I'm trying to prove Kepler's equations

the proof is in my book but I don't understand it

(T1/T2)^2 = (s1/s2)^3

ok first what allows us to rewrite that formula as so???

(s1^3/T1^2)=(s2^3/T2^2)

my book goes about the proof as

m1 ((4 pi^2 r1)/T1^2)

which I get but then

where does this come fromg
it says rearange this to get..

T1^2/r1^3 = ((4 pi^2)/(GMs))

thats don't understand the proof of keplers laws

also what allows us to state that centripetal acceleration is the same thing as gravity???

"Proof" is hardly the term. You are deriviving Keplers law from the universal law of gravitation.

Remember that Kepler's law applies to things in orbit, where the force of gravity supplies enough pull to keep it in orbit,or forcegravity=forcecentripetal.

GMm/r^2= mv^2/r= m (2pi r/T)^2/r
= m (4PI^2 r/T^2)

GM=4PI^2 r^3/T^2

or r^3/T^2= GM/4PI^2 = constant.

Then, since this is a constant for any orbit, any orbiting mass, then for anything,

(r1/r2)^3= (T1/T2)^2

That is one law. Now for equal areas, for circles, the proof is trivial, for areas swept it is a bit more complicated.

As a sidenote, your algebra skills are weak, and you would do well to do a number of exercies (workbooks for various skills are readily available at any bookstore) to improve those skills.

To understand how the equation (T1/T2)^2 = (s1/s2)^3 can be rewritten as (s1^3/T1^2) = (s2^3/T2^2), let's go step by step:

1. Start with the original equation: (T1/T2)^2 = (s1/s2)^3

2. Take both sides of the equation to the power of 2: (T1/T2)^2 = (s1/s2)^3

3. Simplify the left side: T1^2/T2^2 = (s1/s2)^3

4. Multiply both sides of the equation by T1^2: T1^2 = (s1^3/s2^3) * T2^2

5. Divide both sides of the equation by s1^3: T1^2/s1^3 = (s1^3 * T2^2) / (s2^3 * s1^3)

6. Simplify the right side: T1^2/s1^3 = (T2^2 * s1^3) / s2^3

7. Rearrange the terms to get (s1^3/T1^2) = (s2^3/T2^2).

Now, regarding the proof of Kepler's laws you mentioned:

1. The equation you mentioned, m1 ((4 pi^2 r1)/T1^2), is a rearrangement of the equation for the acceleration due to gravity: GMs/r^2, where m1 is the mass of a small object, G is the gravitational constant, M is the mass of the central object (for example, a planet), r1 is the distance between the central object and the small object, and T1 is the orbital period of the small object.

2. Rearranging the equation, we get: T1^2/r1^3 = (4 pi^2)/(GMs). This equation relates the orbital period T1, the distance r1, and the gravitational parameters (G and M) of the system.

3. This equation is derived using Newton's law of universal gravitation, which states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Regarding the relationship between centripetal acceleration and gravity:

1. In circular motion, an object experiences centripetal acceleration, which is directed toward the center of the circular path.

2. The force responsible for the centripetal acceleration in a celestial body's orbit is gravity. According to Newton's law of universal gravitation, the gravitational force acting on an object in orbit provides the centripetal force required to keep it moving in a circular path.

3. Therefore, centripetal acceleration and gravity are directly related in the context of orbital motion.

I hope this explanation helps you understand the proof of Kepler's laws and the relationship between the equations mentioned. If you have any further questions, feel free to ask!