Space Questions - Need some direction (1 Viewer)

[BrotherBread]

Hey guys, for a few days now my friend and I have been stumped as to how to answer these questions we got for holiday homework/revision. Our texts and notes don't seem to be able to help us out and I was wondering if someone could give us a few pointers as to which way to go.

Q1, Using Kepler’s Second Law, show that the velocity of a satellite decreases with the inverse square root of the distance from the centre of a planet. Show also that the quantity v^2r is constant for any given planet.

I know that for the first part it is somewhat similar to inverse square rule for light intensity. However neither myself or my friend can find a way to relate it to Keplers law. Anyone know a way to answer this question?

Q2, Using the equations of projectile motion, show that an object dropped from a given height will always take the same time to fall to the ground, regardless of its initial horizontal velocity. Explain why this occurs. Ignore air resistance and the curvature of the Earth.

I understand what it is asking of us, basically it is saying a ball that is dropped at x meters, will hit the ground in the same time it takes a ball thrown horizontally from x meters. What I am havign trouble with is using equations to show this, and the wording of the explination. I know that because the displacement of the ball thrown is greater than the ball dropped it's velocity must be greater than that of the dropped ball in order for this to occur. This is achieved because gravity is accerating the ball for longer thus it has a greater velocity thna the dropped ball before it hits the ground.

That or I am completely wrong. Any tips pointers would be greatly appreciated, discussion of this would also be very helpful as it helps me and others to understand the concepts fully.

Thanks in advance
Bread

 

[dr baby beanie]

Don't shoot me if I'm wrong - I have a dodgy teacher.

Remember that horizontal velocity and vertical velocity is independant.
A ball dropped vertically and a ball following a projectile path (provided they are dropped from the same height) should hit the ground at the same time (ry=1/2at^2) their ry, 1/2, a, t^2 are the same. The horizontal component just 'pushes' the projectile on and both are zero the begin with (at max height the vertical velocity is zero, so too in this case is the horizontal velocity. So therefore the vertical velocity of the 2 should be the same, thus the time taken.

Heres some direction for now but someone will probs contradict me. I tried.

 

[BrotherBread]

Thanks guys. It is so easy to see now. I must hav ehad amental blank or something and didn't think to make t the subject of the equation.

Any ideas on the first question I am still stumped as a way to relate the inverse square rule to keplers second law. I'm sure that if I could do that I would be able to figure it out.

 

[BrotherBread]

Bump. Anyone have a clue as to how to do the first question using keplers laws

 

[ianc]

I worked it all out here:

(for some reason it doesnt let me embed it into the post in the physics forums, so i had to hyperlink it)

Good luck!

 

[jyu]

I worked it all out here:

http://img412.imageshack.us/img412/4011/keplerslawos4.gif
(for some reason it doesnt let me embed it into the post in the physics forums, so i had to hyperlink it)

Good luck!

You have used Kepler's third law instead of the second law!
Also you have assumed a circular orbit instead of an elliptical orbit.

:wave:

 

[ianc]

lol whoops didnt read the question....

i think thats what the question wanted you to do though, cos youd need some calculus and all sorts of mathematical rubbish to do it using the 2nd law - which isnt really touched by hsc physics.

but in the hsc youd never be asked to do any maths on elliptical orbits.

 

[jyu]

Kepler's second law is about elliptical orbit.

:wave:

 

[BrotherBread]

Thanks heaps man. I got as far as trying to use the formula that you end up with and reverse engineer the equation but I just couldn't see it.

I looked at the question that we were asked in case I transcribed it wrong, but it turns out that they must have made an error. Because it says second law in the question. From memory I can't rember the syllabus saying anything about Keplers second law, but I think it says Keplers law of periods, which is universally accepted as Keplers 3rd law. I then think in the adjacent point the syllabus asks us to be able to answer questions using Keplers 3rd law.

Once again thanks heaps man I spent ages today trying to do it again and got nowhere, you have been a massive help.

 

[jyu]

Thanks heaps man. I got as far as trying to use the formula that you end up with and reverse engineer the equation but I just couldn't see it.

I looked at the question that we were asked in case I transcribed it wrong, but it turns out that they must have made an error. Because it says second law in the question. From memory I can't rember the syllabus saying anything about Keplers second law, but I think it says Keplers law of periods, which is universally accepted as Keplers 3rd law. I then think in the adjacent point the syllabus asks us to be able to answer questions using Keplers 3rd law.

Once again thanks heaps man I spent ages today trying to do it again and got nowhere, you have been a massive help.

Alternative approach:

Satellite (circular orbit) moves in the gravitational field of the planet,

a = g

(v^2)/r = GM/(r^2)

v^2 = GM/r

v = constant * (1/sqrt r)

or (v^2)r = constant

:wave: