# Stating Newton's second law

So I was wondering how did Newton himself state his second law. One account I've read says that he first expressed it in the form that we refer to as impulse and momentum i.e. FΔt=mΔv. Today I was told that Newton never even wrote F=ma, and that the expression a=F/m is a much "Better" way of stating the law. How is this "Better", if different at all?

So how did Newton himself state this law mathematically. Also any links to derivations and other helpful related content would be appreciated.

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ShayanJ
Gold Member
The change of motion is proportional to the [magnitude of the] impressed motive force,
and to be made along the right line by which that force is impressed.
If a force may generate some motion ; twice the force will double it, three times
triples, if it were impressed either once at the same time, or successively and gradually.
And this motion (because it is determined always in the same direction generated by the
same force) if the body were moving before, either is added to the motion of that in the
same direction, or in the contrary direction is taken away, or the oblique is added to the
oblique, and where from that each successive determination is composed.
From "The Mathematical Principles of Natural Philosophy" by Isaac Newton(Translated and annotated by Ian Bruce).

Newton still lived in an age where all of the deductive logic and mathematical proof were based in geometry. I believe that the form F=ma is mostly thanks to Euler. Euler is also responsible for the operational definition of force. In Newton's time force (vis) was still often used to describe a property of motion. Inertia was often called the force of inertia. Also Leibniz definition of 'living force' (vis viva) eventually changed into the modern day expression for kinetic energy.

So I was wondering how did Newton himself state his second law.
Shyan already posted the wording. The corresponding formula in modern notation is F=dp/dt. For constant mass this results in F=m·a.

Today I was told that Newton never even wrote F=ma, and that the expression a=F/m is a much "Better" way of stating the law. How is this "Better", if different at all?
It is not better but rather worse because a=F/m fails for m=0 (not that it would be of any practical relevance). For m>0 both formulas are equivalent.

Sullo
So I was wondering how did Newton himself state his second law. One account I've read says that he first expressed it in the form that we refer to as impulse and momentum i.e. FΔt=mΔv. Today I was told that Newton never even wrote F=ma, and that the expression a=F/m is a much "Better" way of stating the law. How is this "Better", if different at all?

So how did Newton himself state this law mathematically. Also any links to derivations and other helpful related content would be appreciated.
Newton did say that a=F/m is a better way of stating the law. I would guess that, he believed it to be simpler than F=ma, because mass and acceleration are generally, more constant. Acceleration is all about change in velocity. Because of this, it seems that acceleration is dependent on more than mass or force. After all, we define things as concisely and truly as we can; we wouldn't write mass as "m=Fw/g," when asked what mass is dependent on. But......... I'm getting off topic. I hope this helped!

Thanks a lot everyone for the answers, this'll help heaps. :)

tony873004
Gold Member
I teach Conceptual Physics to high school freshmen. Their math background is pretty shaky. I find they understand it better as a=F/m rather than F=ma. It's easier to ask the kids leading questions that help them visualize the formula:

F is in the numerator. If F gets bigger, what happens to a?
m is in the denominator. If m gets bigger, what happens to a?

If they miss the 2nd question, I ask what's bigger, 1/10, or 1/100. They all get that.

My leading question for F=ma is "If I want a mass to accelerate it, what do I have to do?" Force it. They understand that too, but have a more difficult time assembling a formula from the conceptual statement.

a=F/m also makes it easier for them to understand why the big rock and the little rock fall at the same rate. a = 2F/2m.