Proving Elliptical Trajectory Acceleration Vector Passes Through Focus

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In summary, the conversation discusses proving that a particle's acceleration vector follows the focus of an ellipse, as demonstrated by the given equations for its trajectory and the differentiation of its position vector. Ideas regarding how to correctly describe a planet's motion using this information are also mentioned.
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aim1732
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If a particle's trajectory is defined by the law x=acos[pt] and y=bsin[pt] where t is parameter of time then we have to prove that it's acceleration vector passes through the focus of the conic----ellipse in this case as can be clearly seen.
If we write out the position vector in the vector notation and differentiate twice we get a=-p2r and this clearly is directed towards the centre of the axes system and not the focus.Any ideas?
 
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aim1732 said:
If we write out the position vector in the vector notation and differentiate twice we get a=-p2r and this clearly is directed towards the centre of the axes system and not the focus.
Correct.

Any ideas?
Ideas regarding what? I assume you are trying to prove Kepler's first law. The given equation does not describe a planet's motion. You need to find the right equation.
 
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have to prove that it's acceleration vector passes through the focus of the conic----ellipse in this case as can be clearly seen.

I meant this.
 

Related to Proving Elliptical Trajectory Acceleration Vector Passes Through Focus

1. What is an elliptical trajectory?

An elliptical trajectory is a path that an object follows when it is accelerated in a non-circular orbit around a central point, such as a planet or star. It is characterized by an oval shape, with two focal points.

2. What is the focus of an elliptical trajectory?

The focus of an elliptical trajectory is one of the two focal points that define the shape of the path. It is the point at which the acceleration vector passes through, and it is at a fixed distance from the center of mass of the object.

3. How is the acceleration vector of an elliptical trajectory determined to pass through the focus?

The acceleration vector of an elliptical trajectory can be determined by using the laws of motion, specifically Newton's second law which states that the net force acting on an object is equal to its mass multiplied by its acceleration. By calculating the forces acting on the object and the direction of its motion, the acceleration vector can be determined to pass through the focus.

4. Can an elliptical trajectory ever deviate from passing through the focus?

In a perfect system with no external forces acting on the object, the acceleration vector of an elliptical trajectory will always pass through the focus. However, in real-life situations, there may be external forces present that can cause slight deviations from this path. These deviations can be accounted for and minimized through precise calculations and adjustments.

5. Why is it important to prove that the acceleration vector passes through the focus in an elliptical trajectory?

Proving that the acceleration vector passes through the focus is important because it confirms that the object is following a stable and predictable path. It also allows for accurate calculations of the object's motion and trajectory, which is crucial in fields such as astrodynamics and space exploration.

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