Kinetic Energy Calculator

Solves the equation KE = ½mv² for any of its three variables given the other two. Input units default to kilograms and meters per second; output in joules. Optional unit toggles support pounds, mph, and foot-pounds for engineering contexts.

Valid for non-relativistic speeds. Above roughly 10% of c, the relativistic form γmc² − mc² gives noticeably different results.

Kinetic energy is the work required to accelerate a body of mass m from rest to velocity v. The formula KE = ½mv² is derived from the work-energy theorem: integrating force over displacement, with F = ma and a = dv/dt, gives W = ∫F dx = ∫m(dv/dt)(dx/dt)dt = ∫mv dv = ½mv² when integrated from 0 to v. The quadratic dependence on velocity is the most consequential feature: doubling speed quadruples kinetic energy, and the practical implications for vehicle crash safety, projectile damage, and orbital mechanics all flow from this.

A worked example: a 1500 kg car traveling at 60 mph (26.82 m/s) has KE = 0.5 × 1500 × 26.82² = 539,500 joules. Doubling to 120 mph quadruples this to 2,158,000 joules. The same vehicle hitting a wall at 60 mph dissipates 540 kJ; at 120 mph it dissipates 2.16 MJ, four times the energy in the same crumple zone. This is why high-speed crashes are disproportionately fatal: the energy that must be absorbed scales with v², not v.

For smaller-scale examples: a 9 mm bullet (8 g) at 360 m/s has KE = 0.5 × 0.008 × 360² = 518 J. A baseball (145 g) at 90 mph (40 m/s) has KE = 116 J. A 70 kg person walking at 1.4 m/s carries 68.6 J of kinetic energy. The bullet's small mass at high velocity carries more energy than the much larger walking person, illustrating why velocity dominates the equation through its square term.

Limitations come from the non-relativistic assumption. Above approximately 10% of c (30,000 km/s), the Lorentz factor γ = 1/√(1−v²/c²) exceeds 1.005, and the classical formula underestimates true kinetic energy by more than 0.5%. The relativistic kinetic energy is (γ−1)mc², which approaches infinity as v approaches c. For everyday speeds (cars, bullets, satellites in low Earth orbit at ~7.8 km/s), the classical formula is accurate to many decimal places. Particle accelerators and cosmic-ray physics require the relativistic form throughout. Rotational kinetic energy (½Iω²) is a separate quantity; for a rolling object, total KE is the sum of translational and rotational components.