Saturday, July 5, 2008

Clapping for Kepler! - Part I

Have you heard of Kepler's three laws of planetary motion? Do you even know who Johannes Kepler is? Well, he was the great astronomer and mathematician who worked on information on Tycho Brahe (another interesting astronomer!), and realised that planets don't move in perfect circular paths around the Sun!

Well, you might say that's rather commonsense - but try saying it about 250 years ago, when no one knew of Newton's Laws of Motion, and when everyone thought that force was required to sustain motion! Clearly then, it would've been an astronomical task (no pun intended!) to derive and deduce the path taken by the celestial planets.
Let's dive in immediately, because I simply cannot stand dwelling on simple historical facts anymore; to get into the heat of action, let us have Kepler's 2nd Law in a nice statement:

Equal areas are swept out by the radius vector connecting the planet to the Sun in equal times.

Well, I'll explain that in another post, but harkening back to what I'm really interested in for this entry, let us assume for a second that there is no gravitational force or whatsoever in this Universe. Would Kepler's 2nd Law still hold? Well, very much! And I shall endeavour to prove this to you now.

First of all, what does a force do to the state of motion? Easy, it changes either the direction and/or the magnitude of the velocity of the body in motion. Next, where exactly and in what direction does gravity act if you're travelling around the Sun? Easy also: gravity acts also in a straight line towards the centre of the Sun and acts at your centre of mass. Basically this means that the gravitational force is a radial force, a force that acts towards the centre of the mass providing the gravitational field.

So, let us now assume that the Sun is infinitely massive with respect to the Earth or any other object (and it is rather much so!), which is justified by its very much larger mass - if so, then we can neglect the Sun's motion and say that it is effectively in rest in our mathematical treatment. Now, let us have a moving rocket - this rocket moves up, in a straight line, at constant velocity of 10 m/s. And we have three positions outlined below, A, B and C, all in equal intervals of 5 s. So let's refer to the picture below that I've so kindly (ahem!) drawn out for your ease of interpretation of the physical situation at hand:

Now, I've mentioned already this is a thought experiment to see what happens to Kepler's 2nd Law if there were no gravity, and this explains why the rocket has no force on it and continues moving in a straight line with constant speed.

So, the two triangles to compare now, are obviously triangles OAB and OBC; notice that both have the same height h, and because the speed of the rocket is the same, and equal time intervals have passed, we must agree that the rocket travels the same distance in the same amount of time, and therefore AB = BC.

What then is the formula giving the area of triangles? Well, if you've forgotten, it's:

Area of a triangle = 1/2(base)(height)

Since OAB and OBC have the same height h, and the same base (AB = BC), we conclude that they must have the same area! And this must then mean that Kepler's 2nd Law holds true even in the absence of gravity! That equal areas are swept out in equal times. :)

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