Aceleration/Velocity and Integration help

[FLOUR]

Hello, I have expression for the aceleration of a planet in the orbit of another planet and need to get the expression for it's velocity. The aceleration expression came from the Gravitacional Force within the gravitacional field of the planets.

F=ma
F=(-G.m1.m2/r^3) * position_vector

Where r is the scalar value of the position_vector. From here we can get ax and ay.

For the velocity:

1) Is it ok to do it Using
a= Vf-Vi/tf-ti
for each axis (ax ay).?

2) How would it be done using the intergration rules.?

Thanks.

 

[finchie_88]

If I'm understanding correctly, you want an expression for the speed of the planet as it orbits. is that not simply $$\frac{m_2v^2}{r} = \frac{Gm_1m_2}{r^2}$$ canceled down, because that seems far too simple.

 

[topsquark]

If I'm understanding correctly, you want an expression for the speed of the planet as it orbits. is that not simply $$\frac{m_2v^2}{r} = \frac{Gm_1m_2}{r^2}$$ canceled down, because that seems far too simple.

That assumes the orbit is circular. In general they are ellipses.

In general you can't set $$\mathbf{a}=\frac{\mathbf{\Delta v}}{\Delta t}$$ because this implicitly assumes that the acceleration is constant. You need to use $$\mathbf{v}= \int \mathbf{a(r)} \, dt$$ the problem being here that you are integrating over the "wrong" variable. Without dragging the whole Newtonian central force theory into it, I'm not sure what to do to fix that at this point. Off the top of my head I would suggest that perhaps using the facts that the orbit is an ellipse and that equal areas are swept out by the orbit in equal times might be useful.

What subject does this question pertain to? (i.e. is this a Math question or Physics?)

-Dan