Minimum velocity to change radius of orbit

[kraigandrews]

Homework Statement

A rocket is in an elliptical orbit around the earth; the radius varies between 1.12 Rearth and 1.24 Rearth. To put the rocket into an escape orbit, its engine is fired briefly in the direction tangent to the orbit when the rocket is at perigee, changing the rocket's velocity by ∆V. Calculate the minimum value of ∆V to escape from Earth orbit.

Homework Equations

mv1r1=mv2r2
.5mv1^2-(GmM)/r1=.5mv2^2-(GmM)/r2

The Attempt at a Solution

I used energy .5mv1^2-(GMm)/r1=.5mv2^2-(GMm)/r2
then v2=v1(r1/r2) by consv. of angular momentum.
then i solved for v1, this however didnt give me the right answer, I have also tried subtracted v1-v2 for delta_v and still could nt get it. Help?

 

[gneill]

You should only need the conservation of energy for this one. The specific mechanical energy for a bound orbit is negative. Unbound orbits have a specific energy >= zero. At escape velocity the specific mechanical energy is exactly zero.

Do you know how the specific mechanical energy of an orbit is related to its semimajor axis?

 

[kraigandrews]

Not sure if I understand... Because the rocket isn't escaping the orbit, so it would still have gravitational potential.

 

[gneill]

Not sure if I understand... Because the rocket isn't escaping the orbit, so it would still have gravitational potential.

The idea is to give the rocket a sufficient change in velocity in order that it achieves escape speed.

Presumably you've found the speed of the rocket at perigee (before it's changed)? What value did you get?

 

[kraigandrews]

i got 7.505 km/s

 

[gneill]

Okay, looks about right (the precise value depends upon the values used for G, M, R).

Now, the specific mechanical energy for the orbit is given by:
$$\xi = \frac{v^2}{2} - \frac{\mu}{r}$$
As I mentioned, the body becomes unbound (will escape) when $\xi = 0$. So use this expression for specific mechanical energy to compute the required velocity at perigee for escape to occur. Your Delta-V will be the difference between the two velocities.

 

[kraigandrews]

Thanks i appreciate it.