Calculate gravitational potential energy (PE = mgh) and elastic potential energy (PE = ½kx²) with step-by-step physics solutions. Supports multiple unit systems for mass, height, spring constant, and displacement.
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Potential Energy (PE)
0
Joules (J)
Mode
Gravitational
Calculation type used
Alternative Units
—
Converted value
📝 Step-by-Step Solution
Real-World Potential Energy Examples
🏔️ Boulder on a Cliff
Problem: A 250 kg boulder sits on a cliff edge 30 m above the ground. What is its gravitational potential energy? (Use g = 9.81 m/s²)
Solution: Using PE = mgh
PE = 250 × 9.81 × 30 = 73,575 J (73.6 kJ)
That's enough energy to power a 1 kW heater for over a minute! This is why falling rocks are so dangerous.
🏀 Basketball at Peak
Problem: A 0.62 kg basketball reaches a height of 3.5 m above the ground at the peak of its arc. What is its gravitational potential energy?
Solution: Using PE = mgh
PE = 0.62 × 9.81 × 3.5 = 21.29 J
At the peak, all kinetic energy has been converted to potential energy. When the ball falls back down, this energy converts back to kinetic energy.
🧸 Compressed Spring
Problem: A spring with constant k = 500 N/m is compressed by 0.1 m. What elastic potential energy is stored?
Solution: Using PE = ½kx²
PE = ½ × 500 × (0.1)² = 250 × 0.01 = 2.50 J
This energy would be released as the spring returns to its equilibrium position. Springs store energy through elastic deformation.
🎯 Bow and Arrow
Problem: An archer draws a bow with an effective spring constant of 120 N/m, pulling the string back 0.45 m. How much elastic potential energy is stored?
Solution: Using PE = ½kx²
PE = ½ × 120 × (0.45)² = 60 × 0.2025 = 12.15 J
When released, this stored energy converts to kinetic energy of the arrow. The arrow's speed depends on its mass and this energy.
Potential Energy Formulas & Guide
PE = m × g × h
Gravitational Potential Energy
Where m is mass (kg), g is gravitational acceleration (9.81 m/s² on Earth), and h is height above a reference point (m).
PE = ½ × k × x²
Elastic (Spring) Potential Energy
Where k is the spring constant (N/m) and x is displacement from equilibrium (m). The energy scales with the square of displacement.
Key Concepts
📌 What is Potential Energy?
Potential energy is stored energy that has the potential to do work. It depends on the position or configuration of an object within a force field (gravitational, elastic, electric, etc.). It is a scalar quantity measured in joules (J).
📌 Gravitational vs Elastic PE
Gravitational PE depends on mass, height, and gravity (PE = mgh). Elastic PE depends on spring stiffness and displacement squared (PE = ½kx²). Both represent stored energy that can be converted to other forms like kinetic energy.
📌 Conservation of Energy
The total mechanical energy (kinetic + potential) of a system is conserved in the absence of non-conservative forces like friction. As an object falls, its gravitational PE converts to kinetic energy. A compressed spring releases elastic PE as kinetic energy.
📌 Reference Points Matter
Gravitational potential energy is always relative to a chosen reference point (e.g., ground level). The absolute value of PE is not physically meaningful — only changes in PE matter. You can set any height as the zero reference point.
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Gravitational PE Mode
Calculate PE = mgh with mass, height, and customizable gravitational acceleration. Supports kg, g, lb, oz for mass and m, km, ft, in for height.
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Elastic PE Mode
Calculate PE = ½kx² with spring constant and displacement. Supports N/m, N/cm, lb/in for k and m, cm, in, ft for displacement.
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Multiple Units
Results shown in joules (J) with automatic conversions to kJ, calories (cal), and watt-hours (Wh) for gravitational PE or kJ for elastic PE.
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Step-by-Step Solutions
Every calculation comes with a detailed step-by-step breakdown showing the formula, substitution, and final result with proper units.
⚠️ Important Note: Gravitational potential energy is always relative to a chosen reference point. The calculated value assumes a uniform gravitational field (constant g). Elastic potential energy assumes the spring obeys Hooke's Law (linear region) and is not permanently deformed. For real-world systems, factors like air resistance, friction, and non-linear spring behavior may affect actual stored energy.
Frequently Asked Questions
What is potential energy in physics?
Potential energy is the energy stored in an object due to its position or configuration. It is called "potential" because it has the potential to be converted into other forms of energy, such as kinetic energy. The two most common types in introductory physics are gravitational potential energy (energy due to height in a gravitational field) and elastic potential energy (energy stored in a deformed spring or elastic material). The SI unit is the joule (J).
How do you calculate gravitational potential energy?
Gravitational potential energy is calculated using the formula PE = mgh, where m is the mass of the object (in kg), g is the gravitational acceleration (9.81 m/s² on Earth), and h is the height above a reference point (in meters). For example, a 5 kg book on a 2 m high shelf has PE = 5 × 9.81 × 2 = 98.1 J of gravitational potential energy relative to the ground.
What is elastic potential energy and how is it calculated?
Elastic potential energy is the energy stored in a deformed elastic object, such as a stretched spring or rubber band. It is calculated using PE = ½kx², where k is the spring constant (stiffness, in N/m) and x is the displacement from the equilibrium position (in meters). The energy increases with the square of displacement — compressing a spring twice as far stores four times the energy.
What is the difference between potential energy and kinetic energy?
Potential energy is stored energy based on position or configuration (e.g., a ball held at a height, a compressed spring). Kinetic energy is the energy of motion (KE = ½mv²). The key relationship is the conservation of mechanical energy: in the absence of friction, PE + KE = constant. As an object falls, its gravitational PE decreases while its KE increases by the same amount.
Can potential energy be negative?
Yes, potential energy can be negative because it is always measured relative to a reference point. If you set the reference point at a certain height, objects below that point will have negative gravitational potential energy. The absolute value is not physically meaningful — only changes in potential energy matter. For elastic potential energy, the squared term (x²) ensures it is always positive for any non-zero displacement.
How does potential energy relate to work?
The work-energy theorem states that the work done on an object equals the change in its kinetic energy. For conservative forces (like gravity and springs), the work done equals the negative change in potential energy: W = -ΔPE. Lifting an object increases its gravitational PE by the amount of work done against gravity. When the object falls, that potential energy is converted back into work or kinetic energy.