Kinetic and potential energy calculator
kg
m/s
J
Kinetic energy:
No friction. Move the position: kinetic energy (blue) and potential energy (orange) change, but their sum, the total energy, does not. The cart never rises above the starting height. With kinetic energy, the starting height is the drop on Earth (g = 9.81 m/s²) that gives that energy.
Energy in kilojoules:
Drop height on Earth with the same energy:
Calculation:
This kinetic and potential energy calculator solves the two basic formulas of mechanical energy: kinetic, KE = ½ × m × v², and gravitational potential, PE = m × g × h. You can find the energy or solve for the mass, speed, height or gravity, depending on the data you have. An animated roller coaster and energy bars show how the kinetic and potential energy change as the cart moves.
The sum of the two, the total energy, stays constant when there is no friction, and that is conservation of energy. Below you will find the formulas, worked examples and tables.
How to use the kinetic and potential energy calculator
Step 1: Choose the formula: Kinetic energy (uses mass and speed) or Potential energy (uses mass, gravity and height).
Step 2: In What to find, pick the energy or the quantity you want to solve for. That field fills itself in and is shaded.
Step 3: Enter the other values (mass in kg, speed in m/s, height in m, gravity in m/s²) and click Calculate or press Enter. Energy is given in joules (J).
Step 4: Move the Position control along the track: you will see how the total energy splits between kinetic and potential at each point.
Kinetic and potential energy formulas
Kinetic energy is the energy of motion. It depends on mass and, even more, on speed, because speed is squared:
| KE = ½ × m × v² |
Gravitational potential energy is what an object stores by being at a height h above the reference level, with gravity g:
| PE = m × g × h |
The solved forms follow: m = 2 × KE ÷ v² and v = √(2 × KE ÷ m) for kinetic; m = PE ÷ (g × h), g = PE ÷ (m × h) and h = PE ÷ (m × g) for potential. The unit of energy is the joule: 1 J = 1 kg × m²/s².
Without friction, KE + PE is constant. An object that falls from height H loses potential energy and gains the same amount of kinetic energy, and it arrives at the bottom with KE = m × g × H, that is, v = √(2 × g × H). The mass cancels out.
Kinetic and potential energy examples
Example 1: The kinetic energy of a car
A 1000 kg car travels at 20 m/s (44.74 mph):
KE = ½ × 1000 × 20² = 200000 J
That is 200 kJ. At twice the speed it would have four times the energy.
Example 2: The speed of a baseball
A 0.145 kg baseball has 100 J of kinetic energy. We solve for v:
v = √(2 × 100 ÷ 0.145) ≈ 37.14 m/s
That is 83.08 mph.
Example 3: The potential energy of a person climbing
A 70 kg person climbs 3 m (about one flight of stairs) on Earth:
PE = 70 × 9.81 × 3 = 2060.1 J
Example 4: Conservation of energy in a fall
A 2 kg ball is dropped from 5 m. At the top it has only potential energy:
PE = 2 × 9.81 × 5 = 98.1 J
Without friction, all of it becomes kinetic energy at the ground: ½ × 2 × v² = 98.1, so v = 9.905 m/s. With a different mass the speed would be the same.
Kinetic energy by speed
For a 1000 kg car. The energy grows with the square of the speed:
| Speed (m/s) | Speed (km/h) | Kinetic energy (kJ) |
|---|---|---|
| 5 | 18 | 12.5 |
| 10 | 36 | 50 |
| 15 | 54 | 112.5 |
| 20 | 72 | 200 |
| 25 | 90 | 312.5 |
| 30 | 108 | 450 |
How the energy changes in a 10 m fall
A 1 kg ball dropped from 10 m on Earth (g = 9.81 m/s²), without friction:
| Height (m) | Potential (J) | Kinetic (J) | Total (J) | Speed (m/s) |
|---|---|---|---|---|
| 10 | 98.1 | 0 | 98.1 | 0 |
| 7.5 | 73.58 | 24.53 | 98.1 | 7.004 |
| 5 | 49.05 | 49.05 | 98.1 | 9.905 |
| 2.5 | 24.53 | 73.58 | 98.1 | 12.13 |
| 0 | 0 | 98.1 | 98.1 | 14.01 |
Frequently asked questions
What is the formula for kinetic energy?
KE = ½ × m × v². A 1000 kg car at 20 m/s has ½ × 1000 × 20² = 200000 J.
What is the formula for potential energy?
PE = m × g × h. A 70 kg person at a height of 3 m has 70 × 9.81 × 3 = 2060.1 J relative to the ground.
Why is the speed squared?
Because speeding up an object takes work over a distance, and at higher speed that distance grows too. The result is that the energy to stop it quadruples when the speed doubles.
What is conservation of mechanical energy?
It says that, without friction or other losses, the sum of kinetic and potential energy does not change. With friction, part of the energy turns into heat and the cart does not climb as high.
Where is the height measured from?
From the reference level you choose, usually the ground. What matters physically is the difference in height, not the zero.
Learn more
- Acceleration due to gravity in physics
- Solved problems on free fall in physics
- Vertical projectile motion: formulas and examples
- Velocity vs speed in physics
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