# Energy of motion and inertia

At everyday speeds, the energy of motion depends on speed and on mass. Compare two states at the same speed, and any difference in that energy is a difference in mass.

Written for this edition, not translated from Einstein.

## If a body moves at the same speed but carries less energy of motion, what has changed?

A force that pushes a body through a distance does work on it, and the work becomes the body's energy of motion, its kinetic energy. That is separate from energy stored inside the body, as heat or in its chemistry.

For one body at everyday speeds, doubling the speed multiplies the energy of motion by four. At a fixed speed, doubling the mass doubles it. The mass-energy paper's title asks about a body's Trägheit, its inertia: its resistance to a change in its motion, which is measured by pushing it, not by weighing it.

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

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

This formula holds at speeds small compared with the speed of light, and it is not exact close to that speed. So the mass-energy argument works with the exact expression and reads the mass from its low-speed limit, where it takes this form.

## Worked example: Two bodies at the same speed, different masses

1. Two bodies move at 2 metres per second.
2. The 3 kg body has ½ × 3 × 2² = 6 joules of energy of motion; the 2 kg body has ½ × 2 × 2² = 4 joules.
3. The speed is the same, and the energies differ by 2 joules because the masses differ by 1 kg.

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

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

Here the two masses are given, so the example proves nothing about mass and energy. The 1905 argument runs the other way: it computes the energy difference from the light the body gives off, finds that it has this form, and reads the mass difference from the coefficient of ½v².

## Where this lesson stops

At the same low speed, less energy of motion means less mass. The comparison gives only a difference in mass, never the total energy stored inside the body.

## This lesson builds on

- [Squares and square roots](/foundations/bridge-squaring-square-roots/)
- [Fractions and ratios](/foundations/bridge-fractions-ratios/)