Acceleration in Ellipse

In summary: If you want to model the motion of a planet around the sun, you'll need to use a different parameterization, such as the one given in the answer to the previous question.
  • #1
schaefera
208
0
If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus. (Take, for example, t=pi/2... with this, the position is along the y-axis at a distance b, and acceleration points toward the origin, not the ellipse's focus).

That is, the acceleration is always directed normal to velocity, which should only happen in a circle... so what is wrong with my math?
 
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  • #2
schaefera said:
If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus. (Take, for example, t=pi/2... with this, the position is along the y-axis at a distance b, and acceleration points toward the origin, not the ellipse's focus).

That is, the acceleration is always directed normal to velocity, which should only happen in a circle... so what is wrong with my math?

The acceleration is not normal to the velocity except when a = b, the case of the circle.

In the motion of a planet, the parameterization is not the same as the one you have given.
Try solving for the parameters starting with Newton's law of gravitation
 
  • #3
In the parameterization above, isn't velocity normal to acceleration?
 
  • #4
schaefera said:
In the parameterization above, isn't velocity normal to acceleration?

no. the inner product is

a^2sintcost - b^2bsintcost

In planetary motion, the acceleration is not normal to the ellipse except at extreme points.

Generally there is a component of acceleration that is tangent to the ellipse in addition to the normal centripetal component. I think of the direction centripetal component as being rotated by the tangent component so that the total acceleration points towards the focus of the ellipse.
 
  • #5
schaefera said:
If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus.
Yours is but one of many (an infinite number) of parameterizations of an ellipse with the center at the origin. It is not *the* parameterization to use to describe an orbiting body. For one thing, your parameterization describes an ellipse with the center at the origin of the reference frame. The Sun is at one of the foci of an ellipse.
 

1. What is acceleration in an ellipse?

Acceleration in an ellipse refers to the rate of change of velocity at a given point on the ellipse. It is a measure of how much an object's velocity is changing as it moves along the curved path of the ellipse.

2. How is acceleration calculated in an ellipse?

Acceleration in an ellipse can be calculated using the formula a = v^2/r, where a is the acceleration, v is the velocity, and r is the radius of curvature at a given point on the ellipse. This formula follows from Newton's Second Law, which states that force equals mass times acceleration.

3. How is acceleration different in an ellipse compared to a circle?

In a circle, the acceleration is always directed towards the center, while in an ellipse, the acceleration varies in direction and magnitude at different points along the curve. This is due to the changing radius of curvature in an ellipse, whereas a circle has a constant radius of curvature.

4. How does acceleration affect the motion of an object in an ellipse?

The acceleration in an ellipse affects the speed and direction of an object's motion along the curve. As the acceleration changes, the object's velocity also changes, resulting in a curved path. This is why objects in elliptical orbits appear to speed up and slow down as they move around the ellipse.

5. Can acceleration in an ellipse be negative?

Yes, acceleration in an ellipse can be negative if the object is moving in the opposite direction of the changing radius of curvature. This can occur at points where the object is moving away from the center of the ellipse, causing the acceleration to be directed towards the outside of the ellipse.

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