Kinetic and Potential Energy Formulas
Mechanical energy comes in two everyday forms. Kinetic energy is the energy an object has because it is moving. Gravitational potential energy is the energy an object has because of its height in a gravitational field. Both are measured in joules (J), where 1 J = 1 kg·m²/s².
KE = ½ · m · v² PE = m · g · h
- m = mass in kilograms
- v = velocity in meters per second
- h = height above a chosen reference level in meters
- g = standard gravity, 9.80665 m/s² (often rounded to 9.8 or 9.81)
Rearranging gives the other unknowns: m = 2KE / v², v = √(2KE / m), m = PE / (g·h), and h = PE / (m·g).
Worked Examples
Kinetic energy: a 1,000 kg car moving at 20 m/s (72 km/h) has KE = ½ × 1,000 × 20² = 200,000 J (200 kJ). Doubling the speed to 40 m/s quadruples the kinetic energy to 800 kJ, which is why stopping distances grow so quickly with speed.
Potential energy: a 10 kg box lifted 5 m has PE = 10 × 9.80665 × 5 = 490.33 J. If it falls, that energy converts to kinetic energy, so it would hit the ground at about √(2 × 9.80665 × 5) ≈ 9.90 m/s, ignoring air resistance.
Conservation of Mechanical Energy
Without friction or air resistance, the sum KE + PE stays constant. A falling object loses potential energy and gains the same amount of kinetic energy. Setting mgh = ½mv² gives v = √(2gh), which is independent of mass.
Frequently Asked Questions
What is the formula for kinetic energy?
Kinetic energy equals one half times mass times velocity squared: KE = ½mv². With mass in kilograms and velocity in meters per second, the result is in joules.
What is the formula for gravitational potential energy?
PE = mgh, where m is mass in kilograms, g is gravitational acceleration (9.80665 m/s² standard), and h is height in meters above a chosen reference level.
Why does kinetic energy increase so much with speed?
Because velocity is squared. Doubling speed multiplies kinetic energy by four, and tripling speed multiplies it by nine.
Can potential energy be negative?
Yes. Potential energy is measured relative to a reference level you choose. An object below that level has negative gravitational potential energy.
How many joules are in a kilowatt-hour?
One kilowatt-hour equals 3,600,000 joules. One food calorie (kcal) equals 4,184 joules.