Elliptical Motion: Formulating Object Motion & Finding Theta

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The discussion focuses on formulating the elliptical motion of an object perceived to move linearly from an observer's perspective. It emphasizes the use of a position vector that describes this elliptical motion, taking into account the object's velocity and its perpendicular distance from the observer. The conversation suggests deriving the rate of change of the angle theta as the object moves. It also proposes writing the general equation of motion for an ellipse to analyze the lateral component over time. The observer's distance is assumed to be significant, allowing for simplifications in the calculations.
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How can we formulate the elliptical motion of an object apparently observed by us when it moves linearly infront of us. The position vector with respect to us makes an elliptical motion at that time. To start with let's take the velocity as v, the perpendicular distance of the line of motion from observer as x and now find the rate of change of theta with respect to us.
 
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One could right the equation of motion for an ellipse, and then determine the lateral component as a function of time with respect to the line of the observer. Assume that the observer is far way. If \theta is the position angle with respect of the object from some reference, say the major axis, then one could write the general equation for an observer at \theta_o
 

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