Key Takeaways
- Potential energy is stored energy based on an object's position in a gravitational field
- The formula is PE = mgh (mass × gravity × height)
- On Earth, gravity (g) equals 9.8 m/s² or approximately 10 m/s² for quick calculations
- Potential energy is measured in joules (J) in the SI system
- PE depends only on height difference, not the path taken - gravity is a conservative force
What Is Gravitational Potential Energy?
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. When you lift an object against gravity, you do work on it, and that work gets stored as potential energy. This stored energy can be converted back into kinetic energy (motion) when the object falls.
Think of potential energy as "stored work" - the higher you lift an object, the more energy you store in it. A ball held at 10 meters has twice the potential energy of the same ball held at 5 meters. This relationship is what makes potential energy so useful in physics: it's directly proportional to height.
The concept was first formalized in the 19th century by scientists like William Rankine and William Thomson (Lord Kelvin), though the underlying principles were understood by earlier physicists studying mechanics and motion.
PE = m × g × h
How to Calculate Potential Energy (Step-by-Step)
Step-by-Step Calculation Guide
Identify the Mass
Determine the mass of your object in kilograms (kg). If given in grams, divide by 1000 to convert. For example, 500 grams = 0.5 kg.
Measure the Height
Measure the vertical height above your reference point in meters. The reference point (where PE = 0) is arbitrary but typically ground level.
Use Standard Gravity
For Earth calculations, use g = 9.8 m/s². For quick estimates, g = 10 m/s² gives results within 2% accuracy.
Multiply the Values
Calculate PE = m × g × h. For example: 5 kg × 9.8 m/s² × 10 m = 490 Joules.
Express Your Answer
State your answer in joules (J). For very large values, use kilojoules (1 kJ = 1000 J) or megajoules (1 MJ = 1,000,000 J).
Worked Examples
Example 1: Book on a Shelf
A 2 kg textbook sits on a shelf 1.5 meters above the floor. Calculate its potential energy.
PE = 2 kg × 9.8 m/s² × 1.5 m = 29.4 Joules
Example 2: Weightlifter
A weightlifter raises a 100 kg barbell 2 meters above the ground. What is the potential energy stored?
PE = 100 kg × 9.8 m/s² × 2 m = 1,960 Joules (nearly 2 kJ)
Example 3: Hydroelectric Dam
Water (1000 kg) falls from a reservoir 50 meters high. Calculate the potential energy available.
PE = 1000 kg × 9.8 m/s² × 50 m = 490,000 J = 490 kJ. This is how hydroelectric plants generate electricity!
Energy Conservation and Conversion
One of the most important principles in physics is the conservation of energy. Gravitational potential energy doesn't disappear - it transforms into other forms of energy. Understanding these conversions is key to solving many physics problems.
Potential to Kinetic Energy
When an object falls freely (ignoring air resistance), its potential energy converts entirely to kinetic energy. At any point during the fall:
Total Mechanical Energy = PE + KE = constant
This means as height decreases, speed increases proportionally. At the ground (h = 0), all potential energy has become kinetic energy, and the object reaches maximum velocity.
Pro Tip: Finding Final Velocity
To find an object's velocity after falling from height h, use energy conservation: mgh = ½mv². Solving for v gives: v = √(2gh). This formula doesn't depend on mass - all objects fall at the same rate!
Real-World Energy Transformations
- Roller coasters: PE at the top converts to KE going down, then back to PE climbing the next hill
- Pendulums: Continuous exchange between PE (at highest points) and KE (at lowest point)
- Bouncing balls: PE → KE → elastic PE (compressed) → KE → PE (with some energy lost to heat and sound)
- Hydroelectric dams: Water's PE → KE of falling water → rotational energy of turbines → electrical energy
Common Mistakes to Avoid
Watch Out for These Errors
- Wrong units: Always convert to SI units (kg, m, s) before calculating. A mass of 500g should be 0.5 kg.
- Forgetting the reference point: PE = 0 at your chosen reference level. Heights below this point give negative PE values.
- Confusing weight and mass: Weight = mg (force in Newtons), while mass is measured in kilograms.
- Using the wrong g value: Earth's g = 9.8 m/s². The Moon's g = 1.62 m/s². Don't mix them up!
- Path confusion: PE depends only on vertical height, not the path taken. A winding road to a mountaintop gives the same PE as climbing straight up.
Gravity on Different Planets
Gravitational acceleration varies significantly across celestial bodies. This directly affects potential energy calculations:
| Location | Gravity (m/s²) | Relative to Earth | PE of 10kg at 10m |
|---|---|---|---|
| Earth | 9.8 | 100% | 980 J |
| Moon | 1.62 | 16.5% | 162 J |
| Mars | 3.71 | 37.9% | 371 J |
| Jupiter | 24.79 | 253% | 2,479 J |
| Venus | 8.87 | 90.5% | 887 J |
Real-World Applications
Understanding potential energy isn't just academic - it has crucial applications across engineering, sports, and everyday life:
Engineering Applications
- Hydroelectric power: Dams store water at height, converting PE to electrical energy. The Hoover Dam generates over 2,000 MW.
- Pumped storage: Excess electricity pumps water uphill; during peak demand, the water flows down to generate power.
- Structural engineering: Engineers calculate PE to understand forces in bridges, buildings, and towers.
- Crane operations: Lifting heavy loads requires understanding the PE gained and safely controlling descent.
Sports and Recreation
- Skiing and snowboarding: Starting from height provides PE that converts to speed going downhill.
- Bungee jumping: PE at jump height converts to KE, then elastic PE in the cord, creating the bounce effect.
- High diving: Divers convert PE to KE, reaching speeds over 50 km/h from 10-meter platforms.
- Pole vaulting: Athletes convert running KE to elastic PE in the pole, then to gravitational PE as they rise.
Pro Tip: Energy Efficiency
When designing systems that use potential energy (like roller coasters or water slides), account for energy losses to friction and air resistance. Real systems typically achieve 70-90% efficiency in energy conversion.
Types of Potential Energy
While this calculator focuses on gravitational PE, there are several other important types:
- Elastic Potential Energy: Stored in compressed or stretched materials (springs, rubber bands). Formula: PE = ½kx²
- Chemical Potential Energy: Stored in chemical bonds (batteries, food, fuel)
- Electrical Potential Energy: Stored in electric fields between charges
- Nuclear Potential Energy: Stored in atomic nuclei (released in nuclear reactions)
Frequently Asked Questions
The formula for gravitational potential energy is PE = mgh, where PE is potential energy in joules, m is mass in kilograms, g is gravitational acceleration (9.8 m/s² on Earth), and h is height in meters above a reference point. This formula assumes a uniform gravitational field, which is accurate for heights small compared to Earth's radius.
Potential energy is measured in joules (J) in the SI system. One joule equals one kilogram-meter squared per second squared (kg·m²/s²). In some contexts, you may also see energy measured in calories (1 cal = 4.184 J), electron volts (1 eV = 1.6×10⁻¹⁹ J), or foot-pounds in imperial units.
No, gravitational potential energy only depends on the vertical height difference, not the path taken. Whether you climb straight up or take a winding path, the potential energy at a given height remains the same. This path-independence makes gravity a conservative force, which is fundamental to energy conservation in mechanics.
When an object falls, its gravitational potential energy converts to kinetic energy. The total mechanical energy remains constant (ignoring air resistance). At the lowest point, all potential energy has transformed into kinetic energy, and the object reaches its maximum velocity. This principle is expressed as: PE_initial + KE_initial = PE_final + KE_final.
Potential energy is called stored energy because it represents the capacity to do work based on position or configuration. When you lift an object, the work you do against gravity gets "stored" in the object-Earth system. This stored energy can be released as kinetic energy when the object falls, performing work as it descends.
Potential energy varies directly with gravitational acceleration. On the Moon (g = 1.62 m/s²), an object has about 1/6 the potential energy it would have at the same height on Earth. On Jupiter (g = 24.79 m/s²), it would have about 2.5 times more potential energy. This affects everything from how high astronauts can jump to how much energy is needed to launch rockets.
Yes, potential energy can be negative depending on your chosen reference point. If you set ground level as zero, anything below ground (like a basement or mine) has negative potential energy. The choice of reference point is arbitrary - only changes in potential energy have physical significance. This is why we focus on potential energy differences when solving problems.
Potential and kinetic energy are interconvertible forms of mechanical energy. In a closed system without friction or air resistance, their sum remains constant (conservation of energy). As an object falls, potential energy decreases while kinetic energy increases by exactly the same amount. This relationship is described by: PE + KE = Total Mechanical Energy (constant).