Newton's Laws of Motion

Please read Force first.

Sir Isaac Newton

Newton's Three Laws

Back in 1687 Newton wrote three laws about motion, which basically are:


They have wide use today (unless we are dealing with speeds close to the speed of light, or very small things like atoms).

Newton's First Law (Law of Inertia)

Newton originally wrote (translated from Latin):

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it

Since "state of rest" is simply zero velocity we can more simply say:

An object keeps its current velocity unless acted upon by an external force.

In the absence of any external forces, an object will keep doing whatever it was doing.

This is the idea of inertia: matter resists change in its state of motion.


So why do things tend to slow down? Friction!

Friction is a force that opposes current motion. There is so much friction in the world that everything we experience tends to slow down.

car in space


But in space things just naturally drift along.

Newton's Second Law (Law of Acceleration)

Newton: The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

Replacing "change of motion" with acceleration creates this very simple idea:

Force and Acceleration are proportional, and in the same direction.

In fact they are linked by mass: when there is more "stuff" to be accelerated it is harder to get it all moving.

And so we get the most useful equation in all of engineering:

F = ma

Force equals mass times acceleration.

The greater the force applied the greater the acceleration, but greater mass causes less acceleration. See Force for more details.

pulley and elevator

Example: Elevator Ride

You are in a free-running elevator held up by a counterweight on a pulley.

  • You and the elevator together weigh 150kg
  • The counterweight is 125kg

Are you in danger? What will your acceleration be?

Let's start with F = ma

We want to know acceleration so let's rearrange that into:

a = Fm

The total mass that is being accelerated is 150 kg + 125 kg = 275 kg

The force, though, is created only by the difference between weights:

Weight difference = 150 kg − 125 kg = 25 kg

With gravity (g=9.8 m/s2) that makes a force of:

Force = 25 kg × 9.8 m/s2 = 245 N

Let's plug our values in!

a = Fm = 245 N275 kg = 0.89 m/s2

So less than a tenth of gravity's acceleration, just don't fall too far!

(What if the weight was 160 kg?, or 200 kg?)

For more examples see Force Calculations.


Why did Newton not include mass in his law? Perhaps by "motion" he meant momentum.

Momentum (p) is an object's mass (m) times its velocity (v): p = mv.

So maybe this is a better rephrasing:

An object's change in momentum is proportional to the force applied.

Which can be written:

F = ΔpΔt

The "Δ" means "change in", so it says the force is equal to the change in momentum over time.

Both F = ΔpΔt and F = ma are useful! Read momentum to see how they relate.

Newton's Third Law (Law of Action and Reaction)

Newton: To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
stone finger force

Which is usually rephrased as:

For every action there is an equal and opposite reaction.