Kinetic energy is the energy an object possesses due to its motion. It's one of the most fundamental concepts in physics — and the formula is elegantly simple.
The Kinetic Energy Formula
KE = ½ × m × v²
Where:
- KE = kinetic energy in Joules (J)
- m = mass in kilograms (kg)
- v = velocity in metres per second (m/s)
Worked Examples
Example 1: A Moving Car
A 1,500 kg car travelling at 20 m/s (72 km/h):
- KE = ½ × 1,500 × 20²
- KE = ½ × 1,500 × 400
- KE = 300,000 J = 300 kJ
Example 2: A Baseball Pitch
A 0.145 kg baseball thrown at 40 m/s (144 km/h):
- KE = ½ × 0.145 × 40²
- KE = ½ × 0.145 × 1,600
- KE = 116 J
Example 3: A Running Person
A 70 kg person running at 4 m/s (~14.4 km/h):
- KE = ½ × 70 × 16
- KE = 560 J
Units and Conversions
| Unit | Equivalent |
|---|---|
| 1 Joule (J) | 1 kg·m²/s² |
| 1 kilojoule (kJ) | 1,000 J |
| 1 calorie (cal) | 4.184 J |
| 1 kilocalorie (kcal) | 4,184 J |
| 1 watt-hour (Wh) | 3,600 J |
| 1 electron-volt (eV) | 1.602 × 10⁻¹⁹ J |
To convert kinetic energy to calories: KE (cal) = KE (J) ÷ 4.184
The Velocity-Squared Relationship
The most important insight from KE = ½mv² is that kinetic energy scales with the square of velocity:
| Speed Increase | KE Increase |
|---|---|
| 2× faster | 4× more KE |
| 3× faster | 9× more KE |
| 10× faster | 100× more KE |
This is why:
- Doubling highway speed doesn't double stopping distance — it quadruples it
- A bullet at twice the speed carries four times the destructive energy
- Wind turbine power output is proportional to v³ (velocity cubed), not v²
Calculating Velocity from Kinetic Energy
v = √(2 × KE ÷ m)
Example: A 2 kg object has 200 J of kinetic energy. What is its speed?
- v = √(2 × 200 ÷ 2) = √200 = 14.14 m/s
Calculating Mass from Kinetic Energy and Velocity
m = 2 × KE ÷ v²
Example: An object has 500 J of KE and travels at 10 m/s. What is its mass?
- m = (2 × 500) ÷ 100 = 10 kg
The Work-Energy Theorem
The net work done on an object equals its change in kinetic energy:
W = ΔKE = KE_final − KE_initial = ½mv_f² − ½mv_i²
Example: A car accelerates from 10 m/s to 25 m/s. Mass = 1,200 kg:
- ΔKE = ½ × 1,200 × (25² − 10²)
- ΔKE = 600 × (625 − 100)
- ΔKE = 600 × 525 = 315,000 J of work done by the engine
Kinetic vs Potential Energy
| Kinetic Energy | Potential Energy | |
|---|---|---|
| Definition | Energy of motion | Energy of position/configuration |
| Formula | ½mv² | mgh (gravitational) |
| Depends on | Velocity | Height, field strength |
In a closed system with no friction, total mechanical energy is conserved:
KE + PE = constant
½mv² + mgh = constant
A ball falling from height h: as h decreases, v increases — potential energy converts to kinetic energy.
Relativistic Kinetic Energy (High-Speed Objects)
At speeds approaching the speed of light, the classical formula breaks down. Einstein's relativistic formula:
KE = (γ − 1) × mc²
Where γ = 1 ÷ √(1 − v²/c²) is the Lorentz factor. At everyday speeds (v << c), this reduces to the classical ½mv².
Use our speed distance time calculator to work with velocity values, then apply the KE formula to find the energy of any moving object.