Derive equation of trajectory of a body around a fixed body attracted by gravity

In summary, the conversation discusses a homework problem involving finding the equation of trajectory for a body with mass m around a fixed spherical body with mass M. The attempt at a solution involved using Newton's Law of Gravity and integrating, but a solution was not reached. The problem is known as the Kepler problem and the trajectory can be an ellipse, parabola, or hyperbola depending on the given parameters.
  • #1
gupta.shantan
2
0

Homework Statement



There is a fixed spherical body of mass M whose center is to be taken as origin. Another body of mass m whose initial position vector [itex]\vec{r}[/itex] is given. This body is projected with initial velocity [itex]\vec{v}[/itex]. Find the equation of trajectory of body with mass m around the body with mass M.

Homework Equations



Will the trajectory be an ellipse, just like the orbit of Earth around the sun?

The Attempt at a Solution



I tried solving the position using Newton's Law of Gravity. I also tried using the formula a = v dv/dx and integrating but was unable to reach a solution.

Any help is greatly appreciated...
 
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  • #2
Your problem is known as Kepler problem.

It is possible to derive the trajectory with Newton's Law of gravity, but this is an ugly calculation, at least ~2 pages long, involving polar coordinates, some substitutions and messy integrals.

Depending on the velocity, the radius and the masses M and m, the trajectory can be:
- an ellipse
- a parabola
- a hyperpola
which are all conic sections
 
  • #3
thank you mfb
but i am willing to go through the messy mathematics. So can u please help me by giving me a link to where this Kepler problem has been solved.
 

What is the equation of trajectory for a body around a fixed body attracted by gravity?

The equation of trajectory for a body around a fixed body attracted by gravity is known as the Kepler's equation. It is given by r = a(1-e cosθ), where r is the distance between the two bodies, a is the semi-major axis, e is the eccentricity of the orbit, and θ is the angle between the line of apsides and the position of the orbiting body.

What is the significance of the equation of trajectory for a body around a fixed body attracted by gravity?

The equation of trajectory provides a mathematical description of the path that a body takes when orbiting around a fixed body under the influence of gravity. It helps us understand the motion of objects in space and is crucial in predicting the behavior of celestial bodies.

How is the equation of trajectory derived?

The equation of trajectory is derived using the laws of motion and the law of universal gravitation. It involves solving differential equations that describe the motion of the orbiting body and the force of gravity between the two bodies. This results in the Kepler's equation, which is a solution to the problem of two bodies orbiting each other under the influence of gravity.

What factors affect the trajectory of a body around a fixed body attracted by gravity?

The trajectory of a body around a fixed body is affected by several factors, including the mass of the two bodies, the distance between them, the initial velocity of the orbiting body, and the angle of the orbit. These factors determine the shape, size, and orientation of the orbit.

How does the equation of trajectory change for different types of orbits?

The equation of trajectory remains the same for all types of orbits, but the values of the parameters such as eccentricity, semi-major axis, and angle of the orbit will vary. For example, a circular orbit will have an eccentricity of 0, while an elliptical orbit will have an eccentricity between 0 and 1. The equation can also be modified to include other factors such as the effects of other forces or relativistic effects.

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