KE = Kinetic Energy (Joules) • m = mass (kg) • v = velocity (m/s)
What is Kinetic Energy?
Kinetic energy is the energy an object possesses because of its motion. Every moving object — from a rolling tennis ball to a speeding motorway lorry — carries kinetic energy. The word "kinetic" comes from the Greek kinetikos, meaning "to move." In physics, kinetic energy is one of the two primary forms of mechanical energy; the other is potential energy.
The standard unit of kinetic energy is the Joule (J), named after physicist James Prescott Joule. One Joule equals one kilogram metre squared per second squared (1 J = 1 kg·m²/s²). For large values, kilojoules (kJ) or kilowatt-hours (kWh) are commonly used — especially in engineering and energy contexts.
Deriving KE = ½mv² from the Work-Energy Theorem
The kinetic energy formula is not arbitrary — it follows directly from Newton's second law and the work-energy theorem. Work done on an object (W = F × d) equals the change in kinetic energy. Consider an object of mass m accelerating from rest:
- Force: F = ma (Newton's second law)
- Work done: W = F × d = ma × d
- Using kinematics: v² = u² + 2as, so with u = 0: d = v²/(2a)
- Substituting: W = ma × v²/(2a) = ½mv²
Therefore, the work done to accelerate an object from rest to velocity v equals ½mv². This is exactly the kinetic energy stored in the object. The work-energy theorem states: net work done on an object = change in kinetic energy.
How to Use This Calculator
This calculator has three modes. In Find KE mode, enter the mass and velocity with your preferred units to calculate kinetic energy in Joules, kilojoules, kilowatt-hours and calories. In Find Mass mode, enter kinetic energy (J) and velocity (m/s) to find the object's mass using m = 2KE/v². In Find Velocity mode, enter kinetic energy (J) and mass (kg) to find velocity using v = √(2KE/m).
The calculator also computes momentum (p = mv) alongside kinetic energy wherever mass and velocity are both known, illustrating the deep connection between these two quantities.
Real-World Kinetic Energy Examples
Example 1: Person Walking
A person of mass 70 kg walks at 1.4 m/s (a comfortable walking pace):
KE = ½ × 70 × 1.4² = ½ × 70 × 1.96 = 68.6 J
This is roughly the energy in a small bite of food. Walking is very efficient — human legs recover significant energy from each stride, which is why walking 10,000 steps burns only around 300–400 kcal.
Example 2: Car at 30 mph
A typical car (mass 1,500 kg) travelling at 30 mph (13.4 m/s):
KE = ½ × 1500 × 13.4² = ½ × 1500 × 179.6 = 134,700 J = 134.7 kJ
All this energy must be converted to heat by the brakes when the car stops. At 60 mph, the KE is four times greater (539 kJ), which is why stopping distances quadruple when you double your speed.
Example 3: Cricket Ball at 90 mph
A cricket ball (mass 0.156 kg) bowled at 90 mph (40.2 m/s):
KE = ½ × 0.156 × 40.2² = ½ × 0.156 × 1616 = 126 J
Despite the small mass, the high velocity makes this significant — comparable to a 70 kg person walking. This is why protective gear (helmets, pads) is essential in cricket.
Kinetic Energy and Momentum: The Connection
Momentum (p) and kinetic energy are closely related. Momentum is defined as p = mv, while kinetic energy is KE = ½mv². You can express KE in terms of momentum:
KE = p² / (2m)
This relationship is extremely useful in collision problems. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but some kinetic energy is converted to heat, sound or deformation. Crumple zones in cars are a deliberate engineering application of inelastic collisions — they extend the collision time and reduce the peak force, converting KE to deformation energy more gradually.
Conservation of Energy: KE and PE
The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. The classic example is a roller coaster. At the top of a hill (height h), a car has maximum gravitational potential energy (PE = mgh) and minimum KE (assuming it starts at rest). As it descends, PE converts to KE:
- At the top: PE = mgh, KE = 0
- At the bottom: PE = 0, KE = mgh = ½mv²
- Therefore: v = √(2gh) at the bottom
For example, a 40-metre drop gives a speed of v = √(2 × 9.81 × 40) ≈ 28 m/s (about 100 km/h), which matches typical roller coaster speeds. In practice, friction and air resistance reduce this slightly.
Stopping Distances and KE
One of the most important road safety implications of the kinetic energy formula is the relationship between speed and stopping distance. The Highway Code gives typical stopping distances that increase dramatically with speed:
| Speed | Thinking Distance | Braking Distance | Total Stopping Distance |
|---|---|---|---|
| 20 mph | 6 m | 6 m | 12 m |
| 30 mph | 9 m | 14 m | 23 m |
| 50 mph | 15 m | 38 m | 53 m |
| 70 mph | 21 m | 75 m | 96 m |
Braking distance is proportional to KE (and therefore v²). Doubling speed from 30 to 60 mph quadruples braking distance. At 70 mph, braking distance is over five times what it is at 30 mph. This makes excessive speed disproportionately dangerous.
Wind Turbines and Kinetic Energy
Wind turbines extract kinetic energy from moving air. The power available from wind is given by P = ½ρAv³, where ρ is air density (~1.225 kg/m³), A is the swept area of the rotor, and v is wind speed. Because power depends on the cube of wind speed, a wind turbine produces eight times more power in 20 m/s wind than in 10 m/s wind. A modern offshore turbine with a 100-metre radius rotor at wind speed 12 m/s can generate about 3.5 MW of electrical power.
Elastic vs Inelastic Collisions
Understanding kinetic energy is essential for collision physics:
- Elastic collision: Both momentum and kinetic energy are conserved. Example: billiard ball collisions, atomic-scale gas molecule collisions.
- Perfectly inelastic collision: Objects stick together after colliding. Momentum is conserved but KE is not — some converts to internal energy (heat, sound, deformation). Example: car crashes, clay ball dropped on a surface.
- Partially inelastic collision: Most real-world collisions. Some KE is lost but objects do not stick together. Example: tennis ball bouncing (coefficient of restitution < 1).
GCSE and A-Level Physics: Key Points
For GCSE Physics (AQA, Edexcel, OCR), you need to know:
- KE = ½mv² (given on the formula sheet in most specifications)
- GPE = mgh
- Conservation of energy (energy is not lost, just transferred)
- Efficiency = (useful energy output / total energy input) × 100%
For A-Level Physics, you additionally need:
- Work-energy theorem derivation
- Elastic and inelastic collisions
- KE = p²/(2m) and its applications
- Power = rate of energy transfer = Fv
- Relativistic kinetic energy at very high speeds (KE = (γ - 1)mc²)
Unit Conversions for KE
| Unit | Equivalent in Joules | When Used |
|---|---|---|
| 1 Joule (J) | 1 J | SI unit, physics problems |
| 1 kilojoule (kJ) | 1,000 J | Food energy, engineering |
| 1 kilowatt-hour (kWh) | 3,600,000 J | Electricity bills |
| 1 calorie (cal) | 4.184 J | Chemistry, nutrition |
| 1 kilocalorie (kcal) | 4,184 J | Food energy (dietary "Calories") |
| 1 electronvolt (eV) | 1.602 × 10²⁻¹⁹ J | Atomic/nuclear physics |
| 1 BTU | 1,055 J | Heating systems |