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Predicting the future part 2

November 19, 2010
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So when we last left off, my students had just discovered Newton’s 2nd law:

\vec{a}=\frac{\vec{F_{net}}}{m}

They even were able to verify it was right because they remembered using F=ma last year, and algebraically, these are the same thing.

I start by asking them why it is that \vec{a}=\frac{\vec{F_{net}}}{m} is a superior way to learn this relationship. They’re puzzled for a second, but they go along with my penchant for rewriting Newton’s laws. Here are some of the ideas they come up with:

  • It clearly shows the idea that acceleration is directly proportional to force and inversely proportional to mass.
  • It’s a vector law. In eighth grade, you don’t really want to deal with scary vectors.
  • It separates the changes in motion (acceleration) from the causes of those changes (force and inertia).

So from there, I tell them that this law set off a revolution, and showed us we could predict the future. I ask them how.

We start by defining some terms in a slightly new way:

  • position: x, where you are
  • velocity: v, where you are going. In the future, you will be somewhere in this direction.

Already they remember that

x_f=v\Delta t+x_i

And so if we vectorize this they can see this idea:

The velocity predicts the future location of the object.

But the kids quickly point out the flaw in this idea, that this only works for constant velocity. But then I remind them that for small enough \Delta t the velocity is constant, and so this idea will work if limit ourselves to only predicting the future over very small time intervals.

Then I ask the kids what does acceleration tell you, and they see that, acceleration tells you how your velocity will change. At some time in the future, your velocity will point more in the direction of the acceleration, which follows from the equation:

\vec{v_f}=\vec{a}\Delta t+\vec{v_i}

acceleration shows us how the velocity will change

And, again, the kids recognize that the above equation only works for constant acceleration, but over a small enough \Delta t, acceleration will be constant. So again, if we restrict ourselves to predicting the velocity very near in the future, we’re alright.

Now we also know that we can find the acceleration using Newton’s 2nd law, so this gives us a great recipe for predicting the future for an object, so long as we know its initial position and velocity.

  1. Compute the net force acting on the object.
  2. Find the acceleration of the object using N2.
  3. Update the position of the object using, using a very small time interval.
    \vec{r_f}=\vec{v}\Delta t+\vec{r_i}

  4. Update the velocity using
    \vec{v_f}=\vec{a}\Delta t+\vec{v_i}

  5. Repeat the process.

At this point, the kids seem to think that the crystal ball I’m selling them isn’t nearly as cool as they thought it was going to be, so we have to tell a bit of history here. I start by reminding them how we read about the true Romantic Scholar, William Herschel, discoverer of Uranus, and the incredible feat he pulled off discovering the first planet not visible with the naked eye, and this discovery was the culmination of many grueling nights spent painstakingly observing the northern sky.

The rest of the story comes from a wonderful post from Starts with a Bang (the well deserved 2010 winner of best physics blog) on the discovery of Neptune.

Then I had to tell them that before before Newton, Kepler had figured out a few things about the orbits of planets, like that they orbit in ellipses, and later, Newton was able to explain this using the law of gravity (Lots of hand waving here, and saying we’d study this in more detail in the future).

But the key idea was we had scientific laws that were predicting where Uranus should be. And as we took data, we noticed it was slightly off from our predictions. In some places, Uranus was slightly ahead of where it should be, and in others, it was slightly behind. This put science at a crossroads—the choices were:

  • Uranus isn’t a planet
  • Newton’s laws for predicting the motion of objects are flawed
  • Something else is exerting a force on Uranus to account for these differences

And so, scientists began to think the third possibility might be the key—what if there were another planet out there, pulling on Uranus, and at various points in the motion causing it to speed up and slow down, which in turn would account for its discrepancies from Kepler’s and Newton’s predictions?

Now, with the power of these laws, a heap of data on Uranus’s position, and a lot of painstaking mathematics, scientists were able to make predictions of where this new planet might be. I’ll let Ethan Siegel finish the story:

But Le Verrier simply proved to be the better man for this job. After performing his painstaking calculations once, he announced his results publically on August 31, 1846 in front of the French Academy. He then composed a letter detailing his prediction to astronomer Johann Galle at the Berlin Observatory; the letter arrived September 23. That evening, Galle and his assistant, d’Arrest, pointed their telescope towards the exact location Le Verrier predicted.

And right there, less than 1 degree away from the exact spot Le Verrier predicted, was the new planet: Neptune.

I try to emphasize this to them how incredible it must have been in that day to sit down, make a prediction using Newton’s laws of where a new planet should be and then find it exactly where you said it was. It would seem like magic. And it would be a strong signal that the very foundation of our world was changing.

I then show them my favorite quote about this topic, from anothe scientist, Pierre-
Simon Laplace, living at nearly the same time as Le Verrier:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

Now, the kids are hooked. I tell them that Laplace is saying is all I need to do is figure out where every atom in their body is located, and what its velocity is, and I can predict where they will go to college, or go back in time to figure out where they were born. Of course Laplace said that almost 200 years ago, and now we’ve got huge computers capable of doing all sorts of incredible calculations—why can’t we do this?

And here class ends, but almost every stays after to hear about two discoveries that would make this Laplace’s fantasy impossible:

  • Chaos theory: we find that for non linear systems even slight changes in initial conditions of position and velocity can lead to huge differences as a system evolves. This is why we can’t predict the weather far into the future.
  • Uncertainty: With the discovery of quantum mechanics we discovered that we can’t measure some quantities with limitless precision. If we measure the position of an atom extremely well, we make it impossible to know the velocity, and vice vera.

Both of these ideas together tell us that no matter how big the google server farm gets, it will never be able to catalog your atoms and predict your future.

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