Solving GPE of Satellite in Circular Orbit

[crafty2288]

Homework Statement

i) Show that when a satellite (or planet) is in a circular orbit it's kinetic energy (positive) is one-half of it's potential energy (negative).

ii) Show that in order to escape from the Earth you need a speed v=sqrt(2gR) where g=9.8 m/s^2. Neglect Friction and effects of Earth's rotation.

Homework Equations

PE= -GMm/R
KE= (1/2)mv^2

The Attempt at a Solution

I feel like I'm doing some kind of basic algebra wrong here.

I was trying to do part 1 by solving (1/2)KE=PE

So, (1/2)(1/2)mv^2 = GMm/R
(1/4)v^2 = GM/R
v^2 = 4GM/R
and v=sqrt(4GM/R)

So in the process of trying to find the answer of part 1, I got close to the answer for part 2 (If I had made KE=PE), but now I'm just confused. Is the "1/2" in the equation for KE already accounting for this? Should I be looking at it like mv^2=PE?

 

[ideasrule]

Kinetic energy is one-half of potential energy. That means KE=1/2PE, not 1/2KE=PE. However, you shouldn't start off by assuming what you're trying to prove; that just leads to confusing logic. Instead, assume the satellite orbits at a radius R. Find its potential energy, find its kinetic energy, and you'll see that KE=1/2PE.

 

[crafty2288]

So just plug in a random mass and a random radius? I don't have any values to use... I'd need a mass, a radius... Which I could pick anything... But what about velocity? I can't just pick something there. And I can't find velocity without a change in distance and time.

 

[zachzach]

So just plug in a random mass and a random radius? I don't have any values to use... I'd need a mass, a radius... Which I could pick anything... But what about velocity? I can't just pick something there. And I can't find velocity without a change in distance and time.

Just keep your equations with the variables in them such as r and M. For the velocity, do you know what orbital velocity is equal to? Remember it is in circular orbit. What is the force on it?

 

[crafty2288]

Just keep your equations with the variables in them such as r and M. For the velocity, do you know what orbital velocity is equal to? Remember it is in circular orbit. What is the force on it?

I just had to look that up, because I can't find it in the textbook chapter at all.

But, I found V=sqrt(GM/r)
So KE=(1/2)*m*(sqrtGM/r)^2 or (1/2)*m*(GM/r)

So basically that means
PE= -GMm/R
KE=(1/2) GMm/R

I think that's satisfactory for part 1, if I did that right. Because PE is negative, KE is positive, and KE is 1/2 PE.
Im not sure how to approach Part II though.

I think that KE=PE to escape earth... so...
(1/2)mv^2 = GMm/R
v^2 = 2GM/R
and v=sqrt(2GM/R)

But I'm supposed to find v=sqrt(2gR)...Where am I going wrong here? I think I am confused about what needs to happen to escape earth.

 

[zachzach]

I just had to look that up, because I can't find it in the textbook chapter at all.

But, I found V=sqrt(GM/r)



So KE=(1/2)*m*(sqrtGM/r)^2 or (1/2)*m*(GM/r)

So basically that means
PE= -GMm/R
KE=(1/2) GMm/R

I think that's satisfactory for part 1, if I did that right. Because PE is negative, KE is positive, and KE is 1/2 PE.



Im not sure how to approach Part II though.

I think that KE=PE to escape earth... so...
(1/2)mv^2 = GMm/R
v^2 = 2GM/R
and v=sqrt(2GM/R)

But I'm supposed to find v=sqrt(2gR)


...Where am I going wrong here?

Yep looks right to me for the first part. To get the orbital velocity all you do is set the force of gravity (GMm/r^2) equal to ma. But the acceleration is centripetal (a = v^2/r) since it is a circular orbit. Solve for v and voila.

For escape velocity it looks right as well. What does g equal in v=sqrt(2gR)?

 

[crafty2288]

Yep looks right to me for the first part. To get the orbital velocity all you do is set the force of gravity (GMm/r^2) equal to ma. But the acceleration is centripetal (a = v^2/r) since it is a circular orbit. Solve for v and voila.

For escape velocity it looks right as well. What does g equal in v=sqrt(2gR)?

g is 9.8 ms/^2... But I'm not seeing how that makes it equal to v=sqrt(2GM/R)?

 

[zachzach]

g is 9.8 ms/^2... But I'm not seeing how that makes it equal to v=sqrt(2GM/R)?

This is only true on Earth's surface. But we also use it because it is a good approximation when near Earth's surface as well. How do you find g?

 

[crafty2288]

Oooooooook! g = GM/R^2

So v=sqrt(2gR)... v=sqrt(2(GM/R^2)R)...v=sqrt(2GM/R)

Success!

Much thanks. This is a study guide for an upcoming physics exam, so I really appreciate you helping me to understand where everything was coming from.

 

[zachzach]

Yup no problem ;)