Physics I: Mechanics › Forces › full formula sheet

Fnet = maSay it: “the net force on an object equals its mass times its acceleration”

Newton's second law

The bridge between forces and motion — why pushes cause speeding up, and how much.

Notation on this page: Fnet is the vector sum of all forces (bold = vector), m is mass in kg, a is acceleration in m/s².

Where it comes from

Before Newton, the commonsense guess was that force causes velocity — you push a cart, it moves; you stop pushing, it stops. Galileo saw through this: the cart stops because of friction, an opposing force you forgot to count. Left alone, objects keep doing what they are doing. Newton sharpened that into a law: force causes change of velocity — acceleration — and the amount of acceleration is set by how much inertia (mass) the object has.

Before reading on: you push a crate with 10 N and it accelerates at 2 m/s². If you push twice as hard (20 N) with everything else the same, what is the new acceleration? And if you keep 10 N but the crate is twice as heavy?

Newton published the law in 1687 in his Philosophiæ Naturalis Principia Mathematica, in the form “the change of motion is proportional to the motive force impressed.” In modern language, “change of motion” is the change of momentum p = mv, giving the most general form of the law:

Fnet = dp/dtforce is the rate of change of momentum — this is the form Newton actually wroteSay it: “the net force equals the rate of change of momentum”

For the constant-mass objects of everyday physics, dp/dt = d(mv)/dt = m(dv/dt) = ma, and we get the familiar Fnet = ma. The double prediction above resolves cleanly: 20 N on the same crate gives 4 m/s² (force doubled → acceleration doubled), and 10 N on a twice-as-heavy crate gives 1 m/s² (mass doubled → acceleration halved).

Derivation

Newton's second law is a law of nature, not a theorem: it cannot be proved from deeper axioms the way a math formula can. What we can do is show that the familiar F = ma follows from the momentum form Newton actually stated, for the case of constant mass — and read off what one newton really means.

Fnet
=
dp/dt = d(mv)/dt
Step 1 — Newton's statement. Net force is the rate of change of momentum p = mv. This is the empirical law, confirmed by experiment.
=
m · dv/dt
Step 2 — constant mass. If m does not change (no rockets leaking fuel here), it is a constant and slides out of the derivative.
=
ma
Step 3 — recognize acceleration. dv/dt is the acceleration a, by definition. Hence Fnet = ma. ∎

Now read the units off the formula. Mass is in kg, acceleration in m/s², so force comes out in kg·m/s² — and that combination gets its own name:

1 N = 1 kg·m/s²one newton accelerates one kilogram at one metre per second squaredSay it: “one newton equals one kilogram metre per second squared”

When does F = ma fail? Two famous cases: (1) mass changing with time — a rocket burning fuel needs the full F = dp/dt form; (2) speeds near light speed, where momentum is γmv and Newton's law needs Einstein's correction. For this course, mass is constant and speeds are slow: F = ma rules.

How to use it

The procedure, every dynamics problem:

  1. Draw a free-body diagram. Sketch the object as a dot and draw every force on it as an arrow: weight down, normal up, pushes, pulls, friction, tension. Miss a force and the answer is wrong before you start.
  2. Add the forces as vectors to get Fnet. Opposite forces subtract; perpendicular forces combine with Pythagoras (see Example 3).
  3. Write F = ma per axis: ΣFx = max, ΣFy = may. Each axis obeys the law independently — a classic simplification.
  4. Solve and sanity-check. The acceleration points the same way as Fnet. If your a points opposite your net force, something flipped.

Three directions of the formula

The one equation answers three questions: Fnet = ma finds the force from motion; a = Fnet/m finds the motion from forces; m = Fnet/a finds the mass from a force-and-motion pair. Train all three.

Common mistake: plugging in one force instead of the net force. A 10 N push fighting 10 N of friction gives Fnet = 0 and a = 0 — the object does not accelerate at 10/m. Always sum first.

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Example 1 — the basic move: 1200 kg car, a = 2.5 m/s²

  1. Identify what you know. m = 1200 kg, a = 2.5 m/s². We want the net force.
  2. Multiply. Fnet = ma = 1200 × 2.5 = 3000 N.
  3. Check units. kg × m/s² = N. Correct — a force.
  4. Sanity. 3000 N is about the weight of 300 kg — a strong but believable engine push.
Common mistake: dividing instead of multiplying (a/m = 0.002 — nonsense units). Say the formula aloud first: “net force equals mass times acceleration.”
Your turn — a 1500 kg car accelerates at 3 m/s². Net force?

Answer: 4500 N. F = ma = 1500 × 3 = 4500 N.

Example 2 — opposing forces: 40 N east vs 30 N west on 5 kg

  1. Sum first, always. Take east as positive: Fnet = 40 − 30 = 10 N east.
  2. Then accelerate. a = Fnet/m = 10/5 = 2 m/s² east.
  3. Notice: using 40 N alone would have given 8 m/s² — four times too big. The sum-first habit is the whole game.
Common mistake: adding the magnitudes (40 + 30 = 70 N) instead of respecting directions. Vectors subtract when they oppose.
Your turn — 50 N east, 14 N west, mass 6 kg. Acceleration?

Answer: 6 m/s² east. Fnet = 50 − 14 = 36 N; a = 36/6 = 6 m/s².

Example 3 — perpendicular forces: 12 N east, 9 N north on 3 kg

  1. Perpendicular forces need Pythagoras. Fnet = √(12² + 9²) = √(144 + 81) = √225 = 15 N. (A 3-4-5 triangle in disguise.)
  2. Direction: θ = arctan(9/12) = arctan(0.75) ≈ 36.9° north of east.
  3. Accelerate. a = 15/3 = 5 m/s² at 36.9° north of east — same direction as the net force.
Common mistake: adding 12 + 9 = 21 N. The forces are perpendicular, not collinear — draw the arrows and close the triangle.
Your turn — 8 N east, 6 N north, mass 2 kg. Net force and acceleration?

Answer: 10 N, a = 5 m/s² at 36.9° north of east. F = √(64+36) = 10 N; a = 10/2 = 5 m/s²; direction arctan(6/8) = 36.9°.

Before reading on: a 2 kg block is pushed right with 20 N while friction drags left with 6 N. Is the acceleration 10 m/s², 7 m/s², or something else?

Example 4 — friction included: 20 N push, 6 N friction, 2 kg block

  1. Net force first. Fnet = 20 − 6 = 14 N to the right. (Friction is a force — it belongs in the sum.)
  2. Accelerate. a = 14/2 = 7 m/s² to the right.
  3. The lesson: 10 m/s² (ignoring friction) overshoots; the real world always nets first.
Common mistake: “forgetting” friction exists — the most common way students lose marks on F = ma problems. If a surface is mentioned, friction is in the diagram.
Your turn — 4 kg block, 30 N push, 10 N friction. Acceleration?

Answer: 5 m/s². Fnet = 30 − 10 = 20 N; a = 20/4 = 5 m/s².

Memorization tips

  • Say it aloud: “the net force equals mass times acceleration.” Stressing “net” is the memory — the word carries the whole sum-first habit.
  • Three rearrangements: F = ma, a = F/m, m = F/a. Cover two, compute the third — the triangle trick from kinematics works here too.
  • Direction memory: acceleration always points with the net force. If your answer's arrow disagrees with the force arrow, the answer is wrong.
  • Units check: kg × m/s² = N. If your “force” comes out in kg or m/s, you mixed up the formula.
  • Zero-force check: Fnet = 0 → a = 0 → constant velocity. Newton's first law lives inside the second — one law, two moods.
  • Doubling intuition: double the force → double the acceleration; double the mass → halve it. This proportionality is the law in words.

Final challenge

Five mixed questions — basics, directions, and the traps, all in one. Score 5/5 and the second law is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

What is Newton's second law?

Newton's second law says the net force on an object equals its mass times its acceleration: Fnet = ma. The acceleration always points in the direction of the net force.

Why does Newton's second law use the net force instead of a single force?

Acceleration responds to the vector sum of all forces. Individual forces can cancel each other — two equal opposite pushes produce zero net force and zero acceleration, even though each push alone is nonzero.

What is a newton?

The newton (N) is the SI unit of force: 1 N = 1 kg·m/s². It is the net force needed to accelerate a 1 kg mass at 1 m/s².

Is F = ma a definition or a law of nature?

It is a law: an empirical claim about how the world works, confirmed by experiment. The more general form Fnet = dp/dt (force is the rate of change of momentum) reduces to F = ma when mass is constant.

What happens when the net force on an object is zero?

Then a = 0: no acceleration. An object at rest stays at rest, and a moving object keeps its velocity — this is Newton's first law, hiding inside the second.

More from the codex

Live

Physics I formula sheet

All the mechanics formulas — kinematics, forces, energy — printable and quiz-ready.

Open sheet → Live

Formula Sheet Builder

Mix and match any sections into your own printable sheet.

Open tool → Live

Prompt Simulator

Practice prompt engineering with deterministic scoring.

Open tool →