Push something and it speeds up. Push the same thing harder and it speeds up faster; push something heavier with the same force and it speeds up less. F = ma is that sentence written as an equation, and it is the equation the whole of mechanics is built on.
What each letter means
| Symbol | Meaning | Unit |
|---|---|---|
| F | The resultant force — every force on the object added together, direction included | newtons, N |
| m | Mass | kilograms, kg |
| a | Acceleration — the rate at which velocity changes | m/s² |
The word resultant is doing the heavy lifting. A car with 4000 N of driving force and 1500 N of drag has a resultant of 2500 N, and it is 2500 that goes into the equation. Using the driving force alone is the mistake that spoils more of these questions than any other.
A newton is defined by this equation: it is the force that accelerates one kilogram at one metre per second squared. That is why the units work out without conversion, provided the mass is in kilograms.
What the law is really saying
Three things follow from it, and questions test all three:
No resultant force means no acceleration. Not “no motion” — no change in motion. Something moving at a steady speed in a straight line has no resultant force on it, which is the case for a car at constant speed on a motorway, where the engine’s push exactly cancels the drag.
Force and acceleration point the same way. Always. If the resultant force is backwards, the acceleration is backwards, which for a forward-moving object means slowing down.
Mass resists acceleration. Double the mass and the same force produces half the acceleration. This is what mass is, in mechanics — a measure of how hard something is to get moving.
A worked example
A 1200 kg car accelerates from rest to 27 m/s in 12 seconds. What resultant force acts on it?
The force formula needs an acceleration, and we have not been given one, so find it first:
a = change in velocity ÷ time = (27 − 0) ⁄ 12 = 2.25 m/s²
Now apply the law:
F = ma = 1200 × 2.25 = 2700 N
Note what that answer is and is not. It is the resultant force. If the car is also fighting 800 N of drag and friction, the engine must be providing 2700 + 800 = 3500 N to leave 2700 N over. Questions often ask for the driving force having given you the resultant, or the other way round, and the two numbers are never the same in the real world.
Where marks get lost
Using grams instead of kilograms. A 500 g ball is 0.5 kg. Putting 500 in gives an answer a thousand times too large, and it will still look like a number.
Forgetting that F is the resultant. Add up all the forces first, respecting direction, before anything goes into the equation.
Confusing mass with weight. Mass is in kilograms and does not change. Weight is a force, measured in newtons, and equals mg — which is F = ma with gravity supplying the acceleration. A 60 kg person has a mass of 60 kg everywhere and a weight of about 589 N on Earth.
Ignoring direction. Decide which way is positive at the start and stay with it. A negative answer is information, not an error: it means the force or acceleration points the other way.
Using it when the mass is changing. F = ma assumes a constant mass. For a rocket burning fuel it does not hold in that form.
Questions students actually ask
What is the difference between mass and weight?
Mass is how much matter there is, measured in kilograms, and it is the same everywhere. Weight is the force gravity exerts on that mass, measured in newtons, and it changes with where you are — the same person weighs about a sixth as much on the Moon.
Why does my answer come out negative?
Because the force or acceleration points opposite to the direction you chose as positive. For an object travelling forwards, a negative acceleration means it is slowing down. The sign is part of the answer, not a mistake.
Does F = ma work for objects moving in a circle?
Yes, with the acceleration being the centripetal acceleration v²⁄r, directed towards the centre. The resultant force points inwards even though the speed never changes, because the direction is changing.
What if several forces act at once?
Add them as vectors — same direction adds, opposite direction subtracts, and forces at an angle need resolving into components first. The single number left over is the resultant, and that is the F in the equation.
Is a newton a large force?
About the weight of a small apple. Holding a 1 kg bag of sugar against gravity takes roughly 9.8 N, which is a useful reference point when checking whether an answer is sensible.
Related formulas
The equations of motion · Projectile motion · The quadratic formula
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