# Two-dimensional motion under Central Force

• Eagletsam
In summary, the problem is to determine the radial and tangential orbital velocities for a particle under central force where the x and y coordinates are specified as functions of time. The attempt at a solution involves expressing the position vector and evaluating dr/dt, but this leads to messy calculations. Other attempts, such as eliminating t and using coordinate transformations, have also proved challenging. It is possible that the orbit is not elliptical, and may be a parabolic solution to a central force law.
Eagletsam

## Homework Statement

Some students have just drawn my attention to the problem of particle motion under central force where the x and y coordinates are specified as functions of time, such as

x(t) = A [ kt – cos(βt) ]
y(t) = B [ 1 – sin(βt) ],

(here A, B, β and k are constants).

## Homework Equations

The problem is to determine the radial and tangential orbital velocities (orbit assumed to be elliptical), and recover the canonical form of the equation (x2/a2 + y2/b2 = 1).
I have tried to figure this out for a few days without much success. Can anyone assist please?

## The Attempt at a Solution

Express position vector r as

r = [x2 + y2]1/2
and evaluate dr/dt to obtain velocity as function of t. But this leads to messy calculation, which does not yield the angular and radial dependence.

Equally, attempt to eliminate t and obtain an equation in x and y proves quite hard.

We have also thought of coordinate transformation from (x,y,a,b) to (u,v,c,d) where u,v are velocity axes, but could not quite effect that.

Since x(t) is unbounded, this doesn't look like a closed orbit. It's possible that this is a parabolic solution to a central force law, but you might want to verify that.

Thanks, very much, fzero, for early intervention.

I agree entirely that the orbit may not be elliptical!

## What is two-dimensional motion under central force?

Two-dimensional motion under central force refers to the movement of an object in a two-dimensional plane, where the force acting on the object is always directed towards a central point. This type of motion is commonly seen in celestial bodies orbiting around a larger mass, such as planets around the sun.

## What is the difference between centripetal and centrifugal force?

Centripetal force is the inward force that keeps an object moving in a circular path, while centrifugal force is the outward force that acts on an object moving in a circular path. In two-dimensional motion under central force, centripetal force is the force that keeps the object moving towards the central point, while centrifugal force is the reaction force that is equal in magnitude but opposite in direction to the centripetal force.

## What are the key equations for calculating two-dimensional motion under central force?

The key equations for calculating two-dimensional motion under central force are Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration, and the centripetal force equation, which states that the centripetal force is equal to the mass of the object multiplied by its velocity squared, divided by the radius of its circular path.

## How does the central force affect the path of an object?

The central force affects the path of an object by constantly changing the direction of its velocity towards the central point, while the magnitude of its velocity remains constant. This results in a circular or elliptical path, depending on the strength and direction of the central force.

## What are some real-life examples of two-dimensional motion under central force?

Some real-life examples of two-dimensional motion under central force include the orbits of planets around the sun, the motion of satellites around the Earth, and the motion of electrons around the nucleus of an atom.

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