What is the ratio between the speeds before and after a short rocket thrust?

[alex3]

Homework Statement

A satellite is in a circular orbit around the sun, radius [tex]r_{e}[/tex], the radius of Earth's orbit. After a short rocket thrust parallel to its velocity, it is in a new orbit with aphelion at Jupiter's orbital radius, [tex]r_{j}[/tex]. What is the ratio of the speed's just before and after the thrust?

2. The attempt at a solution

I'm thinking like this:

• The thrust is parallel to the velocity and is for a very short time, so the radii before and after the thrust are equal.
• The aphelion is at Jupiter's orbital radius: the satellite is now describing an elliptical orbit, with an aphelion at Jupiter's orbital radius, and an perihelion at the thrust point; Earth's orbital radius.
• We need the speed just after the thrust, so we need what the speed would be at the perihelion of the orbit.

Ok so far? I've worked out the eccentricity by using the following logic:

• The sun is at one focus.
• The major axis is the [tex]A = r_{j} + r_{e}[/tex], semi-major is [tex]a = \frac{A}{2}[/tex].
• The distance from the centre of the ellipse to the focus is [tex]ae[/tex], so the eccentricity can be calculated using [tex]a - ae = r_{e}[/tex] and solving for [tex]e[/tex].

I calculated 0.677 for [tex]e[/tex], using

[tex]r_{j} = 7.8 \times 10^{11}[/tex]
[tex]r_{e} = 1.5 \times 10^{11}[/tex]

But how can I deduce the perihelion speed with this data?

EDIT

Is it just as simple as, by energy conservation, using the vis viva equation and setting r to r_e, or will I need some additional calculation?

 

[Filip Larsen]

Vis-viva should do fine.