How Do You Calculate the Semi-Major Axis of a New Elliptical Orbit?

[teme92]

Homework Statement

(a) A satellite is initially in a geostationary orbit around the Earth, so that the satellite always remains above the same point on the Earth’s equator. Show that the radius of the orbit is $4.22\times 10^7$(b)An engine is briefly fired in the direction of the satellite’s motion, making the speed of the satellite suddenly increase to a speed $v_p$ and sending the satellite into an elliptical orbit with eccentricity $e = 0.35$. What is the semi-major axis of the new orbit?

Homework Equations

Period: $T=2\pi\sqrt{\frac{r^3}{GM_E}}$

Escape Velocity: $V_{esc}=\sqrt{\frac{2GM_E}{r}}$

Eccentricity: $e=\sqrt{1+\frac{2{\epsilon}L^2}{(GM_E)^2}}$

$e=\frac{r_a-r_b}{r_a+r_b}$ where $r_a=a(1-e)$ and $r_b=a(1+e)$

The Attempt at a Solution

Done the (a) part straight forward enough with the period formula.
So I subbed in my values to get:

$r=4.22\times 10^7 m$

$V_{esc}=4345 ms^{-1}$

What I don't understand is how to get the semi major axis. If someone could point me in the right direction that would be great. I think I have to get the new period of the orbit but I don't know how.

 

[SteamKing]

Homework Statement

An engine is briefly fired in the direction of the satellite’s motion, making the speed of the satellite suddenly increase to a speed $v_p$ and sending the satellite into an elliptical orbit with eccentricity $e = 0.35$. What is the semi-major axis of the new orbit?

Homework Equations

Period: $T=2\pi\sqrt{\frac{r^3}{GM_E}}$

Escape Velocity: $V_{esc}=\sqrt{\frac{2GM_E}{r}}$

Eccentricity: $e=\sqrt{1+\frac{2{\epsilon}L^2}{(GM_E)^2}}$

$e=\frac{r_a-r_b}{r_a+r_b}$ where $r_a=a(1-e)$ and $r_b=a(1+e)$

The Attempt at a Solution

So I subbed in my values to get:

$r=4.22\times 10^7 m$

$V_{esc}=4345 ms^{-1}$

What I don't understand is how to get the semi major axis. If someone could point me in the right direction that would be great. I think I have to get the new period of the orbit but I don't know how.

Have you posted the complete problem statement and/or all the information you were furnished?

 

[teme92]

Hey SteamKing I edited my post and put in more,I didn't think it was necessary sorry.

 

[SteamKing]

Hey SteamKing I edited my post and put in more,I didn't think it was necessary sorry.

It normally wouldn't make a difference if this were Psychic Forums, rather than Physics Forums.

But, the additional information clarifies this thread greatly.

 

[teme92]

Ouch

 

[andrevdh]

Hint: The transfer occur at perihelion.

 

[teme92]

What do you mean by transfer?

 

[andrevdh]

From the one orbit to the other.