KE = 1/2 m v2 - kinetic energy of a moving object with multiple unit options.
KE = ½ · m · v². Doubling velocity quadruples kinetic energy — the safety reason why a small speed reduction makes a big crash difference.
Kinetic energy is the canonical physical quantity: how much energy is stored in a moving object, computable from just its mass and velocity. It explains why a 50 km/h crash is so much more dangerous than a 25 km/h crash (4× the energy, not 2×), why bullets are tiny but devastating (low mass, very high velocity → squared), why high-speed trains need elaborate brakes (huge mass × moderate speed), and why a flywheel is a useful energy store (rotational KE, same formula). The classroom formula KE = ½ m v² packs more counterintuitive consequences than almost any other equation in physics. This calculator computes KE in joules, kilojoules, kilocalories and watt-hours from mass and velocity entered in any of the common units, plus shows momentum, plus visualizes the parabolic KE-vs-velocity relationship.
Kinetic energy (translational, non-relativistic): KE = ½ · m · v², with m in kg and v in m/s yields KE in joules. The calc converts inputs to SI internally:
Output conversions: - Joules → kilojoules (÷ 1 000). - Joules → food calories (÷ 4 184) — useful as a tangible reference. - Joules → watt-hours (÷ 3 600) — useful for comparison to electrical energy.
Momentum = m · v (kg·m/s) is shown as a side metric — momentum is conserved in collisions while KE is not (some KE converts to deformation, heat, sound).
The chart plots KE vs velocity from 0 to 1.5× the input velocity. The parabolic shape (KE doubles when v increases by ~41 %, quadruples when v doubles) is the visual lesson — the safety implication of speed limits is geometric, not linear.
Enter the mass of the moving object in your preferred unit (kg, g, t, lb). Enter the velocity in km/h, m/s, mph, or knots. The calc returns:
Car at 50 km/h, mass 1 500 kg.
Same car at 100 km/h: v = 27.78 m/s, KE = 0.5 × 1500 × 771.6 = 579 kJ — 4× the energy at 50 km/h.
A 9 mm bullet, mass 8 g, velocity 360 m/s.
A 70 kg sprinter at 36 km/h (10 m/s).
Non-relativistic only. ½ m v² assumes v ≪ c (speed of light, 3 × 10⁸ m/s). For relativistic speeds, KE = (γ − 1) m c², where γ = 1 / √(1 − v²/c²). At 1 % of c (3 000 km/s), the relativistic correction is 0.005 %; ignorable. At 50 % of c, γ ≈ 1.155, and the difference between non-relativistic and relativistic KE is +30 %. Particle physics (electrons, protons in accelerators) absolutely requires relativistic.
Translational only. Rotational KE = ½ I ω², where I is moment of inertia and ω angular velocity. A spinning flywheel has rotational KE in addition to any translational KE. Out of scope here.
Reference frame matters. KE is frame-dependent. Two cars at 50 km/h moving toward each other have closing speed 100 km/h in each other's reference frame, so the head-on collision releases 4× the per-vehicle KE per car. In the road's frame, each car has KE_50 and the total is 2 × KE_50; in either car's frame, the other car has KE_100 and the cost is the same. The formula gives the magnitudes; the safety lesson is the asymmetry.
Air resistance and rolling friction. Real-world objects don't conserve KE freely — air drag converts some to heat continuously. KE = ½ m v² is the instantaneous energy; the energy needed to accelerate an object from rest to v is at least ½ m v² but is usually more.
Mass at velocity vs rest mass. Special-relativity convention: m here is rest mass; relativistic KE adds γ. Don't confuse with "relativistic mass" (m_rel = γ m), an outdated concept that introduces errors.
Not the same as work. Work-energy theorem: net work done on an object = change in KE. A car coasting at constant speed has a net work of 0 (drag balances drive). Work depends on path; KE is a state function.
Bullet penetration is not all KE. Wound ballistics depends on KE transfer into tissue, not just KE delivered. A high-velocity bullet that passes through cleanly transfers less KE than a low-velocity slug that flattens.
Energy ≠ damage / impact. KE is one input to crash damage, but vehicle structure, deceleration distance, and angle of impact dominate the practical outcome. A 50 km/h crash into a tree is much worse than a 50 km/h crash into a deformable barrier — same KE, different deceleration.