Newton's Laws of Motion: F = m·a Explained

Nothing changes without a push

Slide a puck across smooth ice and it glides a long way. Slide it over carpet and it stops almost at once. The ice is not helping the puck move; it just gets out of the way. Newton's first law says that a body stays at rest, or keeps moving in a straight line at constant speed, unless a net force acts on it. This stubbornness is called inertia. In deep space, where there is almost nothing to rub against, a switched-off probe coasts for millions of kilometres at the same speed. In the animation, watch the puck: its speed barely drops, because almost nothing pushes against it.

F = m·a

The second law connects a push to its effect: F = m·a. F is the net force in newtons (N), m is the mass in kilograms and a is the acceleration in m/s². One newton is the push that gives 1 kg an acceleration of 1 m/s²; a 100 g apple weighs about 1 N. Rearranged, a = F / m. A 10 N push on a 2 kg cart gives a = 10 / 2 = 5 m/s². The same 10 N on a 5 kg cart gives only a = 10 / 5 = 2 m/s². Heavier means harder to speed up.

How fast after three seconds?

Acceleration tells how quickly the speed changes: starting from rest, v = a·t. The light cart, after 3 s, moves at 5 × 3 = 15 m/s. The heavy one has only reached 2 × 3 = 6 m/s. In the same time the light cart has also gone farther: s = a·t² / 2 gives 22.5 m against 9 m. Use the sandbox under the picture. Set the force and the mass, then compare the acceleration and the speed after 3 s with your own numbers.

Friction pushes back

Friction is a force, so the second law covers it too. It acts against the motion, and for a sliding body it is about f = μ·m·g, where μ (the Greek letter mu) is the friction coefficient of the two surfaces, m is the mass and g = 9.81 m/s². Take a 5 kg cart and μ = 0.2: f = 0.2 × 5 × 9.81 = 9.81 N. Push it with 20 N and the net force is 20 − 9.81 = 10.19 N, so a = 10.19 / 5 ≈ 2.04 m/s². Without friction the same push would give 4 m/s². Switch friction on and off in the sandbox and watch the difference.

Every push pushes back

The third law: when body A pushes on body B, B pushes back on A with an equal force in the opposite direction. Two skaters stand face to face and one pushes the other. Both roll away. The forces are equal, but the accelerations are not, because the lighter skater gains more speed. With 50 kg and 70 kg, for example, the lighter one leaves at 1.4 m/s and the other at 1.0 m/s, since 50 × 1.4 = 70 × 1.0. A rocket works the same way: it throws hot gas backwards, and the gas pushes the rocket forwards. It needs no air to push against, which is why rockets work in space.

The Principia

In 1687 Isaac Newton published Philosophiæ Naturalis Principia Mathematica. In it, three laws and a law of gravity are enough to describe a cart on a table, a falling stone, the tides and the planets. Before him, the heavens and the Earth seemed to obey different rules. After him, the same equation F = m·a worked for a hockey puck and for the Moon. The book was not an easy read, but its method became a model for physics: state simple laws in mathematics, then test them against measurement. The next chapters use these laws to follow balls, pendulums and planets.

Newton's three laws of motion