• Support PF! Buy your school textbooks, materials and every day products Here!

Two-dimensional motion under Central Force

  • Thread starter Eagletsam
  • Start date
  • #1
2
0

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.

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
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.
 
  • #3
2
0
Thanks, very much, fzero, for early intervention.

I agree entirely that the orbit may not be elliptical!
 

Related Threads on Two-dimensional motion under Central Force

Replies
0
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
2K
Replies
10
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
2
Views
3K
Replies
1
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
1
Views
866
Top