# Motion of particles close to the Earth

• flyleaf_
In summary, the conversation discusses two different solutions to a problem involving equations of motion in a rotating reference frame. The first solution uses Lagrangian mechanics and the second uses a rotating reference frame. However, there is uncertainty about whether the equations of motion from the two solutions are the same. The next steps involve using conservation laws and numerical methods to solve the problem, but there are some complications and difficulties in doing so. Additionally, it is mentioned that solving this problem analytically in the rotating reference frame may be challenging.
flyleaf_
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.

v<v_escape makes sure it lands again, but there are useful approximations that only work if v<<v_escape. Without these things get complicated.

Solving this analytically in the rotating reference frame will be really messy or even impossible.

The equations of motion in your first approach are simply from Newtonian mechanics. They work, of course. In practice you probably want to use the knowledge that free-fall trajectories are ellipses, and calculate parameters of these ellipses.

## 1. How does gravity affect the motion of particles close to the Earth?

Gravity is the force that pulls all objects towards the center of the Earth. This means that particles close to the Earth will experience a gravitational force that will affect their motion. The greater the mass of the Earth, the stronger the gravitational force will be.

## 2. What is the difference between weight and mass when it comes to the motion of particles close to the Earth?

Weight is a measure of the gravitational force acting on an object, while mass is a measure of the amount of matter in an object. When particles are close to the Earth, their weight will be affected by the gravitational force, but their mass will remain the same.

## 3. Can the motion of particles close to the Earth be affected by other forces besides gravity?

Yes, particles close to the Earth can also be affected by other forces such as air resistance, friction, and electromagnetic forces. These forces can either increase or decrease the speed or direction of the particles' motion.

## 4. How does the Earth's rotation and revolution affect the motion of particles close to the Earth?

The Earth's rotation and revolution have a minimal effect on the motion of particles close to the Earth. This is because the particles are also rotating and revolving with the Earth, so their motion is relative to the Earth's motion.

## 5. Is the motion of particles close to the Earth always constant?

No, the motion of particles close to the Earth can change depending on the forces acting on them. If there is a net force acting on the particles, their motion will change. However, if there is no net force, the particles will continue to move at a constant speed and direction.

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