Mechanics - Orbits under Central Force

[Master J]

Homework Statement

A particle moving in a plane is attracted towards a point (fixed) by a force zr^-5. The particle is projected from an apse at distance a with speed SQRT(z/(2a^4)).

Show that the orbit is r=acos(theta)

Using the Orbit Eq: d^2 u/dC^2 + u = za^-5/(hu)^2

h = angular momentum, or r(speed) in this case, u = 1/r



I get down to (du/dC)^2 + u^2 = -4/[(a^3)u] + A

where C = theta, A = integration constant.

Am I correct so far?

 

[Dick]

za^-5/(hu)^2 should be zu^5/(hu)^2 shouldn't it? Rather than solving the equation, why don't you just substitute u=1/(a*cos(theta)) and see if the solution works?

 

[Master J]

Why is that?

 

[Dick]

Because if the radial force per unit mass is z/r^5 and r=1/u that turns into zu^5.