Newton's Laws of Motion

Three sentences, the mechanics of the universe

5 July 1687. Philosophiæ Naturalis Principia Mathematica is published. Isaac Newton is 44 years old. The book sums up all everyday-scale motion, from the heavens to daily life, in three sentences. More than three centuries later, as long as speeds stay well below the speed of light and objects stay well above the size of atoms, these laws are still the backbone of engineering, sport and space travel.

First law: inertia

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

This is Newton's most revolutionary idea. It turns 2000 years of Aristotle's physics upside down.

Aristotle believed the "natural" state of an object was rest, and that a continuous force was needed to keep it moving. Newton says: the opposite. An object keeps its speed and direction unless a force acts on it.

So why does a pencil you push across a table stop after a few seconds? Because friction is a force. Push the same pencil on ice and it goes much further. In space, with no friction, it would go on forever.

The Voyager 1 space probe was launched in 1977. For nearly 50 years it has been travelling at about 17 km per second with no engine running: living proof of Newton's first law.

Second law: F = m · a

The acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.

This law makes physics quantitative: measurable with formulas. The link between the three variables is very simple:

$F = m \cdot a$

With the same force, if the mass doubles, the acceleration halves. Why does a Formula 1 car accelerate so quickly? Because its engine produces a large force while the car (~750 kg) is relatively light. A truck (~10 tonnes) accelerates much more slowly with the same force.

Units:

• Force: the newton (N); 1 N is the force that gives a 1 kg object an acceleration of 1 m/s²
• An apple is pulled toward the ground with a force of about 1 N

An everyday example: you pull a 30 kg dog by its lead and it accelerates slowly. Apply the same force to a 3 kg toy and it accelerates 10 times as fast.

Third law: action and reaction

For every action there is an equal and opposite reaction.

Simple at first sight, but a law with remarkable consequences when you look closely. If one object exerts a force on another, the second object exerts a force of the same size in the opposite direction on the first.

Examples:

• A rifle recoils. The force pushing the bullet forward (action) → an equal force pushing the rifle back (reaction).
• Rocket thrust. The force pushing exhaust gas backward (action) → an equal force pushing the rocket forward (reaction). NASA's Saturn V burned about 13 tonnes of propellant every second at lift-off.
• Swimming. The arm pushes water backward (action) → the water pushes the swimmer forward (reaction).
• When you stand on the floor, the floor pushes back on you, otherwise you would sink into it.

The mathematical consequence of this law is conservation of momentum: in a closed system the total momentum stays constant. When two objects collide, whatever momentum one loses, the other gains.

Newton's limits

The three laws are extraordinarily accurate, but they do not hold everywhere:

• Very fast: when an object's speed becomes a sizeable fraction of the speed of light (above roughly 10%), Einstein's special relativity takes over.
• Very small: subatomic particles need quantum mechanics.
• Very strong gravity: around massive objects, Einstein's general relativity corrects Newton (GPS satellites, black holes and so on).

Still, for 99.99% of everyday life, Newton is enough. Aircraft, bridges, footballs and cars are all designed with Newtonian physics.

One step further

Newton's laws lead naturally to the ideas of energy and work:

• $W = F \cdot d$: work is force times displacement
• $E_k = \frac{1}{2} m v^2$: kinetic energy
• $E_p = m \cdot g \cdot h$: potential energy

These give rise to the conservation of mechanical energy, another wonderfully simple law.

Sources

• Stanford Encyclopedia of Philosophy: Newton's Philosophy
• NASA Apollo Engineering: Saturn V
• Royal Society: Principia (original manuscript)