Solving Satellite Collision Homework

[vivekrai]

Homework Statement

A satellite is revolving around the Earth in a circular orbit with radius of 2R, where R is radius of earth.If suddenly,its velocity becomes zero in the orbit due to collision with some inter-stellar object(like a satellite),find the time which it takes to hit surface of the earth. Ignore the effect of atmosphere and treat Earth to be perfect sphere.

Homework Equations

Necessary formulas. . all are at hand.

The Attempt at a Solution

There are a no. of confusions I'm having. Firstly, What is aftermath motion of the satellite. Does it swirl inwards towards the Earth or fall directly.
Secondly, Since there is no energy dissipation the energy remains conserved ie., Since PE decreases the KE should increase. Also the Angular Momentum should remain conserved about the earth.

Will it suffice to solve this? Or Am I missing something?

 

[tiny-tim]

hi vivekrai!

you're reading too much into this!

what is complicated about? …

… its velocity becomes zero …

… zero velocity = zero KE = zero angular momentum

 

[vivekrai]

So Initial Angular momentum is zero which implies that the satellite must fall in such a way that it's Angular Momentum remains zero for along the motion. This is only possible if it falls in a straight line joining the Center of Earth and the satellite?

The problem then reduces to just finding the time taken by the satellite to hit the surface of Earth under a variable force of gravitation due to the earth. Does it? This would however require Integration and I'' handle it.

Now Is there any other method based purely on Kepler's Laws which can help to solve this without the maths? [Just for Knowledge]

 

[tiny-tim]

hi vivekrai!

So Initial Angular momentum is zero which implies that the satellite must fall in such a way that it's Angular Momentum remains zero for along the motion. This is only possible if it falls in a straight line joining the Center of Earth and the satellite?

yes!

Now Is there any other method based purely on Kepler's Laws which can help to solve this without the maths?

well, you could get if from kepler's third law …

the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit​

… if you had another orbital period to compare it with!

 

[vivekrai]

… if you had another orbital period to compare it with!

In this case, the ellipse that we thought for its motion becomes flat ie., Converges to a straight line. Now what after this?

 

[tiny-tim]

you still need another orbital period to compare it with

 

[vivekrai]

Some more clues please please!

 

[tiny-tim]

there isn't a way using kepler's laws!

use the way you mentioned earlier …

So Initial Angular momentum is zero which implies that the satellite must fall in such a way that it's Angular Momentum remains zero for along the motion. This is only possible if it falls in a straight line joining the Center of Earth and the satellite?

The problem then reduces to just finding the time taken by the satellite to hit the surface of Earth under a variable force of gravitation due to the earth.

 

[vivekrai]

Sir Please tell me that you're not hiding something. If there is really , then please tell. I've come to know that indeed there (based on some assumption) , which I'm not getting.

 

[tiny-tim]

it falls in a straight vertical line

just find the time taken by the satellite to hit the surface of Earth under the variable force of gravitation