Gravity and velocity of a body in an elliptical orbit

In summary, there are two methods to determine the velocity of a body at its farthest point from a star: one involving angular momentum conservation and the other involving centrifugal and gravitational forces. The first method is more accurate for elliptical motion, while the second method is more applicable for circular motion. However, the second method does not take into account the gravitational potential energy and the kinetic energy of the central body. Thus, the first method is preferred when considering a general case.
  • #1
Suraj M
Gold Member
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consider a body revolving around a star and having a velocity v when closest to the star ( distance r) then the velocity of the body at a point farthest ( distance R) is?
1)by angular momentum conservation ::
r × mv = R× mV
»V = (r/R).v

2) By centrifugal and gravitatinal force
v²/r = GM/r²
v² α 1/r
in this we can get V = (r/R)½ v

the angular momentum method is the right method ... but what am i missing out on from method no. 2??
 
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  • #2
Suraj M said:
v²/r
That's the centripetal acceleration for circular motion, not elliptical motion in general.
 
  • #3
A.T. said:
That's the centripetal acceleration for circular motion, not elliptical motion in general.
considering a small sector of the particles motion ..can't we use this
if not then how do we solve this through gravitational force??
 
  • #4
Suraj M said:
considering a small sector of the particles motion ..can't we use this
No.

Suraj M said:
if not then how do we solve this through gravitational force??
Gravitational potential energy vs. kinetic energy?
 
  • #5
A.T. said:
No.

Gravitational potential energy vs. kinetic energy?
how'd you do that? Is the sum of the KE and GPE const.?? that's not necessary is it?
 
  • #6
Suraj M said:
Is the sum of the KE and GPE const.?? that's not necessary is it?
If M >> m then it's approximately true in the approximately inertial rest frame of M. Otherwise you have to consider the KE of M too.
 
  • #7
A.T. said:
If M >> m then it's approximately true in the approximately inertial rest frame of M. Otherwise you have to consider the KE of M too.
but in that way i would need the mass of the central body to find V ...and that's not a general case..
 

1. What is the difference between gravity and velocity in an elliptical orbit?

Gravity is the force of attraction between two objects, in this case, the body and the object it is orbiting. Velocity is the speed and direction at which the body is moving while in orbit. In an elliptical orbit, the body experiences both gravity and velocity, with gravity keeping the body in orbit and velocity determining the shape and speed of the orbit.

2. How does the gravity and velocity of a body change in an elliptical orbit?

In an elliptical orbit, the gravity of the body remains constant, while the velocity changes as the body moves closer or farther away from the object it is orbiting. As the body moves closer, the velocity increases, and as it moves farther away, the velocity decreases.

3. Why does a body in an elliptical orbit have a faster velocity at perihelion than at aphelion?

Perihelion is the point in an elliptical orbit where the body is closest to the object it is orbiting, while aphelion is the point where the body is farthest away. At perihelion, the body is experiencing stronger gravitational pull, resulting in a higher velocity. At aphelion, the body is experiencing weaker gravitational pull, resulting in a slower velocity.

4. How does the shape of an elliptical orbit affect the gravity and velocity of a body?

The shape of an elliptical orbit affects the strength of the gravitational force and the speed of the body. A more elongated orbit, also known as a high eccentricity, results in a larger difference between the velocity at perihelion and aphelion, while a more circular orbit, or a low eccentricity, results in a smaller difference.

5. Can the gravity and velocity of a body in an elliptical orbit be calculated?

Yes, the gravity and velocity of a body in an elliptical orbit can be calculated using mathematical formulas such as Newton's law of universal gravitation and Kepler's laws of planetary motion. These calculations take into account the mass and distance of the two objects involved in the orbit.

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