Satellite moving around a planet

[utkarshakash]

Homework Statement

A satellite is describing a circular orbit around a massive planet of radius R. The altitude of the satellite above the surface of planet is 3R and its speed is v_0. To place the satellite in an elliptical orbit which will bring it closer to the planet, its velocity is reduced from v_0 to βv_0., when β<1. The smallest permissible value of β if satellite is not to crash on the surface of planet is √(2/K), find K.

The Attempt at a Solution

I think the angular momentum shall be conserved.

$16mR^2 v_o /4R = mR^2 \beta v_0 / R$

But this equation gives incorrect value of β.

I also tried using conservation of energy but the expression for β does not come out to be in the same format as asked in the question.

 

[Rellek]

Ok, so what would the second radius of the equation be? Are we supposed to assume the satellite comes right to the surface of the planet without actually touching it?

 

[nasu]

Why do you think that the angular momentum is the same on the two trajectories?
The velocity is decreased by a factor of β. Is the distance from the planet increased by the same factor during the deceleration ?

The angular momentum is conserved on each trajectory but not between them. A tangential force (so a torque) was applied to reduce the velocity.

 

[Rellek]

R₁mV₀ + ∫Mdt = R₂mβV₀ ?

 

[nasu]

I would assume that the radius is the same after the velocity is reduced. The equation you wrote is OK in principle. But not useful here.
Focus on conservation laws written for the elliptic trajectory itself.

The circular orbit is just to find a relationship between vo and R.

 

[utkarshakash]

I would assume that the radius is the same after the velocity is reduced. The equation you wrote is OK in principle. But not useful here.
Focus on conservation laws written for the elliptic trajectory itself.

The circular orbit is just to find a relationship between vo and R.

Will applying energy conservation help me in this case?

 

[nasu]

You are not assuming correctly.
The two positions are on the opposite sides of the major axis. The minor axis is irrelevant here.
Draw a diagram.

 

[utkarshakash]

You are not assuming correctly.
The two positions are on the opposite sides of the major axis. The minor axis is irrelevant here.
Draw a diagram.

Is the length of major axis 5R?

 

[nasu]

You edited your post after I answered?
Didn't you say that you assume semi-minor axis is R?

 

[utkarshakash]

You edited your post after I answered?
Didn't you say that you assume semi-minor axis is R?

I later realized that it is wrong and that I don't need to know the length of minor axis for this question.

I think that the centre of planet will be one of the focii of the ellipse. If I apply conservation of angular momentum around the focus, I can write

$4Rv_0 = R \beta v_0$

But I don't think that my logic is correct as the above equation still gives me the wrong answer. :(

 

[nasu]

Please don't delete parts of your posts after someone else posted. It makes the replies to your post look confusing or silly.

Yes, the planet is in one of the focal points of the ellipse.
The equation you wrote is not correct though.
At one end the satellite has velocity βvo and radius 4R. At the other end the velocity is some value, v2 (unknown yet) and the radius is R.

You need two more equations to solve the problem:
1. Conservation of energy between the two extreme point on the ellipse
2. Newton's second law for the initial circle

 

[utkarshakash]

Please don't delete parts of your posts after someone else posted. It makes the replies to your post look confusing or silly.

Yes, the planet is in one of the focal points of the ellipse.
The equation you wrote is not correct though.
At one end the satellite has velocity βvo and radius 4R. At the other end the velocity is some value, v2 (unknown yet) and the radius is R.

You need two more equations to solve the problem:
1. Conservation of energy between the two extreme point on the ellipse
2. Newton's second law for the initial circle

Thank you so much !