Kinetic Energy Calculator

About the Kinetic Energy Calculator

Kinetic energy is the energy an object has because it moves. For something travelling in a straight line it is K = ½mv²; for something spinning it is K = ½Iω², where I is the moment of inertia. This calculator solves either formula for any one unknown — the energy, the mass (or moment of inertia) or the speed — and converts between joules, foot-pounds, kilocalories, kilowatt-hours and electronvolts, kilograms, pounds and grains, and m/s, km/h, mph and ft/s.

Every straight-line answer is also checked against Einstein’s relativistic formula K = (γ − 1)mc². At everyday speeds the two agree to many decimal places; once an object moves faster than about a tenth of the speed of light the classical formula is noticeably low, so the calculator switches to the relativistic value and tells you by how much ½mv² is off. The spinning mode works out I for ten standard shapes and can add the forward motion of a ball, wheel or pipe that rolls without slipping.

Common uses

• Physics homework on kinetic energy, the work–energy theorem, rotation and rolling, with the substitution shown.
• Archery: arrow energy in foot-pounds from the arrow weight in grains and the chronograph speed in ft/s.
• Comparing the energy of a vehicle at two speeds — doubling the speed means four times the energy to dissipate when braking.
• Particle physics: the speed of an electron or proton from its kinetic energy in eV, keV or MeV.
• Flywheels, wheels and rotors: the energy stored at a given rpm.

The formulas

• Moving object: K = ½ m v², so m = 2K ÷ v² and v = √(2K ÷ m). Energy grows with the square of the speed: double the speed and the energy goes up four times.
• Spinning object: K = ½ I ω² with ω in radians per second (1 rpm = 2π/60 rad/s).
• Rolling without slipping: the centre moves at v = ωR, so K = ½ I ω² + ½ M v². The split is fixed by the shape: a solid sphere keeps 2/7 of its energy in rotation, a solid cylinder 1/3 and a hoop 1/2.
• Relativistic: K = (γ − 1) m c² with γ = 1 ÷ √(1 − v²/c²) and c = 299,792,458 m/s. From an energy, γ = 1 + K ÷ (mc²) and v = c √(1 − 1/γ²).

When the classical formula stops being accurate

The relativistic energy is always the larger one. How far it exceeds ½mv²:

• 1 % of c (3,000 km/s): 0.0075 %
• 10 % of c: 0.76 %
• 50 % of c: the relativistic value is 23.8 % higher (γ = 1.155)
• 99 % of c: 12.4 times higher (γ = 7.089)

The calculator uses ½mv² while it is within 1 % (below about 11.5 % of c, where the difference passes 1 %) and mentions the correction from 0.01 % on. A mass can never reach c: the energy needed grows without limit, so speeds of c or more are rejected.

Moments of inertia used for shapes

From OpenStax University Physics Vol. 1, Figure 10.20 (M = mass):

• Solid cylinder or disc about its axis: ½MR² (rolls)
• Hoop or thin ring about its axis: MR² (rolls)
• Thick ring or pipe about its axis: ½M(R₁² + R₂²) (rolls on the outer radius)
• Solid sphere about a diameter: ⅖MR² (rolls)
• Thin spherical shell: ⅔MR² (rolls)
• Thin rod about its centre: ML²/12; about one end: ML²/3
• Hoop about a diameter: ½MR²
• Solid cylinder about a central diameter: ¼MR² + ML²/12
• Rectangular slab about a perpendicular axis through its centre: M(a² + b²)/12

For anything else, type I directly. Moments of inertia add, so several identical parts can be entered as one part with their combined mass when they share the same axis and shape.

Units for archery and ballistics

Arrow and projectile weights are usually given in grains (1 gr = 1/7000 lb = 64.79891 mg exactly) and speeds in feet per second. With exact conversions, KE (ft·lbf) = grains × (ft/s)² ÷ 450,437. The widely quoted archery shortcut divides by 450,240 instead, which gives values 0.04 % higher — the difference is far smaller than chronograph error. Momentum p = mv is shown in kg·m/s and in lbf·s (the same as slug·ft/s), the unit archery charts use.

Sources

• OpenStax, University Physics Volume 1 (CC BY 4.0): §7.2 Kinetic Energy (Eq. 7.6, Examples 7.6–7.8) and §10.4 Moment of Inertia and Rotational Kinetic Energy (Figure 10.20, Examples 10.9 and 10.10); §11.1 Rolling Motion (v = Rω).
• OpenStax, University Physics Volume 3 (CC BY 4.0): §5.9 Relativistic Energy (Example 5.12).
• BIPM, The International System of Units (SI Brochure, 9th ed., 2019): exact c, h and e (1 eV = 1.602176634 × 10⁻¹⁹ J).
• NIST, CODATA 2022 recommended values: electron, proton, neutron and alpha-particle masses, atomic mass constant.
• NIST, SP 811 Appendix B.8: grain, foot-pound, calorie, Btu and other conversion factors.

Limitations

• Speeds are relative to whatever frame you measure them in; kinetic energy depends on that choice (OpenStax Example 7.7).
• The spinning mode assumes a rigid body turning about a fixed axis; for an axis that does not pass through the centre of mass, add Md² to I (parallel-axis theorem) and enter I directly.
• Rolling assumes no slipping and no deformation; a skidding or bouncing object does not obey v = ωR.
• The moments of inertia assume uniform density and ideal shapes (a thin rod has negligible thickness, a hoop negligible wall).
• Rest energy mc² and kinetic energy are different things: this tool reports only the kinetic part, (γ − 1)mc².
• Very close to the speed of light a typed speed pins the energy down poorly (each extra 9 in 0.999… c multiplies it by about 3.2), so for fast particles enter the energy and solve for the speed.

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