Mechanics⏱ 6 min read

Newton's Second Law

F = ma

What is the Newton's Second Law?

Newton's Second Law says that an object's acceleration is directly proportional to the net force acting on it and inversely proportional to its mass. Push harder and it accelerates faster; make it heavier and the same push accelerates it less. It's the second of Isaac Newton's three laws of motion, published in 1687 in the Principia Mathematica, and it's the one that actually lets you calculate motion numerically — the first law just describes what happens with zero net force (nothing, or constant velocity), and the third describes force pairs between two objects.

F here means net force — the combined effect of every force acting on the object, not just one applied push. If a box is being pushed forward while friction pulls it backward, F is the difference between the two, not the push alone. This is the single biggest source of setup errors in mechanics problems: students often plug in one applied force and ignore gravity, normal force, friction, or tension acting at the same time. Before using F = ma, it usually helps to draw a free-body diagram listing every force on the object, then sum them (accounting for direction) to get the true net force.

The law also implies something often missed at first: an object moving at constant velocity, even a very high one, has zero acceleration and therefore zero net force acting on it — that's not the same as "no forces," just forces that cancel out.

What Each Variable Means

Net forceThe vector sum of every force acting on the object. (newtons (N))

MassHow much matter the object has — not its weight. (kilograms (kg))

AccelerationThe rate at which the object's velocity changes. (meters per second squared (m/s²))

Units

Quantity	Symbol	Unit
Force	`F`	newton (N)
Mass	`m`	kilogram (kg)
Acceleration	`a`	meter per second squared (m/s²)

When to Use It

Calculating the force needed to accelerate a known mass
Finding acceleration when the net force and mass are known
Analyzing motion problems in introductory mechanics

Where This Formula Comes From

Start from momentum

Momentum is mass times velocity.

p = mv

Newton's Second Law is really about the rate of change of momentum

Force is how fast momentum changes over time.

F = dp/dt

Expand using the product rule

For an object with constant mass, only velocity changes with time.

F = d(mv)/dt = m(dv/dt)

Recognize dv/dt as acceleration

By definition, acceleration is the rate of change of velocity.

F = ma

Step-by-Step Example

Problem: A 12 kg box is pushed across a frictionless floor with a net force of 30 N. Find its acceleration.

Write down what's known

Identify the given values and what's being asked for.

F = 30 N, m = 12 kg, a = ?

Rearrange the formula

Solve F = ma for acceleration.

a = F / m

Substitute and calculate

Divide the net force by the mass.

a = 30 / 12 = 2.5

✓

Answer: a = 2.5 m/s²

Interactive Calculator

m (mass) (kg)a (acceleration) (m/s²)

Result will appear here

Solving for Other Variables

m = F / aSolve for mass when force and acceleration are known.

a = F / mSolve for acceleration when force and mass are known.

Common Mistakes

Mistake: Using weight (mass × gravity) as if it were the net applied force.
Fix: F in F = ma is the net force — sum every force acting on the object (including gravity and friction where relevant) before applying the formula.
Mistake: Mixing grams with newtons.
Fix: Convert mass to kilograms first — F = ma only gives force in newtons when mass is in kilograms and acceleration in m/s².

Practice Questions

What net force is needed to accelerate a 5 kg object at 4 m/s²?
Hint: F = ma.
A 1,500 kg car accelerates at 3 m/s². What net force acts on it?

Frequently Asked Questions

Is F = ma only true for constant mass?

Yes — for systems where mass changes over time (like a rocket burning fuel), the more general form F = dp/dt is needed instead.

What's the difference between mass and weight here?

Mass (kg) is how much matter an object has and stays constant. Weight is the force of gravity on that mass (weight = mg) and changes depending on gravity — F = ma uses mass, not weight.

Can acceleration be negative?

Yes — negative acceleration just means the net force points in the negative direction of whatever axis you've chosen, which often means the object is slowing down.