# Energy of motion and inertia

Authored explanation; editorial review pending.

Work transfers energy when a force acts through a displacement. The energy associated with the motion of a body is called kinetic energy. It is not the same as the energy of its internal heating, chemistry, or other stored processes.

For the same body at ordinary slow speeds, doubling speed multiplies its energy of motion by four. At a fixed speed, doubling inertial mass doubles that energy. The word inertia describes resistance to a change of motion, not a measurement of gravitational weight.

$$
K=\frac{1}{2}mv^2
$$

In Newtonian mechanics, kinetic energy is one half times inertial mass times speed squared.

The formula is the low-speed rule. It cannot be assumed exact for a traveler moving at a substantial fraction of light speed. To identify inertia in a relativistic comparison, examine the coefficient as speed approaches zero.

## Worked example

Compare two bodies at 2 metres per second. In the Newtonian model a body of 3 kilograms has 6 joules of energy of motion; a body of 2 kilograms has 4 joules. The speed is unchanged, but the energy of motion differs by 2 joules.

$$
K_0-K_1=\frac{1}{2}(m_0-m_1)v^2
$$

At the same low speed, the kinetic-energy drop equals one half times the mass decrease times the squared speed.

This example assumes independently specified masses; it is not evidence for mass–energy equivalence. The mass-energy argument instead calculates an energy difference from emitted light and then identifies its low-speed coefficient.

Less energy of motion at the same low speed means a smaller inertial mass within the Newtonian approximation. The comparison alone does not supply an absolute internal energy.