Angular Momentum Conservation in Spacecraft Orbits

[Like Tony Stark]

Homework Statement

A spacecraft designed to reach Mars has mass=2000 kg. The first stage moves in an almost parabolic orbit. When it reaches the closest point (5 times Mars radius), it starts moving in a circular orbit. After that, it moves in an elliptical orbit and lands. Determine
A) what magnitudes of the spacecraft are conserved
B) angular momentum in each orbit
C) velocity when it reaches the closest point in the parabolic orbit
D) energy needed to change to the circular orbit, and then to the elliptical one.
E) velocity when it lands

Relevant Equations

$E_M=E_C+E_P$
$L=rxv$

Tell me if I'm right:
A) Angular momentum is conserved because there are no external torques. Linear momentum isn't conserved because gravity is acting on the spacecraft . Mechanical energy isn't conserved because it has to change between different orbits.
B) Parabolic orbit: $L=mv_1.5r_{Mars}$
Circular: $L=mv_2 5r_{Mars}$
Elliptical: $L=mv_3r_{Mars}$

I don't know if I should consider $v_1$ as given because that's what I'm asked in C. If I should't then I don't know how to calculate $L$.

D) I know that $E_M=E_C+E_P$, so I have to do $\Delta E$. Kinetic energy is simple, but what about potenial energy? From circular to elliptical we have $E_{cir}=-GmM/5r_m$ and $E_{elip}=-GmM/2(r_a+r_p)$, where $r_p=r_{Mars}$ but what about $r_a$?

E) That's just the velocity at perigee.

 

[jbriggs444]

A) Angular momentum is conserved because there are no external torques

Would you expect the rocket to fire its thrusters to change from parabolic orbit to circular orbit?

 

[PeroK]

Is the mass of the spacecraft conserved?

 

[jbriggs444]

As I read the question, the intent is to model the spacecraft as a payload of fixed mass to which propulsion is applied by an unspecified external force. This force provides (or absorbs) the required change in payload energy at the two points specified in:

D) energy needed to change to the circular orbit, and then to the elliptical one.

If we decide to model the entire space craft, including the not-yet-burned fuel, the situation is more difficult. Since we are not given an exhaust velocity, I would think that it best to stick with the simpler model.

 

[Like Tony Stark]

If we decide to model the entire space craft, including the not-yet-burned fuel, the situation is more difficult. Since we are not given an exhaust velocity, I would think that it best to stick with the simpler model.

So angular and linear momentum, and mechanical energy are not conserved.
But then I have different unknown variables, because I don't have $dm/dt$ or initial velocity for example.
How should I go on?
And when I want to calculate the change in energy between the orbits what formulas should I use? (Because $GmM/2a$ is used for elliptical or circular orbits, but what about parabolic or hyperbolic orbits?)

 

[jbriggs444]

So angular and linear momentum, and mechanical energy are not conserved.
But then I have different unknown variables, because I don't have $dm/dt$ or initial velocity for example.
How should I go on?
And when I want to calculate the change in energy between the orbits what formulas should I use? (Because $GmM/2a$ is used for elliptical or circular orbits, but what about parabolic or hyperbolic orbits?)

For a circular orbit, can you calculate kinetic energy and potential energy? What fraction is the one of the other?

 

[Like Tony Stark]

For a circular orbit, can you calculate kinetic energy and potential energy? What fraction is the one of the other?

Yes, the kinetic energy of a satellite in a circular orbit is half its gravitational energy

 

[jbriggs444]

Yes, the kinetic energy of a satellite in a circular orbit is half its gravitational energy

Now, is there anything special about a parabolic orbit? How much kinetic energy is left at infinity in such an orbit?

 

[Like Tony Stark]

Now, is there anything special about a parabolic orbit? How much kinetic energy is left at infinity in such an orbit?

In parabolic orbits you have $GmM/r = 1/2 m v^2$

 

[jbriggs444]

In parabolic orbits you have $GmM/r = 1/2 m v^2$

So you have everything you need to compute the velocity before and after the transition from parabolic to circular orbit, yes?

 

[Like Tony Stark]

I

So you have everything you need to compute the velocity before and after the transition from parabolic to circular orbit, yes?

I've understood
thanks!