00:01
In this problem, we're going to be deriving some formulas for the velocity at perihelion and apheelion of an object in orbit using conservation of angular momentum and conservation of energy.
00:15
So for the first part, part a, we're trying to prove this equation, given right here, for the ratio of the velocity of perihelian, v .p over the velocity at epihelion, v .a.
00:26
Try to prove that it equals 1 plus e divided by 1 minus e.
00:30
The tools that we're going to use to get there are angular momentum, which is the position vector r cross the momentum, which is m times v.
00:40
And then we also know that at perihelian, that position vector is a times 1 minus e, and at at helion, the radius vector is a times 1 plus e.
00:52
So those are the things that are going to be useful for us.
00:54
Okay, because we know that angular momentum is conserved, then that means that the angular momentum at parahelian should be equal to the angular momentum at apheelium.
01:07
So that's lpp is equal to l a.
01:10
Because at parahelan and apheelian, the velocity is directly perpendicular to the position vector r.
01:18
That means that this equation just becomes l equals r times m times v.
01:26
So what we can plug in for the angular momentum at perihelian and aphealian is this.
01:32
The radius at perichelian times mass times that velocity at perichelian should be equal to the radius at apheelian times mass times the velocity of apheelian.
01:44
The mass is constant, of course, so those cancel out.
01:48
And next i'm going to substitute in our formulas from above.
01:53
So the radius at perihelian is a times 1 minus e times the velocity at perihelian is equal to the position vector r at apulian is a times 1 plus e times velocity at apulian va.
02:14
All right.
02:16
Next i'm going to basically just divide, do a little bit of algebra rearranging things.
02:22
So if i divide both sides by va, then i'm going to have vp over va over here and then divide both sides also by that term right there.
02:34
So that the right side of my equation ends up being what i already had, a times 1 plus e, over that term a times 1 minus e.
02:50
My a is cancel, and i end up back with that equation that we were trying to get.
02:56
Velocity perennelion over velocity of apheelian is equal to 1 plus e over 1 minus e.
03:03
In the next part of the problem, we are trying to find formulas for the square of the velocities at periculine and apculine using conservation of energy.
03:15
The energy during orbit is going to be the sum of the gravitational potential energy u plus the kinetic energy k.
03:23
And there's a formula in the textbook given, which is over here on the right that the gravitational potential energy is gmm over r, and the kinetic energy is one -half.
03:36
Mv squared.
03:37
So the sum of those two terms is the total energy.
03:40
Because the total energy is conserved, we can set those two equal to each other at perihelian and apheelian in order to derive this equation that we're looking for for vp squared and va squared.
03:52
Along the way, we're also going to need the results from our previous problem, which is the ratio, vp over va, is equal to one plus e over one minus e.
04:04
Okay, so let's go.
04:06
At, perihelian we have negative g capital m lowercase m over r and it's going to be rp plus one half m vp squared and that's going to be equal to the energy at app hulian which is negative g m over r a plus one half m v a squared.
04:38
Now i'm going to use two things here i'm going to let's try to solve for first the velocity squared at perihelian vp.
04:46
So in order to do that, i'm going to need something to substitute in for va in terms of vp.
04:55
So i'm just going to solve this equation over here for vp.
04:58
Actually, other way right, i'm going to solve it for va in terms of vp.
05:02
That's what i want.
05:03
Sorry.
05:05
So that should be va equals.
05:09
I'm essentially going to multiply va over to the right and then take this other term that's currently on the right and divide it off to the left.
05:16
So i will have, because i divided this term, it's now the reciprocal of what it originally was, 1 minus e over 1 plus e times vp.
05:27
That's just rearranging this equation right here to solve for va.
05:32
All right, so i'm going to substitute that in for va, and then i'm also going to substitute in the formulas, same ones that we used, in the previous part of the process.
05:41
For the position vector r at perihelian, which is a times 1 minus e.
05:47
And at apheelian, a times 1 plus e.
05:55
Okay, now lots of substitution.
05:58
One thing i can also do is my masses all cancel out.
06:01
If you notice every single term has that lowercase m, so i can cancel all of those out.
06:06
It just saves me a little bit of writing.
06:07
All right, so i'm going to have negative g, capital m, over rp is a times 1 ,000, minus e plus, let's see, vp squared.
06:21
That's what i want.
06:22
I'm going to leave it as is over two.
06:25
That's the one -half, is the over two.
06:27
And then i have negative g capital m over our a.
06:32
I'm going to replace with a times one plus e plus.
06:39
All right.
06:39
So now i've got my va, which again, i'm going to replace with this term over here.
06:45
So it's going to, excuse me, it's going to now be one, my, minus e over 1 plus e, all of that squared, vp, also squared, and then all of that is over two.
07:03
So, or one half of that, i can say.
07:07
Stick that one half out in front.
07:13
Next, i'm going to rearrange my equation.
07:16
I'm going to try and get both my vp terms on the left side of the equation and both of my gm terms on the right side of the equation.
07:23
So here's vp squared.
07:24
And then this one -half times all that business squared times vp squared.
07:29
I'm going to subtract that off over here.
07:31
So minus one -half times this is 1 -e over 1 -plus -e, and all of that is squared, v -p squared.
07:44
Okay, equals my original term on the right was negative gm over a times 1 -plus -e, and then the similar term with the 1 -1 -e that was originally on the left side of the equation, i'm now adding over to the right side of the equation.
07:59
So now it's positive gm over a times 1 minus e.
08:05
Okay.
08:07
Now let's do some algebra.
08:08
I'm going to factor some stuff out.
08:10
So on the left side of the equation, i'm going to factor out of vp squared over 2 from both times vp squared over 2 with that factored out is 1 minus.
08:22
Okay.
08:23
And then i'm actually going to expand the stuff i have in parentheses there.
08:27
So 1 minus e squared is 1 minus 2e plus e squared over 1 plus 2e plus e squared.
08:43
That's going to be useful in my next step.
08:46
So just hang on and you'll see about that.
08:49
Equals.
08:50
And then from both of these i can factor out in negative gm over a times.
08:57
Okay.
08:58
I'm going to eventually want to combine these two fractions in my algebra.
09:02
So i'm going to give them like denominators while i'm also doing this factory, kind of doing multiple things in the same step.
09:08
So originally this is 1 over 1 plus e.
09:11
I'm going to multiply numerator and denominator by 1 minus e.
09:15
So now that'll be 1 minus e over.
09:19
Well, 1 plus e times 1 minus e.
09:21
That's just 1 minus e squared.
09:24
So that's the term on the left.
09:27
Plus, okay, originally the denominator was 1 minus e.
09:31
So i'm going to multiply numerator and denominator by 1 plus e.
09:36
And again, the denominator is now that like term 1 minus e squared.
09:42
So 1 minus e times 1 plus e.
09:48
All right.
09:48
Next step, vp squared over 2.
09:53
And then i'm going to do something similar with this fraction.
09:56
I'm going to take my 1, and i'm going to make it 1 plus 2e plus e squared in the numerator and the denominator.
10:03
So that i can combine this fraction.
10:05
So 1 plus 2e plus e squared is now the numerator and the denominator.
10:15
1 plus 2e plus e squared.
10:18
So that's really just 1.
10:20
And then that's minus all of this business that was originally in that numerator, which is 1 minus 2e plus e squared.
10:33
Okay.
10:34
So that's the left side of my equation.
10:37
It's getting bigger, but it'll eventually get smaller.
10:40
And then we have negative gm over a is our coefficient on the right times.
10:45
Now that these have the same denominator, i can combine their numerators.
10:48
So 1 plus 1 is 2.
10:50
And negative e plus positive e.
10:55
Those e's just cancel out.
10:56
So it's really just 2 over 1 minus e squared.
11:04
Simplifying the left side of my equation, i've got vp squared over 2 times.
11:09
Let's see.
11:10
So 1 minus 1 cancels.
11:13
And then positive 2e and then minus a negative is going to be plus 2e.
11:20
So i actually can't cancel that.
11:22
It's 2e plus 2e.
11:23
So now it's 4e.
11:27
And then positive e squared minus e squared.
11:31
So those do cancel out.
11:33
So it's just 4e in the numerator over.
11:35
I'm going to collapse this back down to what it originally was, which is 1 plus e squared.
11:43
Okay, equals negative j.
11:47
Gm over a times 2 over 1 minus e squared.
11:56
Okay.
11:59
You know what? i just realized we're supposed to actually have an e over here on the right side of the equation.
12:04
I dropped it a couple of terms back.
12:07
Did you catch my mistake? good for you if you did.
12:10
So here, when i factored out the gm, i actually should not have factored out that negative because only one of the two terms was negative.
12:20
So the first term, this one, one minus e, that one was negative...