Elliptical Motion: Formulating Object Motion & Finding Theta

[vaishakh]

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.

 

[Astronuc]

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 $\theta_o$

 

[charng]

I would first like to clarify that elliptical motion refers to the path followed by an object with a constant speed, where the distance from a fixed point (the focus) to the object is always proportional to the distance from the object to a fixed line (the directrix). This type of motion is commonly seen in celestial bodies, such as planets orbiting around a star.

To formulate the elliptical motion of an object observed by us when it moves linearly in front of us, we need to consider the position vector of the object with respect to us. This position vector will describe the location of the object at any given time, and it will change as the object moves.

If we consider the velocity of the object as v and the perpendicular distance of the line of motion from the observer as x, we can use trigonometric functions to describe the position vector and its relationship to the angle theta. The position vector can be represented as a function of theta, which will give us the distance of the object from the observer at any given angle.

To find the rate of change of theta with respect to us, we can use the chain rule of differentiation. This will give us the angular velocity of the object, which describes how quickly the angle theta changes with respect to time. This angular velocity will be constant for an object moving in an elliptical motion.

In conclusion, by considering the position vector, velocity, and angle theta, we can formulate the elliptical motion of an object observed by us when it moves linearly in front of us. This type of motion can be described using mathematical equations and can be used to accurately predict the path of the object in the future.