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/ |
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