Orbital Motion Mechanics

In summary, the conversation discusses a problem involving a particle in a plane, projected from an apse, with a specific speed and a repulsive force. The goal is to find the equation for the particle's motion in terms of theta. The conversation goes on to explain the process of solving the problem, including integrating both sides of the equation and using trigonometric functions to find the final equation for u.
  • #1
Master J
226
0
Looking over past problems and these types of ones always bothered me. Really need the help here guys.

A particle moves in a plane, projected from an apse, with speed SQRT(t/b), at a distance b, under a repulsive force t(u)^2 + 2bt(u)^3.

Show the motion is u = (-1/3b) + (4/3b)COS(SQRT(3)O)

O is theta, the angular term. u = 1/r. t and b are constants.


So the orbit equation is:

second derivative (du/dO) + u = [-F(1/u)]/(h^2)(u^2) per unit mass

the right side I work out at the start as: (1/b) + 2u

There is a trick then to multiply across by 2(du/dO), which allows one to integrate. But after that I don't know where to go. I get a horrible integral!

(du/dO)^2 - u^2 = (2/b)u - (3/b^2) Is this right first of all??<<<<<<<



Cheers
 
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  • #2
in advance!!!Yes, that is correct. Now you can integrate both sides with respect to θ to get the equation for u in terms of θ. On the left side, you will have an integral of (du/dθ) and on the right side, you will have an integral of u:Integral (du/dθ) = Integral u = [(1/3b) + (4/3b)cos(SQRT(3)θ)] So the motion is u = (-1/3b) + (4/3b)cos(SQRT(3)θ).
 
  • #3
for reaching out and asking for help. Orbital motion mechanics can be quite challenging, so it's great that you're seeking assistance.

Firstly, let's break down the given information. We have a particle moving in a plane, projected from an apse (a point where the orbit is closest to the attracting object), with a speed of SQRT(t/b) at a distance b. This particle is under the influence of a repulsive force given by t(u)^2 + 2bt(u)^3, where t and b are constants. The goal is to show that the motion of this particle can be represented by the equation u = (-1/3b) + (4/3b)COS(SQRT(3)O).

To begin, we can use the orbit equation, which describes the motion of a particle under the influence of a central force. This equation is given by:

(d^2u)/(dO^2) + u = (-F(1/u))/m(u^2)

where F is the magnitude of the central force, m is the mass of the particle, and O is the angular position. In this case, the central force is the repulsive force described above, so we can substitute it into the equation:

(d^2u)/(dO^2) + u = [-(t(u)^2 + 2bt(u)^3)]/(m(u^2))

Next, we can simplify the right side by factoring out a u^2 and rearranging:

(d^2u)/(dO^2) + u = -(1/m)[t + 2bu](u^2)

Now, we can use the given information that the particle's speed is SQRT(t/b) at a distance b to find the values of t and b. We know that the speed is equal to the derivative of the position with respect to time, so we can set up the following equations:

SQRT(t/b) = du/dt

b = u

Solving these equations, we get t = u^2 and b = u. Substituting these values into the equation above, we get:

(d^2u)/(dO^2) + u = -(1/m)[u^2 + 2u^3](u^2)

Next, we can use the given information that O is theta, the angular term, and u = 1/r. This allows us to rewrite
 

What is orbital motion mechanics?

Orbital motion mechanics is the study of the motion of objects in orbit around a larger body, such as planets orbiting the sun or satellites orbiting Earth. It involves understanding the forces and equations that govern the motion of these objects.

What are the two main factors that affect orbital motion?

The two main factors that affect orbital motion are the gravitational force between the two bodies and the initial velocity of the orbiting object. These factors determine the shape and size of the orbit.

How is orbital speed calculated?

Orbital speed is calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of the larger body, and r is the distance between the two bodies. This formula shows that the orbital speed is inversely proportional to the distance from the center of mass.

What is Kepler's First Law of Planetary Motion?

Kepler's First Law of Planetary Motion states that all planets in the solar system move in elliptical orbits, with the sun at one focus of the ellipse. This means that the distance between the planet and the sun varies throughout its orbit.

How does orbital mechanics impact space missions?

Orbital mechanics is crucial for planning and executing space missions. It helps scientists and engineers calculate the necessary trajectory, launch velocity, and fuel requirements for a successful mission. It also helps with orbital maneuvers, such as rendezvous and docking with other spacecraft, and reentry into Earth's atmosphere.

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