Calc III Lab #5: Escape Velocity
Again return the inital condition to the point
(x,y)=(0,4). Let the initial velocity be
(vx,vy)=(1,0) and compute a trajectory with these initial
conditions. You may have to zoom out
in order to see what is happening to the trajectory. Try other values
for vx, but keep vy=0 for now.
The numerical experiment suggests that somewhere between
vx=0.1 and vx=1.0, a transition occurs: for small
values of vx, the trajectory is a bounded orbit.
But for larger values of vx, the trajectory is
unbounded. This critical value of vx is called
escape velocity.
Long before calculus (and differential equations) were discovered, Kepler proved that all orbits of the type we
are investigating are conic sections: ellipses, circles, parabolas,
and hyperbolas.
Question #3:
Experimentally estimate the value of escape velocity for the current
dynamical system. You do not need to be very accurate; just estimate
the value to within 0.05.
Question #4:
The moral of this next question is don't trust everything that the
computer shows you.
Return the
viewing range of the TwoD View Window to its default range.
Set the initial conditions in DsTool to be (x,y)=(0,4) with
velocity (vx,vy)=(0.001,-0.099).
- What trajectory does the particle travel? (Hint: you will have to
compute about 5000 time steps of the trajectory and then
zoom in on the apparent "line".)
- Is this a conic section?
Think about what the gradient vector looks like near the origin and try to determine
what might be the cause of this numerical error. We will
discuss the source of this error in a future class.
Go To
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Mon Aug 15 07:48:30 2005