Vector Calculus: When Should the Enterprise Fire?

Date: 7/21/96 at 14:48:53
From: Anonymous
Subject: Vector Calculus: When Should the Enterprise Fire?

I'm working on a problem that I just can't solve and need some 
help. 

The starship Enterprise has been captured by a tractor beam and is 
being held in an elliptical orbit around a planet by the beam. 
If F represents the focus of the ellipse and P is the position of 
the ship on the ellipse, then the vector FP can be written as a sum 
vector T + vector N where vector T is tangent to the ellipse and 
vector N is normal (perpendicular) to the ellipse.  The engines must 
be fired when the ratio |vector T|/|vector N| is equal to the 
eccentricity of the ellipse.  Your mission is to save the Enterprise.

This is all of the information given in the problem.  We weren't 
given the constant of eccentricity, or a directrix of the ellipse. 
We also weren't given the mass of the planet or anything like that.  
We were told that this problem is also given to the instructors in 
beginning level college calculus classes.  Help, any clues would be a 
great help.

So far...

I've tried representing the problem in a polar coordinate system, but 
get lost in the conversion from an ellipse in the x^2/a^2 +... into 
the rcos( ), rsin( ). What I get is [a cos ( )]^2 + [b sin ( )]^2 = 1   
where ( ) represents angle theta. 

If this is right, I don't know what to do with it. I know that 
eccentricity in a polar conic system is r= 1/[1+ e cos( )] where e is 
eccentricity and ( ) is the angle.  Because the orbit is elliptical, I 
assume that e < 1.  I'm wondering how the foci of an ellipse are 
represented in the graph of a polar equation (they aren't the center, 
are they?)  

Without a given directrix, or the value of eccentricity, I can't get a 
representation of "where on the ellipse" the ship should be.  I know 
that T and N are the component vectors of FP.  I'm pretty sure that 
using the polar coordinate system is the way to go, but I just can't 
get started. 

If you can just give me some clues, I may be able to finish this off, 
but I don't know where to start.

Thanks in advance,
Chris Oliphant
Student of Mechanical Engineering and Mathematics
California State University, Long Beach
Colleges of Engineering and Natural Science

Date: 7/21/96 at 20:32:24
From: Doctor Anthony
Subject: Re: Vector Calculus: When Should the Enterprise Fire?

What you require here is the (p,r) equation of the ellipse, since you 
are given the ratio of lengths of the normal from F to the tangent, 
and the length along the tangent from foot of this normal to the point 
P.  Any standard textbook will show you that the equation of the 
ellipse with a focus as origin, and using (p,r) coordinates is:

  b^2/p^2 = 2a/r - 1   ....(1)   

where 2b = minor axis, 2a = major axis, r= distance from origin 
(focus) to point P, and p = perpendicular distance from the origin to 
the tangent at P.

The length from foot of normal along the tangent can be found using 
pythagoras, and is given by  sqrt(r^2 - p^2), and we require the ratio 
of this length to the length 'p' to equal the eccentricity of the 
ellipse. So squaring we have 

   (r^2-p^2)/p^2  = e^2

    r^2 - p^2 = e^2*p^2

          r^2 = p^2(1+e^2)   and so  p^2 = r^2/(1+e^2)

We now substitute for p^2 in the equation of the ellipse, shown as (1) 
above. 

  b^2(1+e^2)/r^2 = 2a/r - 1

  b^2(1+e^2) = 2ar - r^2

  r^2 - 2ar + b^2(1+e^2) = 0

So r = [2a +or- sqrt{4a^2 - 4b^2(1+e^2)}]/2

Now we use relationship b^2 = a^2(1-e^2)

r = [2a +or- sqrt{4a^2 - 4a^2(1-e^2)(1+e^2)}]/2

  = [2a +or- sqrt{4a^2 - 4a^2(1-e^4)}]/2

  = [2a +or- sqrt{4a^2*e^4}]/2

  = a +or- ae^2

So two solutions are r = a(1+e^2)  and r = a(1-e^2)

The rockets must be fired when the Enterprise is at either of these 
two distances from the focus of the ellipse. We must assume that the 
lengths of the major and minor axes of the ellipse (and hence the 
eccentricity) are known.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/22/96 at 4:8:28
From: Jackie Puppet
Subject: Is there another way to represent this answer?

I was wondering if there was a way to solve the problem without 
assuming that the lengths of the major and minor axes of the ellipse 
are known (where the answer isn't in terms of the eccentricity and 
length of the minor axis). In the problem given to us, it said nothing 
about eccentricity, major or minor axis, or anything but the ratio 
being known. In other words... is there a way to represent the answer 
in terms of the vector magnitudes or something? Is there a relation 
that can be formulated between the vectors as given and the 
eccentricity, without having to know the major and minor axis? Is 
there a way to use them, then somehow cancel them or do a substitution 
so they aren't in the answer equation "R = A*(1 +OR- E^2)"?

Thanks so much for your time,
Chris

Date: 7/22/96 at 9:2:4
From: Doctor Anthony
Subject: Re: Is there another way to represent this answer?

If you imagine yourself sitting in the spacecraft, then the time to 
fire the rockets is when something you can observe is at its correct 
value. The only thing you can observe is your distance from the 
planet. The ratio of the vectors along and perpendicular to the 
tangent are ratios of lengths of an unseen imaginary triangle in space 
- even the direction of the tangent of your path is not easily 
determined - and if you were to try to use the ratios of these sides 
of the triangle you require to know the eccentricity, since that is 
what the ratio has to be. In practice the eccentricity is known by 
knowing the lengths of the major and minor axes, so the answer 'Fire 
the rockets when distance from the planet is a(1 +or- e^2)' is the 
only realistic solution.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/