Gravitational Potential Energy & Kinetic Energy
What is Gravitational Potential Energy?
Imagine you’re holding a book in your hand. The book has the ability to fall because it is higher than the ground. That ability is called gravitational potential energy (GPE). The higher you lift an object, the more “potential” it has to do work when it falls. 🌄
The Key Formula
In a uniform gravitational field (like near the Earth’s surface), the change in gravitational potential energy when you move an object of mass \$m\$ by a vertical distance \$\Delta h\$ is:
\$\Delta E_{\text{P}} = m\,g\,\Delta h\$
- \$m\$ = mass of the object (kg)
- \$g\$ = acceleration due to gravity (\$9.8\ \text{m/s}^2\$ on Earth)
- \$\Delta h\$ = change in height (m). Positive if you lift the object, negative if you lower it.
Why It Works – A Simple Analogy
- Think of a stretched rubber band. The higher you pull it, the more energy it stores. The formula is like measuring how much “stretch” (height) you gave the band.
- When you release the band, that stored energy turns into motion (kinetic energy). Similarly, when you drop an object, its GPE turns into kinetic energy.
Quick Reference Table
| Symbol | Meaning | Units |
|---|
| \$m\$ | Mass of the object | kg |
| \$g\$ | Acceleration due to gravity | m/s² |
| \$\Delta h\$ | Change in height | m |
| \$\Delta E_{\text{P}}\$ | Change in gravitational potential energy | J (joules) |
Example 1 – Lifting a Water Bottle
- Mass of bottle: \$m = 0.5\ \text{kg}\$
- Lift height: \$\Delta h = 2.0\ \text{m}\$
- Compute \$\Delta E_{\text{P}}\$:
\$\Delta E_{\text{P}} = 0.5 \times 9.8 \times 2.0 = 9.8\ \text{J}\$
- Interpretation: You’ve stored 9.8 J of energy in the bottle. If it were to fall, that energy would convert into kinetic energy (plus some losses).
Example 2 – Lowering a Book
- Mass of book: \$m = 1.2\ \text{kg}\$
- Lowered by: \$\Delta h = -1.5\ \text{m}\$ (negative because it goes down)
- Compute \$\Delta E_{\text{P}}\$:
\$\Delta E_{\text{P}} = 1.2 \times 9.8 \times (-1.5) = -17.64\ \text{J}\$
- Interpretation: The book loses 17.64 J of potential energy, which becomes kinetic energy as it falls.
Practice Problems
- A 2 kg crate is lifted 3 m above the ground. What is the change in its gravitational potential energy?
- A 0.8 kg toy car is dropped from a height of 4 m. Calculate the kinetic energy it would have just before hitting the ground (ignore air resistance).
- Someone lifts a 1.5 kg backpack 5 m. How much work is done against gravity?
- Calculate the change in GPE for a 3 kg ball that is raised 0.5 m and then lowered 0.5 m back to its original position.
Key Take‑aways
- The change in gravitational potential energy depends only on mass, gravity, and height change.
- Positive \$\Delta h\$ → energy stored (lifting). Negative \$\Delta h\$ → energy released (lowering).
- In a uniform field, \$g\$ is constant, so the formula is straightforward.
- Energy conservation: the lost GPE becomes kinetic energy (plus other forms like heat).