Orbit Intersection Question (classical mechanics)

[^_^physicist]

Homework Statement

A comet is going in a parabolic orbit lying in the plane of Earth's Orbit. Regarding Earth's orbit as circular of radius "a," show that the points where the comet intersects Earth's orbit are given by:

cos(theta)= -1 + (2*p)/a where p is the perihelion distance of the comet defined at (theta)=0

Homework Equations

Circular orbit: eccenticinty = 0
Parabolic orbit, Energy = 0, eccentincity = 1

Differential equation of an orbit:

d^2(u)/d(theta)^2 + u = -1/(m*l^2*u^2)*f(u^-1), where f is the function of the central force, u= 1/r, where r is the radius, and l is the angular momentum per mass.

Another representation of the differential equation of an orbit (using energy, and only for an inverse-squared central attractive force):

(du/d(theta))^2 + u^2 = 2E/ml^2 + 2ku/(ml^2); conditions are the same, except k= GM.

The Attempt at a Solution

Ok my thought was the solve the differential equations, once I have the solutions set the equations equal to one another and show that the points are predicted by the equation given; however, I don't have enough information given to solve the diff. eqs.

Any other ideas? I am quite stumped.

 

[D H]

Any other ideas? I am quite stumped.

Stop using physics. This is a simple geometry problem: Find the intersection of a parabola and a circle. The only physical result that is needed is that the center of the circle and the focus of the parabola are the same point.

 

[Dick]

You don't have to solve the equations of motion. The geometry of the orbits is already given to you. So it's just a geometry problem. Just write down the polar coordinate representations of the two curves and intersect them. It's REALLY easy.

 

[^_^physicist]

Thanks for the tips; and yes you were right it was so easy to just think of it in geometric terms.