If the first law tells you what happens when forces are balanced, the second law tells you what happens when they’re not. A resultant force makes an object accelerate — and the bigger the force (or the smaller the mass), the bigger that acceleration. All of it lives in one equation: F = ma.
📘 What you need to know
The law: the resultant force on an object is directly proportional to its acceleration → F = ma.
F = resultant force (N), m = mass (kg), a = acceleration (m s⁻²).
The acceleration is always in the same direction as the resultant force.
For a fixed mass, acceleration is proportional to force; for a fixed force, more mass means less acceleration.
A force opposing the motion causes the object to decelerate — not suddenly travel backwards.
The law and the equation
Newton’s second law describes the change in motion produced when a resultant force acts on an object:
Newton’s second lawF = ma
Read it as: a resultant force F acting on a mass m gives it an acceleration a. Because force and acceleration are both vectors, a always points the same way as the resultant force.
Resultant force along the motion → the object speeds up or slows down.
Resultant force at an angle to the motion → the object changes direction.
If the resultant force opposes the motion, the object decelerates; it does not instantly reverse.
The acceleration is proportional to the resultant force and points in the same direction.
How force, mass and acceleration relate
Rearranging F = ma shows the two relationships you’ll be tested on:
For a fixed mass: a = F / m, so acceleration is proportional to the resultant force.
For a fixed force: a heavier object accelerates less; mass measures how much an object resists changes in motion (its inertia).
For a constant mass, a plot of acceleration against resultant force is a straight line through the origin — its gradient is 1/m.
Worked examples
WE 1
A launching rocket
A rocket produces an upward thrust of 15 MN and has a weight of 8 MN. In flight it experiences 500 kN of air resistance (downwards). (a) Find the resultant force. (b) The rocket’s mass is 0.8 × 10⁵ kg — find its acceleration.
(a) Take up as positive, add the forcesF = 15 − 8 − 0.5 (in MN)resultant = 6.5 MN upwards(b) Use a = F / ma = (6.5 × 10⁶) ÷ (0.8 × 10⁵)a ≈ 81 m s⁻² upwards
WE 2
The biggest possible acceleration
Three forces — 4 N, 8 N and 24 N — all act on a 5 kg object. What is the maximum acceleration these forces could give it?
Largest resultant = all forces in the same directionF(max) = 4 + 8 + 24 = 36 NThen use a = F / ma = 36 ÷ 5a(max) = 7.2 m s⁻²any acceleration above this is impossible for these three forces
WE 3
A quick rearrange
A 0.45 kg football is kicked and accelerates at 240 m s⁻² while the boot is in contact with it. Find the average force on the ball.
Use F = maF = 0.45 × 240F ≈ 108 N
Quick reference:F = ma • a = F/m • m = F/a. Acceleration points the same way as the resultant force; a bigger mass resists acceleration more.
💡 Top tips
Use the resultant force, not a single force — add all the forces (with signs) first, then apply F = ma.
Pick a positive direction (usually the direction of motion) and keep your signs consistent throughout.
Watch your powers of ten. Convert MN and kN to newtons before dividing — that’s where marks are lost.
Air resistance is fluid resistance — the IB calls it that and uses the drag symbol Fd.
With no drag, falling objects accelerate at the same rate regardless of mass (the hammer-and-feather result).
⚠ Common mistakes
Putting a single force into F = ma instead of the resultant of all the forces.
Thinking a backward force sends the object backwards. It causes deceleration first — direction only reverses if it keeps acting.
Mixing up mass and weight. Mass (kg) goes in F = ma; weight (N) is itself a force, Fg = mg.
Forgetting acceleration is a vector — its sign tells you the direction relative to your chosen positive.
The second law can also be written in terms of momentum: the resultant force equals the rate of change of momentum. That version handles cases where the mass changes too — we’ll meet it again in Force & Momentum. Up next: Newton’s Third Law, the forces that come in equal and opposite pairs.
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