What Is the Eccentricity of an Orbit Given Vmin and Vmax?

[HHveluj]

Homework Statement

A satellite in an orbit around Earth has velocities between Vmin=V-V0 and Vmax=V+V0. Find the eccentricity of the orbit.

Homework Equations

rmin=l^2/(Gm1m2*mu*(1+e))
E=(Gm1m2)^2*mu*(e^2-1)/2l^2
where mu = reduced mass = m1m2/m1+m2

The Attempt at a Solution

I know all this formulas for eccentricity, but they all involve disctances/or energy, but not velocities! I don't even know how to start solving this... Any help/hints?

Thank you very much in advance!

 

[nrqed]

Homework Statement

A satellite in an orbit around Earth has velocities between Vmin=V-V0 and Vmax=V+V0. Find the eccentricity of the orbit.

Homework Equations

rmin=l^2/(Gm1m2*mu*(1+e))
E=(Gm1m2)^2*mu*(e^2-1)/2l^2
where mu = reduced mass = m1m2/m1+m2

The Attempt at a Solution

I know all this formulas for eccentricity, but they all involve disctances/or energy, but not velocities! I don't even know how to start solving this... Any help/hints?

Thank you very much in advance!

You may use conservation of angular momentum to relate the speed at aphelion to the speed at perihelion. Recall, ${\vec L} = {\vec r} \times {\vec p}$. Since at aphelion and at perihelion the motion is perpendicular to the position vector, you get that $r_{min} v_{max} = r_{max} v_{min}$. Since $r_{min} = a(1-e)$ (*if* I recall correctly) and $r_{max} = a(1+e)$ , you get a simple equation relating the max and minimum speeds. You will get a simple expression for the eccentricity in terms of V and V_0.

Hope this helps.

Patrick