Posts Tagged ‘experiment’

Bounce, Part 6

January 11, 2010

Last time, we looked at what Galileo had to say about free fall. This time, we’ll take one more example from his dialog and try to squeeze a little moral out of it.

Galileo presents his ideas through the character Salviati, who explains them to his companions Sagredo and Simplicio. Salviati’s interlocutors raise all manner of objection to his theories, but Salviati answers them and convinces everyone of his point all the more surely in the process. One such objection is given by Sagredo, who doesn’t believe that the velocity of a falling object increases evenly with each second of falling:

So far as I see at present, the definition might have been put a little more clearly perhaps without changing the fundamental idea, namely, uniformly accelerated motion is such that its speed increases in proportion to the space traversed; so that, for example, the speed acquired by a body in falling four cubits would be double that acquired in falling two cubits and this latter speed would be double that acquired in the first cubit.

Sagredo is suggesting that rather than Galileo’s law

$v \propto t$,

that the velocity of a falling body increases the same amount each second, we should instead have

$v \propto x$,

the the velocity increases the same amount each meter the body falls. These are different hypotheses, and so we need to distinguish between them. Given that Salviati states he is not interested in examining the fundamental cause of gravity, and only in characterizing its behavior, there is only one way to do this – experiment.

Instead, Salviati offers the following retort:

…that motion should be completed instantaneously; and here is a very clear demonstration of it. If the velocities are in proportion to the spaces traversed, or to be traversed, then these spaces are traversed in equal intervals of time; if, therefore, the velocity with which the falling body traverses a space of eight feet were double that with which it covered the first four feet (just as the one distance is double the other) then the time-intervals required for these passages would be equal. But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous [discontinuous] motion;

What a strange counterargument! It makes absolutely no sense. Gaining an even increment of speed for each unit of time is a perfectly consistent mathematical law, and does not at all imply instantaneous motion. We can write this law as

$\frac{dx}{dt} = c(x - x_0)$,

which implies that the distance fallen increases exponentially with time. This is completely contrary to observation, and it would be hard to build a unified mechanics like Newton’s that respects this law, but it isn’t logically impossible for the reasons Salviati cites.

And how do Salviati’s friends respond to this argument? Do they rip it apart, or restate their objection more clearly, or request further detail?

Sagredo replies,

You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion.

I guess it’s easier to convince the people you’re arguing with when they’re fictional characters you invented yourself!

Everyone makes mistakes, and they hadn’t gotten around to inventing peer review in the sixteenth century, so let’s forgive Galileo, and take a further look at this hypothesis.

Suppose we have a projectile with constant horizontal velocity and vertical velocity that changes according to the distance traveled in the vertical direction.

What happens if we shoot it up out of a cannon? We know from experience that the cannonball slows down, so it must be losing a constant amount of velocity for each unit height it gains.

The cannonball slows down its vertical velocity, but as it does so, its vertical position changes less. Since its vertical position changes less, the change in its vertical velocity slows down more. In fact, the cannonball asymptotically reaches a certain height above ground, and then stays there!

If the vertical height isn’t changing, then according to this law the vertical distance traveled is zero, and because vertical velocity only changes when vertical height changes, the vertical velocity stays zero. This cannonball would never come down.

On the other hand, if it were pushed down just a little bit, it would gain speed very rapidly, falling exponentially back toward Earth. The motion under Sagredo’s law is absurd, but I wonder why Galileo brought it up at all, only to miss the point.

Too much philosophizing can be dangerous, but this sort of philosophy – extracting results from speculative physical laws – is exactly what theoretical physicists do. The name of the game for a theoretical cosmologist, for example, is to come up with some crazy ideas about how the universe might work, the way Sagredo came up with an idea about falling bodies. Then, the cosmologist tries to work out the consequences of the theory, for example that cannonballs ought to hang in midair until a slight breeze comes along and gives them a downward tap, and they come plummeting back to Earth extremely quickly. If the theory doesn’t agree with observation, it’s wrong.

One difference between the Renaissance and Internet Age versions is that Sagredo’s and Salviati’s theories about falling are easy to test. The experiment doesn’t require any equipment. You just drop something. If you don’t have a thing, you can try jumping instead. But with advanced theoretical ideas, it can be very difficult to make the required observations. That’s why we need giant particle accelerators and kilometer-long interferometers and thirty-meter telescopes and ridiculously-good gyroscopes.

But another problem is that working out the consequences of modern theories is hard. We saw an example of Galileo failing to work out the consequences of a theory, but that was a simple mistake, and if someone had brought his attention to it, he’d have been able to fix it. Some of today’s new ideas about physics are so complicated that even if we can state the law (the equivalent of Sagredo’s idea about velocity being proportional to distance fallen), we may not know how to get to the conclusion (cannonballs hanging over our heads).

We’ve come long way since Galileo, figuring out lots of ways to check ourselves and test our ideas. But there’s a much longer way left to go.