# Why the F is not equal to ma

Those of you who know me will also know that I’m a big fan of crossword puzzles. Maybe it’s the rotational and reflection symmetry in American style crosswords, but I make a point to do all of the late-week puzzles. I even competed in the American Crossword Puzzle Tournament a few years ago, only to have my ass handed to me by my mom, who also competed.

A few days ago, the New York Times Sunday Crossword had the clue:

F=ma formulator

with the anticipated answer: “Newton”.

Or, at least I think it was supposed to be Newton. I may have gotten a few other clues wrong, and if the real answer was “Bortax” you can ignore the rest of this rant.

Shame on you, Will Shortz!

Newton said no such thing, and it’s given high school physics students a skewed perception of reality for a very long time now. The problem with F=ma is that it only works when you’re talking about things that move much, much less than the speed of light. We normally teach physics under the assumption that virtually everyone taking the class is going to grow up to be an engineer. F=ma is fine for building bridges or racecars, but is decidedly less good if we’re talking about the physics in the LHC.

So if he didn’t say F=ma, what did Newton actually say?

In his Principia Mathematica, Newton actually wrote:

Law II: The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’d.

which sounds kind of vague, except in so far as Newton then continues to give a bunch of examples. Essentially, what his verbiage amounts to is:

$F=\frac{\Delta p}{\Delta t}$

where $p$ is the momentum, and where I’ve put in the “technical” tag because of the equations. Now, as you learned it in high school,

$p=mv$

This is the form you learned in high school — the form that Newton derived — and assuming that mass is a constant, it leads pretty quickly to:

$F=m\frac{\Delta v}{\Delta t}$

which is the old familiar (but wrong) form of Newton’s 2nd law.

Rearranging our Newtonian equation gives:

$v=\frac{F}{m}\Delta t$

for a particle starting at rest.

Suppose you had a 10kg particle that you wanted to accelerate at roughly earth normal gravitational acceleration. You might apply a force of 100 N. The acceleration is: $10 m/s^2$. So, after

 Time Final Speed 1s 10 m/s 2s 20 m/s 10s 100 m/s 100s 1,000 m/s $10^7\simeq 4$ months $10^8 m/s=0.33 c$ $10^8\simeq 3$ years $10^9 m/s=3.33c$

In other words, if you take it too literally, this whole F=ma business leads you inexorably to faster-than-light speeds. In a quirky coincidence, accelerating at 1g for almost exactly a year would get you to the speed of light.

Of course, Newton was wrong, but that’s why we needed Einstein.

But Newton wasn’t as wrong as he could be. F=ma is wrong, but $F=\Delta p/\Delta t$ is actually correct. Newton’s problem is that he didn’t know what momentum really is. So what is it?

$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}$

where the extra correction is known the “gamma-factor.”
or, if you want to invert the equation, you get:

$v=\frac{p/m}{\sqrt{1+\frac{p^2}{m^2c^2}}}$

The point is, the momentum gets higher and higher as a particle gets closer and closer to the speed of light, but to actually get to the speed of light requires a literally infinite amount of momentum.

When you take all of that into account, the “constant acceleration” produces a considerably different result:

 Time Final Speed 1s 10 m/s 2s 20 m/s 10s 100 m/s 100s 1,000 m/s $10^7\simeq 4$ months $31.6\% c$ $10^8\simeq 3$ years $95.8\% c$ $10^9\simeq 30$ years $99.96\% c$

Protons in the LHC are zipping around at 99.99994% the speed of light. It doesn’t make sense to simply round up to c. The difference between 99.99994% and 99% is a huge difference. The former is about 130 times as energetic as the latter.

And that’s the sort of discrepancy that comes into play if you make the same sort of errors as Will Shortz.

-Dave

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