — a short paper with a devastating argument.
Leibniz's Killer Experiment¶
Leibniz reasoned like this: imagine dropping a 1 kg ball from a height of 4 meters, and a 4 kg ball from 1 meter. Which has more "force" when it hits the ground?
Descartes' \(mv\) gives us the answer — but first we need the velocities. From Galileo's law of free fall (\(v^2 = 2gh\)):
- 1 kg from 4 m: \(v = \sqrt{2g \cdot 4} = \sqrt{8g}\), so \(mv = 1 \cdot \sqrt{8g}\)
- 4 kg from 1 m: \(v = \sqrt{2g \cdot 1} = \sqrt{2g}\), so \(mv = 4 \cdot \sqrt{2g}\)
These are different! \(4\sqrt{2g} \neq \sqrt{8g}\) (actually \(4\sqrt{2g} = 4\sqrt{2g}\) while \(\sqrt{8g} = 2\sqrt{2g}\), so the 4 kg ball has twice the Cartesian "force").
But here's the catch: it took the same amount of effort to lift 1 kg by 4 meters as it takes to lift 4 kg by 1 meter. Any builder with a pulley can tell you that. So shouldn't they have the same "force" when they fall?
Leibniz argued that the true "living force" (vis viva) was \(mv^2\):
- 1 kg from 4 m: \(mv^2 = 1 \cdot 8g = 8g\)
- 4 kg from 1 m: \(mv^2 = 4 \cdot 2g = 8g\)
Equal! The energy stored by lifting an object depends on \(mh\), and Leibniz's \(mv^2\) preserves this symmetry while Descartes' \(mv\) does not.
The Woman Who Proved It: Émilie du Châtelet¶
The debate raged for decades. Then in the 1740s, Émilie du Châtelet — mathematician, physicist, and translator of Newton's Principia into French — found the experimental evidence that settled it.
She analyzed experiments performed by Willem 's Gravesande, who dropped brass balls into soft clay from different heights and measured how deep they sank:
- Drop from height \(h\) → dent of depth \(d\)
- Drop from height \(2h\) → dent of depth... \(2d\)? No. Only about \(2d\).
- To get a dent of depth \(4d\), you need height \(4h\)
Wait — that doesn't sound right for \(v^2\). But here's the key: the velocity at impact is \(v = \sqrt{2gh}\), so \(v^2 = 2gh\). Double the height means double \(v^2\), which means double the depth of the dent. The dent depth was proportional to \(v^2\), not \(v\).
This proved Leibniz right. The energy of a falling object depended on \(v^2\), and therefore the energy stored by lifting it depended on \(mh\).
From "Vis Viva" to Potential Energy¶
It took another century for the terminology to catch up with the physics.
In 1829, Gaspard-Gustave de Coriolis introduced the factor \(\frac{1}{2}\) and formalized kinetic energy as \(\frac{1}{2}mv^2\) (half of Leibniz's vis viva). In the 1850s, William Rankine coined the term "potential energy" to describe energy stored in position.
The conceptual leap was this: when you lift a stone, you're not creating "force." You're doing work against gravity to store energy that can later be released as kinetic energy.
The Derivation¶
From Work Done Against Gravity¶
The derivation is beautifully direct. Start with the definition of work:
When you lift an object straight up, the force you need to apply equals the object's weight (to overcome gravity):
The distance you lift it is the height \(h\). So the work you do is:
This work doesn't disappear. It gets stored as gravitational potential energy:
That's it. No tricks, no hidden steps. The formula is a direct consequence of "work = force × distance" applied to gravity.
Why This Only Works Near Earth's Surface¶
There's a subtlety. We assumed \(g\) is constant — that gravity pulls with the same force whether you're 1 meter or 100 meters up. Near Earth's surface, this is an excellent approximation (gravity at the top of the Eiffel Tower is only 0.03% weaker than at the base).
But if you go very high — satellite altitudes — gravity gets weaker with distance, and you need the more general formula:
The \(mgh\) version is the "near Earth" approximation of this deeper truth. It works because for small heights compared to Earth's radius (6,371 km), \(g\) barely changes.
Variables Explained¶
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \(U\) | Gravitational potential energy | Joules (J) | Energy stored by an object's height |
| \(m\) | Mass | Kilograms (kg) | How much stuff the object contains |
| \(g\) | Gravitational acceleration | m/s² | \(\approx 9.81\) on Earth's surface |
| \(h\) | Height | Meters (m) | Vertical distance above reference point |
Worked Examples¶
Example 1: A Book on a Shelf¶
A 0.5 kg book sits on a shelf 2 meters above the floor.
If the book falls, it hits the floor with 9.81 J of kinetic energy — about the same as a gentle slap. That's why a falling book startles you but doesn't injure you.
Example 2: The Leibniz Test¶
Let's verify Leibniz's argument with real numbers.
1 kg lifted 4 m:
4 kg lifted 1 m:
Identical. When either object falls, it hits the ground with 39.24 J of kinetic energy. Descartes' momentum would say they have different "quantities of motion" — and he'd be right about momentum, but wrong about energy. Both quantities are real; they measure different things.
Example 3: Hydroelectric Power¶
The Hoover Dam holds water at about 180 meters above the turbines. For 1,000 kg (one cubic meter) of water:
That's enough to power a 100-watt lightbulb for about 5 hours. Multiply by the millions of cubic meters flowing through, and you get enough power for 1.3 million homes. All from \(mgh\).
Why Linear in \(h\)? Why Not \(h^2\)?¶
Because gravity near Earth's surface is essentially constant. When a force is constant, the work done is simply force times distance — a linear relationship. No squaring needed.
Compare with kinetic energy (\(\frac{1}{2}mv^2\)), where the \(v^2\) comes from the fact that accelerating objects cover more distance at higher speeds (the Galilean \(d = \frac{1}{2}at^2\) relationship). Potential energy doesn't have that complication. You're just fighting a constant force over a distance.
If gravity weren't constant (like far from Earth), the relationship becomes nonlinear — which is why the general formula \(U = -GMm/r\) is an inverse, not linear.
Common Mistakes¶
- Forgetting that height is relative: \(U = mgh\) depends on where you define \(h = 0\). A ball on a table has \(U = 0\) if the table is your reference, but positive \(U\) if the floor is your reference. Only changes in potential energy (\(\Delta U\)) matter physically.
- Thinking potential energy is "in" the object: It's not stored inside the object. It's a property of the system — the object and the Earth together. Without Earth's gravity, the object has no gravitational potential energy regardless of its "height."
- Using \(mgh\) at large distances: At satellite altitudes or beyond, \(g\) is no longer approximately constant. Use \(U = -GMm/r\) instead.
- Confusing energy with force: Lifting a 10 kg weight 1 meter stores 98.1 J of energy. Lifting it 2 meters stores 196.2 J. The force required is the same (98.1 N) in both cases — but the energy stored is different because you push for a longer distance.
Related Formulas¶
- Kinetic Energy — where potential energy goes when the object falls
- Newton's Laws — the framework that makes this derivation possible
- Measuring the Constants — how \(g\) and \(G\) were actually measured
History¶
- 1686 — Leibniz publishes Brevis demonstratio, arguing that "vis viva" (\(mv^2\)), not momentum (\(mv\)), is the true measure of a body's energy
- 1720s — 's Gravesande's brass ball experiments provide physical evidence for the \(v^2\) dependence
- 1740s — Émilie du Châtelet analyzes 's Gravesande's data, publishes Institutions de physique, and translates Newton's Principia into French with extensive commentary supporting Leibniz's view
- 1829 — Coriolis introduces the \(\frac{1}{2}\) factor, giving us the modern \(\frac{1}{2}mv^2\)
- 1853 — Rankine coins the term "potential energy," finally giving a name to the energy stored in position
- 1847 — Helmholtz publishes On the Conservation of Force, unifying kinetic and potential energy into a single conservation law
References¶
- Leibniz, G. W. (1686). Brevis demonstratio erroris memorabilis Cartesii
- du Châtelet, É. (1740). Institutions de physique
- Smith, George E. "The vis viva dispute." Physics Today, 2006.
- Feynman, R. P. The Feynman Lectures on Physics, Vol. 1, Ch. 4 & 13.