Potential Energy Calculator — Step-by-Step Solutions

The Potential Energy Calculator solves energy problems in physics with full step-by-step working. Calculate gravitational potential energy (PE = mgh), elastic potential energy (PE = 1/2 kx squared), kinetic energy (KE = 1/2 mv squared), and energy conservation between potential and kinetic energy.

Enter your values in any mode and get instant results. The calculator includes detailed formula breakdowns and interpretations for each calculation.

Mass (kg)

Height (m)

Gravity (m/s²)

Your calculations will appear here

How to Use the Potential Energy Calculator

Select a calculation mode: Gravitational PE, Elastic PE, Kinetic Energy, or Energy Conservation.
For Gravitational PE, enter mass (kg), height (m), and gravity (default 9.81 m/s squared). The calculator computes PE = mgh.
For Elastic PE, enter the spring constant k (N/m) and displacement x (m). The calculator computes PE = 0.5 times k times x squared.
For Kinetic Energy, enter mass (kg) and velocity (m/s). The calculator computes KE = 0.5 times m times v squared.
For Energy Conservation, enter mass, initial height, initial velocity, and final height. The calculator finds the final velocity assuming no energy losses.

Formula and Theory

Potential energy is stored energy due to an object's position or configuration. Gravitational potential energy (PE = mgh) depends on mass, gravitational field strength, and height above a reference point.

Elastic potential energy (PE = 0.5 times k times x squared) is stored in a deformed elastic object such as a spring. It depends on the spring constant k (stiffness) and the displacement x from the natural length.

Kinetic energy (KE = 0.5 times m times v squared) is the energy of motion. It depends on mass and the square of velocity, so doubling the speed quadruples the kinetic energy.

The law of conservation of energy states that in an isolated system, total energy remains constant. For mechanical systems without friction, the sum of potential and kinetic energy stays the same: PE_i + KE_i = PE_f + KE_f. This lets you find unknown velocities or heights.

Worked Examples

Example 1: Gravitational PE

Problem: A 10 kg object is 5 m above the ground. What is its gravitational PE? (g = 9.81)

Solution: PE = mgh = 10 * 9.81 * 5 = 490.5.

Answer: 490.5 J

Example 2: Elastic PE

Problem: A spring with k = 200 N/m is compressed by 0.1 m. What is the elastic PE?

Solution: PE = 0.5 * k * x^2 = 0.5 * 200 * (0.1)^2 = 0.5 * 200 * 0.01 = 1.

Answer: 1 J

Example 3: Kinetic energy

Problem: A 2 kg ball moves at 8 m/s. What is its kinetic energy?

Solution: KE = 0.5 * m * v^2 = 0.5 * 2 * 64 = 64.

Answer: 64 J

Example 4: Energy conservation

Problem: A 3 kg ball is dropped from 20 m. What is its speed just before hitting the ground?

Solution: mgh = 0.5mv^2. v = sqrt(2gh) = sqrt(2 * 9.81 * 20) = sqrt(392.4) = 19.81.

Answer: 19.81 m/s

Frequently Asked Questions

What is the difference between potential and kinetic energy?

Potential energy is stored energy based on position or configuration (height, spring compression). Kinetic energy is energy of motion. They can convert back and forth, as when a ball falls (PE converts to KE).

Does the mass cancel in energy conservation problems?

When only gravity does work and you are comparing PE and KE, mass cancels: mgh = 0.5mv squared simplifies to v = sqrt(2gh). The final speed is independent of mass.

What if there is friction?

With friction, some mechanical energy is converted to heat. The energy conservation equation becomes PE_i + KE_i = PE_f + KE_f + W_friction. This calculator assumes no friction.

Can potential energy be negative?

Yes. Gravitational PE depends on the reference point. If the object is below the reference height, PE is negative. Only differences in PE have physical meaning.

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