# How to discover the $F = m\cdot a$ equation from scratch? [duplicate]

Is there a paper or a book that conceptually 'discover' the Newton's famous formula?

Did Newton empirically noticed that when we apply an acceleration to a body, then the accelerationon the body that has contact with it, will be proportional either to the mass or to the acceleration? Then, he simply called it 'force'?

OMG i'm so confused :c I can understand differential equations, integrals, and so, but I can't understand, from scratch, how to build up Newton's laws in my mind. I need good papers or books.

• It might help you to think of force as rather the time derivative of momentum. Then maybe it's a little easier to visualize. It's also important to realize that we don't really know what a "force" is. We can see the effects of a force, but we don't know exactly what it is. Jul 3, 2014 at 19:15
• @jhobbie I would like to know more about this 'derivative'. You have some good papers? Jul 3, 2014 at 19:17
• What do you mean? Do you not know what the derivative is? Jul 3, 2014 at 19:18
• Also, is it how Newton though? Jul 3, 2014 at 19:18
• possible duplicate of How did Newton discover his second law?
– Danu
Jul 3, 2014 at 19:54

Imagine an object sitting in space (so no friction, etc. to worry about). You can push on it (exert some force on it), and see what happens to it. When you push on it with a constant force, you see the object start to accelerate, which means its velocity increases linearly with time. If the object is moving at 10 m/s after 1 second, then it will be moving at 20 m/s after 2 seconds. Now you try pushing on the object twice as hard, and you notice it accelerates twice as fast. If you push on it five times as hard, it accelerates five times as fast. It quickly becomes obvious to you that force and acceleration have a linear relationship, so if you multiply the force by some constant $\alpha$, the acceleration gets multiplied by the same value. Furthermore, since the object doesn't accelerate at all when you aren't exerting any force on it, you know that the two quantities (force and acceleration) must be related by an equation of the form $F=\kappa a$, where $\kappa$ is some constant. We call this constant mass, and denote it by $m$, but mass (or at least inertial mass, the kind that shows up in Newton's second law) is really just a measure of how much an object accelerates when you exert a force on it.