Foundation lesson

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}\,m\,v^{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

Two bodies move at 2 metres per second.
The 3 kg body has ½ × 3 × 2² = 6 joules of energy of motion; the 2 kg body has ½ × 2 × 2² = 4 joules.
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}\,\left(m_0 - m_1\right)\,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.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: a before-and-after ledger

A steady push along the motion does work equal to the force times the distance, and the work becomes energy of motion. Choose the body, how fast it is already going, and the push. The last choice changes where every energy is counted from.

Mass

Speed before the push

Force

Distance pushed

Count every energy from

The push does 3 N × 2 m = 6 J of work, and the 3 kg body goes from 0 to 2 m/s. Its energy of motion rises by exactly the work.

The ledger, in joules
Entry	Energy of motion	Counted from zero
Before the push, ½mv²	0	0
Work done, F × d	+ 6	+ 6
After the push	6	6

What it shows, in words

A push of 3 N through 2 m does 6 J of work. A 3 kg body starting at rest ends with 6 J of energy of motion, which at ½mv² is a speed of 2 m/s, the lesson's worked example read the other way. From rest, doubling the distance doubles the work but raises the speed only by a factor of √2, about 1.41, because energy of motion grows with the square of the speed. Counting every energy from 100 J or from a million joules changes both totals by the same amount and changes no difference, so nothing that can be measured changes. That is why a comparison of energies gives a difference in mass, and never the total energy stored inside a body.

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.

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