You already know two big ideas. First, that when the forces on an object don't cancel, there is a leftover resultant force. Second, that accelerating means changing velocity — speeding up, slowing down, or turning. Newton's Second Law is the bridge between them: it says exactly how big an acceleration a given resultant force produces.

Push a shopping trolley and it speeds up. Push harder and it speeds up faster. Load it with bricks and the same push barely gets it going. Everyone knows this in their bones — Newton's genius was to write it down as a clean, exact rule you can put numbers into. Two things decide how quickly something accelerates: how hard you push it (the resultant force) and how much stuff it is made of (its mass).

The law itself

Newton found that the acceleration of an object is set by the resultant force acting on it and by its mass, tied together in one short equation. In words: force equals mass times acceleration.

Every symbol has a unit that must be used for the numbers to come out right:

• F — the resultant force, measured in newtons (N).
• m — the mass, measured in kilograms (kg).
• a — the acceleration, measured in metres per second squared (m/s²).

In fact a newton is defined by this law: 1 N is exactly the resultant force that gives a 1 kg mass an acceleration of 1 m/s². So the units lock together perfectly — 1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2 — and if you feed in kilograms and metres-per-second-squared, the answer is guaranteed to be in newtons.

Reading the two stories inside F = m\,a

A single equation, but it carries two everyday truths at once. Keep one of the three quantities fixed and watch what the other two do.

Same mass, bigger force → bigger acceleration. If the mass stays put, then a is proportional to F: double the resultant force and you double the acceleration; treble it and you treble the acceleration. Push a 2 kg mass with 4 N and it accelerates at 2\ \text{m/s}^2; push the same mass with 8 N and it accelerates at 4\ \text{m/s}^2. Twice the shove, twice the pick-up.

Same force, bigger mass → smaller acceleration. If instead the force stays put, then a is inversely proportional to m: double the mass and you halve the acceleration. A 12 N force on a 2 kg mass gives 6\ \text{m/s}^2; the very same 12 N on a 6 kg mass gives only 2\ \text{m/s}^2. More stuff to shift, so the same push achieves less.

Rearranging the law

F = m\,a is a triangle of three quantities: give any two and you can always find the third. Rearranged, the same law reads:

Use whichever version fits the question. If you know the force and the mass and want the acceleration, reach for a = F/m. If you know the force and the acceleration and want the mass, use m = F/a. And if you already have the mass and the acceleration and want the force, the original F = m\,a is ready to go.

A quick tip for getting them right: cover the quantity you want in the words "F over m-a". Cover F and you see m \times a; cover m and you see F over a; cover a and you see F over m.

Worked examples

Example 1 — find the force. A 2\ \text{kg} ball is accelerated at 3\ \text{m/s}^2. What resultant force is needed? Use F = m\,a directly:

Example 2 — find the acceleration. A 1200\ \text{N} resultant force acts on a 800\ \text{kg} car. Rearrange to a = F/m:

Example 3 — find the mass. A resultant force of 50\ \text{N} gives a trolley an acceleration of 2.5\ \text{m/s}^2. Rearrange to m = F/a:

Example 4 — friction first, then the law. A 5\ \text{kg} sledge is pulled with 40\ \text{N} while friction resists with 15\ \text{N}. First find the resultant:

and only then apply the law:

Example 5 — the proportion at work. A 10\ \text{N} force on a 4\ \text{kg} mass gives a = 10/4 = 2.5\ \text{m/s}^2. Double the force to 20\ \text{N} and the acceleration doubles to 5\ \text{m/s}^2; instead double the mass to 8\ \text{kg} and the acceleration halves to 1.25\ \text{m/s}^2. Two ways to change a, pulling in opposite directions.

Try it: push a box

Here is a box on the ground with a single resultant force arrow pushing it to the right. Set the force F in newtons and the mass m in kilograms with the sliders, and the figure works out the acceleration a = F/m live, drawing an acceleration arrow whose length grows with a. Turn the force up and watch a climb; then pile on mass with the force held fixed and watch the very same push produce less and less acceleration.

Weight is Newton's Second Law in disguise

Drop anything near the Earth and it accelerates downwards at about g = 10\ \text{m/s}^2 (more precisely 9.8). What resultant force produces that acceleration? By F = m\,a, a mass m falling at g must feel a force

That downward force is exactly the object's weight. So W = m\,g is not a separate rule to memorise — it is just F = m\,a with the acceleration set to gravity's value. A 2\ \text{kg} bag weighs W = 2 \times 10 = 20\ \text{N} on Earth; take it to the Moon where g \approx 1.6\ \text{m/s}^2 and its weight drops to about 3.2\ \text{N}, even though its mass — the amount of stuff — hasn't changed at all.

Inertial mass: mass is resistance to acceleration

Look again at a = F/m. The mass is the thing that resists being accelerated: for a given force, the bigger the mass, the smaller the acceleration you get. Seen this way, mass is a measure of an object's reluctance to change its motion. Physicists call this the inertial mass, and Newton's Second Law is really its definition — mass is precisely F/a, the force needed per unit of acceleration.

This is why a fully loaded lorry pulls away so ponderously from the lights while an empty one leaps forward on the same engine, and why it is so much harder to get a heavy shopping trolley rolling — or to stop it once it is. The engine or your arms supply roughly the same force either way; it is the mass that decides how much acceleration that force can buy.