Kinetic Energy

Insert the mass in kilograms and the speed in metres per second, square the speed, multiply by the mass and halve it: E_kin = ½·m·v². The result comes out in joules, since 1 J = 1 kg·m²/s². A car of 1500 kg at 100 km/h, that is 27.8 m/s, has E_kin = ½·1500·27.8² ≈ 579 kJ. The most important step is converting km/h into m/s by dividing by 3.6. Also do not forget the factor ½, which is often dropped in exams and would otherwise wrongly double the result.

In E_kin = ½·m·v² the speed enters quadratically. Doubling v gives (2v)² = 4v², so the energy becomes four times as large. At triple speed it is even nine times. This quadratic relationship is why braking distance rises steeply with speed: twice as fast means four times the kinetic energy, which the brakes must dissipate as heat, and therefore roughly four times the braking distance. Mass, by contrast, enters only linearly, so a body twice as heavy has merely twice the kinetic energy at the same speed.

First multiply both sides by 2, then divide by the mass and take the square root: v = √(2·E_kin/m). The speed is therefore the square root of twice the energy divided by the mass. Example: a body of 2 kg with 100 J of kinetic energy has v = √(2·100/2) = √100 = 10 m/s. Watch the order of operations; the 2 belongs under the root, not in front of it. Insert the energy in joules and the mass in kilograms so the speed comes out in metres per second. Convert the result to km/h with a factor of 3.6 if needed.

E_kin = ½·m·v² is a non-relativistic approximation and is very accurate as long as the speed is small compared with the speed of light c. Only when v reaches roughly a tenth of c or more does the classical formula deviate noticeably and you need the relativistic expression E_kin = (γ−1)·m·c². In particle physics, for example with electrons in accelerators, this is the normal case. For all everyday and school problems involving vehicles, balls or falling bodies, however, v is far below c, so the classical formula gives exact results.

Kinetic energy is the energy of motion and depends on mass and speed, E_kin = ½·m·v². Potential energy is stored positional energy and, in a gravitational field, depends on mass, the local factor and height, E_pot = m·g·h. During a free fall potential energy converts into kinetic energy: the height the body loses it gains as speed. Their sum, the total mechanical energy, stays constant without friction. That is the conservation of mechanical energy. In exams you often use it, via m·g·h = ½·m·v², to find the final speed without knowing the exact fall time.