Potential Energy Calculator
Calculate gravitational (PE = mgh), elastic (PE = ½kx²), and electric potential energy.
Potential Energy Type
Formula:
PE = m × g × h
Gravitational Potential Energy
6.87 kJ
Energy Conversions:
Energy Conservation
If converted to kinetic energy (no losses):
v = 14.01 m/s
Energy Comparisons
Apple falling 1m
2 J
3500.0x your value
Book on desk (1kg, 1m)
10 J
700.0x your value
Person on 2nd floor (70kg, 3m)
2.1 kJ
3.3x your value
Car on hill (1500kg, 10m)
147.2 kJ
0.0x your value
Elevator rise (1000kg, 30m)
294.3 kJ
0.0x your value
Roller coaster drop (500kg, 50m)
245.3 kJ
0.0x your value
Types of Potential Energy
Gravitational PE
Energy due to position in a gravitational field. PE = mgh. Increases with height and mass.
Elastic PE
Energy stored in compressed/stretched springs. PE = ½kx². Proportional to displacement squared.
Electric PE
Energy between charged particles. PE = kq₁q₂/r. Positive for like charges (repulsion).
What Is Potential Energy?
Potential energy is stored energy based on an object's position or configuration within a force field. Unlike kinetic energy (energy of motion), potential energy is "waiting" to be converted into other forms. A book on a shelf, a stretched spring, and charges in an electric field all possess potential energy that can be released to do work.
The concept of potential energy is fundamental to understanding energy conservation. In any closed system, energy transforms between potential and kinetic forms while the total remains constant—the principle that governs everything from roller coasters to planetary orbits.
| Type | Source | Formula | Applications |
|---|---|---|---|
| Gravitational PE | Height in gravity field | PE = mgh | Dams, roller coasters, falling objects |
| Elastic PE | Deformed springs/materials | PE = ½kx² | Springs, bows, trampolines |
| Electric PE | Charge separation | PE = kqQ/r | Batteries, capacitors |
| Chemical PE | Molecular bonds | Bond energies | Food, fuel, explosives |
| Nuclear PE | Nuclear binding | E = mc² | Nuclear reactors, stars |
Gravitational Potential Energy
Where:
- PE= Potential energy (Joules)
- m= Mass (kg)
- g= Gravitational acceleration (9.8 m/s² on Earth)
- h= Height above reference point (m)
Gravitational Potential Energy
Gravitational potential energy is the energy stored due to an object's height in a gravitational field. The higher an object, the more potential energy it possesses. This form of PE is crucial for hydroelectric power, pendulums, and understanding projectile motion.
An important concept: only changes in potential energy matter physically. You can choose any reference point as "zero height"—the ground, sea level, or the floor. What matters is the height difference, not the absolute height.
| Object/Scenario | Height | Mass | Gravitational PE |
|---|---|---|---|
| Person on 10th floor | 30 m | 70 kg | 20,580 J |
| Water in reservoir | 100 m | 1,000 kg | 980,000 J |
| Skydiver at altitude | 4,000 m | 80 kg | 3,136,000 J |
| Apple on table | 1 m | 0.2 kg | 1.96 J |
| Airplane cruising | 10,000 m | 80,000 kg | 7.84 GJ |
For heights comparable to Earth's radius, use the general formula: PE = -GMm/r, where r is distance from Earth's center.
Elastic Potential Energy
Elastic potential energy is stored in deformed elastic materials—stretched or compressed springs, bent bows, rubber bands, and bouncy balls. This energy is governed by Hooke's Law, which states that force is proportional to displacement within the elastic limit.
The quadratic relationship (x²) means doubling the stretch quadruples the stored energy—just like kinetic energy's relationship with velocity. This principle is used in catapults, vehicle suspension, and countless mechanical systems.
| System | Spring Constant (k) | Typical Displacement | Energy Stored |
|---|---|---|---|
| Soft spring (pen) | 100 N/m | 0.02 m | 0.02 J |
| Car suspension | 30,000 N/m | 0.1 m | 150 J |
| Trampoline | 5,000 N/m | 0.5 m | 625 J |
| Archery bow | 500 N/m | 0.7 m | 122.5 J |
| Garage door spring | 10,000 N/m | 0.3 m | 450 J |
Elastic Potential Energy
Where:
- PE= Elastic potential energy (Joules)
- k= Spring constant (N/m)
- x= Displacement from equilibrium (m)
Conservation of Mechanical Energy
In systems with only conservative forces (gravity, springs), mechanical energy is conserved: the sum of kinetic and potential energy remains constant. This principle simplifies many physics problems—instead of tracking forces and accelerations, we can equate energies at different points.
E_total = KE + PE = constant (in conservative systems)
| Scenario | At Start | At End | Conservation Equation |
|---|---|---|---|
| Falling object | PE max, KE = 0 | KE max, PE = 0 | mgh = ½mv² |
| Pendulum at bottom | KE max, PE min | PE max, KE = 0 | ½mv² = mgh |
| Roller coaster | PE at top | Mix of KE and PE | mgh₁ = ½mv² + mgh₂ |
| Spring launch | ½kx², KE = 0 | KE max, PE = 0 | ½kx² = ½mv² |
| Projectile at apex | KE + PE | PE max, KE (horizontal) | E₁ = E₂ |
Non-conservative forces (friction, air resistance) convert mechanical energy to heat, reducing total mechanical energy over time.
Choosing Reference Points
Potential energy requires a reference point where PE = 0. This choice is arbitrary but affects your calculations. The key insight: only differences in PE have physical meaning, so any consistent reference works.
Common conventions include ground level for gravitational PE, equilibrium position for springs, and infinity for gravitational/electric fields. Choose whichever makes your calculations simplest.
| Situation | Recommended Reference | Why |
|---|---|---|
| Objects falling to ground | Ground = 0 | Final PE = 0, simplifies calculation |
| Pendulum swinging | Lowest point = 0 | KE maximum at reference point |
| Table to floor | Floor = 0 | Natural endpoint |
| Orbital mechanics | Infinity = 0 | Standard convention, PE becomes negative |
| Spring systems | Equilibrium = 0 | Natural rest position |
Warning: Never mix reference points in the same problem. Stay consistent throughout your calculation.
Potential Energy Diagrams
Potential energy diagrams plot PE against position, revealing stable and unstable equilibrium points. These diagrams help visualize how objects behave—like balls rolling in valleys (stable equilibrium) or balancing on hilltops (unstable equilibrium).
The slope of the PE curve gives the force: F = -dPE/dx. Objects naturally move toward lower PE (downhill on the diagram). Points where the slope is zero are equilibrium positions.
| Feature on Diagram | Physical Meaning | Object Behavior |
|---|---|---|
| Valley (minimum) | Stable equilibrium | Returns if displaced |
| Peak (maximum) | Unstable equilibrium | Falls away if displaced |
| Flat region | Neutral equilibrium | Stays where placed |
| Steep slope | Strong force | Rapid acceleration |
| Gentle slope | Weak force | Slow acceleration |
| Turning point | Total energy = PE | KE = 0, direction reverses |
Real-World Applications of Potential Energy
Potential energy concepts underpin many technologies and natural phenomena. Understanding PE helps engineers design efficient systems and explains everyday experiences from bouncing balls to hydroelectric dams.
| Application | PE Type | Energy Conversion | Efficiency |
|---|---|---|---|
| Hydroelectric dam | Gravitational | PE → KE → Electrical | 85-90% |
| Grandfather clock | Gravitational | PE → KE (weights descend) | Nearly 100% |
| Bow and arrow | Elastic | PE → KE (arrow) | 70-80% |
| Pogo stick | Elastic | KE ↔ PE cycling | ~50% |
| Battery | Chemical | Chemical PE → Electrical | 80-95% |
| Pumped hydro storage | Gravitational | Electrical → PE → Electrical | 75-80% |
| Roller coaster | Gravitational | PE ↔ KE cycling | N/A (designed loss) |
Energy storage technologies often utilize potential energy because it's stable and predictable—batteries store chemical PE, dams store gravitational PE, and flywheels store rotational KE.
Worked Examples
Gravitational PE Calculation
Problem:
A 5 kg bowling ball is held 1.5 m above the floor. Calculate its gravitational potential energy relative to the floor.
Solution Steps:
- 1Identify values: m = 5 kg, g = 9.8 m/s², h = 1.5 m
- 2Choose reference: floor = 0 (PE = 0)
- 3Apply formula: PE = mgh
- 4Substitute: PE = 5 × 9.8 × 1.5
- 5Calculate: PE = 73.5 J
Result:
The bowling ball has 73.5 Joules of gravitational potential energy. When dropped, this converts to kinetic energy, reaching v = √(2gh) = 5.42 m/s at impact.
Elastic PE in a Spring
Problem:
A spring with k = 800 N/m is compressed 0.15 m. How much energy is stored? What velocity can it give a 0.2 kg ball?
Solution Steps:
- 1Calculate elastic PE: PE = ½kx²
- 2Substitute: PE = ½ × 800 × (0.15)²
- 3Calculate: PE = 0.5 × 800 × 0.0225 = 9 J
- 4For velocity, use energy conservation: ½kx² = ½mv²
- 5Solve: v = √(kx²/m) = √(800 × 0.0225/0.2) = √90 = 9.49 m/s
Result:
The spring stores 9 Joules of elastic potential energy, which can launch a 0.2 kg ball at 9.49 m/s (34 km/h).
Energy Conservation on a Slide
Problem:
A child (30 kg) starts from rest at the top of a 3 m high slide. Ignoring friction, what is their speed at the bottom?
Solution Steps:
- 1At top: PE = mgh = 30 × 9.8 × 3 = 882 J, KE = 0
- 2At bottom: PE = 0, KE = 882 J (energy conserved)
- 3Use KE formula: 882 = ½ × 30 × v²
- 4Solve for v²: v² = 882 × 2/30 = 58.8
- 5Calculate: v = √58.8 = 7.67 m/s
Result:
The child reaches 7.67 m/s (27.6 km/h) at the bottom. This equals √(2gh), independent of mass—all objects slide at the same final speed regardless of weight.
Tips & Best Practices
- ✓Choose your reference point strategically—usually where the object starts or ends, making one PE value zero.
- ✓Only changes in potential energy matter physically; absolute values depend on your arbitrary reference choice.
- ✓For springs, energy quadruples when displacement doubles (PE ∝ x²), just like kinetic energy with velocity.
- ✓Use energy conservation to avoid complex force/acceleration calculations when you only need initial and final states.
- ✓Near Earth's surface, use PE = mgh; for space applications, use PE = -GMm/r.
- ✓In potential energy diagrams, valleys are stable equilibria and peaks are unstable—like a ball in a bowl vs. on a hill.
- ✓Total mechanical energy (KE + PE) is conserved only when no non-conservative forces (friction, air resistance) act.
Frequently Asked Questions
Sources & References
Last updated: 2026-01-22