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flyleaf_

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- Homework Statement
- Problem: A spherical body is rotating with angular velocity (=Omega constant) on the z-axis. A person throws 3 balls with initial speeds v0:

1-) First ball to the north with angle pi/4 to the surface

2-) Second ball pi/2 degrees to the surface (right upwards)

3-) Third ball to the south with angle pi/4 to the surface

Find where the balls will land. Assume v<v_escape at any time.

- Relevant Equations
- v<v_escape at any time;

Radius of the planetary body = a;

Initial latitude = theta_0

I tried to solve the problem in 2 ways, first using lagrangian mechanics and second by putting a rotating reference frame on the initial take-off point.

However I cannot be sure if the equations of motion for the two solutions came out the same.

A-) Equations of motion from Lagrangian Solution:

d/dt(dx/dt) = -G*M_earth*x/((x^2+y^2+z^2)^3/2);

d/dt(dy/dt) = -G*M_earth*y/((x^2+y^2^z^2)^3/2),

d/dt(dz/dt) = -G*M_earth*z/((x^2+y^2+z^2)^3/2);

B-) Equations of motion from the rotating reference frame solution:

d/dt(dx/dt) = 2*omega*(dy/dt*cos(theta)-dz/dt*sin/theta);

d/dt(dy/dt) = -2*omega*dx/dt*cos(theta) - omega^2*a*sin(theta)^2-g;

When I use the equations from Lagrangian, I feel like I omitted something leaving some terms relating the orbital motion out. I am not sure if any of the two solutions are true. After I find the equations of motion, I will use the equations and the initial conditions to numerically solve the problem. Then I will use conservation laws at the problem to choose the best numerical method to solve this problem. However I am stuck at this stage. I would be glad if anyone helped.

However I cannot be sure if the equations of motion for the two solutions came out the same.

A-) Equations of motion from Lagrangian Solution:

d/dt(dx/dt) = -G*M_earth*x/((x^2+y^2+z^2)^3/2);

d/dt(dy/dt) = -G*M_earth*y/((x^2+y^2^z^2)^3/2),

d/dt(dz/dt) = -G*M_earth*z/((x^2+y^2+z^2)^3/2);

B-) Equations of motion from the rotating reference frame solution:

d/dt(dx/dt) = 2*omega*(dy/dt*cos(theta)-dz/dt*sin/theta);

d/dt(dy/dt) = -2*omega*dx/dt*cos(theta) - omega^2*a*sin(theta)^2-g;

When I use the equations from Lagrangian, I feel like I omitted something leaving some terms relating the orbital motion out. I am not sure if any of the two solutions are true. After I find the equations of motion, I will use the equations and the initial conditions to numerically solve the problem. Then I will use conservation laws at the problem to choose the best numerical method to solve this problem. However I am stuck at this stage. I would be glad if anyone helped.