Calculating the velocity of an elliptical orbit

Question

I need an explanation as to why my solution to this problem doesn't work (please do not give me an answer or tell me exactly how to do it as this is a homework question that I need to do myself, hints are allowed though).

I am given the velocity for a given distance from the sun in an elliptical orbit, and need to calculate the velocity at another given distance.

So I thought to use the fact that

$T\propto r^{\frac{3}{2}}$

$T^2\propto r^3$

$\therefore \frac{T^2}{r^3}=k$,

where $k$ is some constant.

This means that

$\frac{T_1^2}{r_1^3}=\frac{T_2^2}{r_2^3}$.

Substituting $T=\frac{2\pi r}{v}$ gives

$\frac{4\pi^2r_1^2}{v_1^2r_1^3}=\frac{4\pi^2r_2^2}{v_2^2r_2^3}$

$\frac{r_1^2}{v_1^2r_1^3}=\frac{r_2^2}{v_2^2r_2^3}$

$v_2=\sqrt{\frac{v_1^2r_1}{r_2}}$.

When I plug in my values and enter my answer, it says that it is wrong.

Why is this solution wrong?

I've completed this now using conservation of momentum, but still why is this solution wrong?

Answer 1

Your equation relates the period of the orbit to the length of the semi-major axis, not to the absolute distance at any point. You can use the Vis-viva equation if you have more information. But you don't have the semi-major axis length or other details about the orbit.

As you suggest, conservation of energy is the simpler way forward.

Answer 2

It seems to me you may be misunderstanding the problem as stated. You are assuming you are being asked about two different objects (planets?) in different orbits; but I think from reading the question that you are being asked about the same object at different points in its elliptical orbit.

For an object in an elliptical orbit, conservation of angular momentum tells you what the tangential velocity needs to be as a function of distance; and if the eccentricity of the orbit is small, so the radial velocity can be neglected, then the solution is found trivially.

If you cannot ignore the radial velocity, you would actually need to know at what point of the orbit you are in order to complete the calculation - because if you do not know the eccentricity of the orbit you can't simply calculate the ratio of velocities at two different points in an orbit if you only have the radii. For orbits where $r_1$ and $r_2$ correspond to the perigee and apogee (farthest and closest point) the radial velocity is zero at both points and conservation of angular momentum can be used trivially; while if you had a different orbit with the same perigee, but a different eccentricity (and thus there would be a radial component of velocity at $r_2$) then the ratio of velocities MUST be different.

Conservation of energy should be able to help you here - sum the potential and kinetic energies at the two points.

The reason your initial attempt does not work is that you apply Kepler's law inappropriately - you are using instantaneous distance $r$ as though it is equal to the semi major axis. Which is only (generally) true for a circular orbit in which case $r_1=r_2$ and your expression says the velocities remain equal...

Answer 3

In simple words, as the orbit is elliptical, you need to know that the velocity is not constant at any point in the orbit. It keeps changing.

T=2πr/v is valid only for a circular orbit where the speed at every point in the orbit is const.

Energy conservation : (v^2 / 2) - (G.M/r) = -(GM/2a) where, G = Gravitational constant M = mass of the earth m = mass of the planet (or whatever is orbiting) a = length of semi major axis of the elliptical orbit v = speed at a point which is at at distance r from the sun

Instead, you should be using energy conservation formula to find the length of the semi major axis first, and then using the energy conservation formula again to find the required velocity ( or rather speed).