There are two identical uniform spherical planets of radius R. The first has its center at the origin of the xyz coordinate system. The second has its center at (2R, 0, 0). The planets are touching.
A projectile is launched from the "North Pole" of the first planet at (0, 0, R) with its initial velocity pointed in the direction of the vector (1, 0, 1).
Let the escape speed relative to the planet's surface be ve. Note that here, the escape escape is for a single planet in isolation (following the typical convention).
With the given launch vector, let v0 be the minimum launch speed for the projectile to reach the "North Pole" of the second planet at (2R, 0, R).
How are the two speeds ve and v0 related?
There may be a way to exploit the symmetries in the problem -
say for example if the
trajectory is symmetric like the gravitational potential around a line between the spheres.....
and if the apex is exactly on this line then the x accelerations going up and coming will cancel. But I am not sure about this.
This idea can be tested with a "3-body" simulation with two fixed point masses (so really a 1-body simulation). The method is standard: at each time step the force on the projectile is computed. Dividing by its mass gives its acceleration, and multiplied by delta_t gives the velocity increment, and finally the new velocity x delta_t gives the position increment. For fun we choose Earth parameters.
Well I wrote a simulation
here with output
here (with 11 different V0's between 95% and 105% Vesc. Lo and behold Vesc worked best.
(A couple of the columns suffer overflow when the projectile gets too close to the point mass at the center of the planet and slingshots....)
While plotting these would be interesting the better question is Why Vesc? (Or very very near Vesc? ) I believe the answer lies in the system's symmetry where a symmetric trajectory will wind up with exactly the same E_kin and E_pot with E_Pot caused at the N pole of either N pole being identical. Not a very scientific proof, I realize.
I suspect in the syms Vx returns to zero as well.
Edited on June 27, 2019, 2:41 am
one method
