REFUEL
Rendezvous for aircraft refueling
BRIEF ABSTRACT
This is an open-ended problem for differential calculus (using notions of position and velocity) in which students determine the flight position of a refueling plane.
GENERAL INFORMATION
FileName:    REFUEL
Full title:     Rendezvous for aircraft refueling
Last Update: 6/3/96
Developer: 
Kimberly Foltz, Mathematics and Computer Science Division,
Indiana Academy for Science, Mathematics, & Humanities, Muncie IN  47306  USA
Contact:  
Aaron Klebanoff, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. 
Phone:  812-877-8151.  Email:  Klebanoff@rose-hulman.edu.  
FAX:  812-877-3198.
Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
STATEMENT OF PROBLEM 
A jet plane is in flight with a constant velocity of 300 mph headed due north.  A fuel plane leaves from a base 200 miles directly south of the jet, flying the same course as the jet.  The fuel plane must catch up with the jet in order to perform an in-air refuel.   The refueling process will take 30 minutes to complete.  Assume that each plane will be at the appropriate relative altitude to perform the refueling.

In order to facilitate the refuel procedure, the speeds of the jet and the fuel plane must match at the instant when the fuel plane meets the jet.  Also, once contact is made between the two planes, the fuel plane must hold (i.e., maintain the same velocity and relative position as the jet) with  the jet throughout the 30 minutes of the refueling.   The fuel plane must then return to the base so that its total flight time is 3 hours.

Calculate an appropriate position function for the fuel plane so that its position meets the specified conditions.
KEYWORDS  
Position, velocity, continuity, differentiability, modeling.
TEACHER NOTES
ISSUES RELATED TO THE PROBLEM 
The problem statement purposely leaves students room to offer many different reasonable solutions.  It's important for students to know that there is no single correct answer, so they must explain their assumptions and show all work.
Prerequisites
Differentiation, velocity.
Time allotment - time management
This could be used as an extended assignment over a two week period.  Students might be assigned to work on this project in groups of three.  This allows students to work cooperatively and makes the grading easier for the instructor.
Expectations 
If students need a hint to start the modeling process, a suggestion might be to begin by looking for a second degree polynomial to model the first portion of the flight of the fuel plane.  
Future payoffs
Extensions
The solution put forth has continuous position and velocity.  Modify your solution to make acceleration continuous as well.  Further, are the g-forces acceptable to the pilots?
References and Sources
The problem was inspired by "The Bicycle Race" problem in the book Student Research Projects in Calculus (Cohen, Gaughan, Knoebel, Kurtz, and Pengelley, The Mathematical Association of America, Spectrum Series, 1991, p. 94).  In this problem, students are given the cubic function which models the position of the support car and must find the appropriate constant velocity for the biker in order to facilitate the instantaneous hand-off of a cold drink.
POSSIBLE SOLUTION(S)
The jet is 200 miles north of the base.  Thus, relative to the fuel plane, the jet has an initial position of 200 miles.  Also, the jet is flying with a constant speed of 300 mph, so its position can be expressed as a linear function of the form  s(t) = v0 t + s0.   
jet[t_]:=300 t+ 200
There are at least three constraints that govern the first portion (take-off to contact) of the travel of the fuel plane:  (1) initial position, (2)  position and (3) velocity at the time of "contact".  We also consider (4) the final return position, as a fourth constraint to assure that the contact time occurs early enough.  This allows for a four parameter, one of which will be time of contact, model.  Thus, the position of the fuel plane can be modeled by...
f[t_] = a t^2+ b t + c 
             2
c + b t + a t
The time of contact for the planes and the parameters a, b, and c of the quadratic which satisfy the stated conditions can be found by simultaneously solving the system of four equations.
sol=Solve[{f[t]==jet[t],f'[t]==jet'[t],f[3]==0,
			f[0]==0},{a,b,c,t}]//N
{{c -> 0, b -> 120.639, a -> -40.213, t -> -2.23014}, 
 
  {c -> 0, b -> 746.028, a -> -248.676, t -> 0.896805}}
Applying these values to the quadratic model produces a quadratic function which meets the necessary conditions at the time of contact.
fuel[x_]=f[x]/.sol[[2]]
                     2
746.028 x - 248.676 x
Plot[{fuel[t],jet[t]},{t,0,5},
			AxesLabel->{"Time","Distance North of Base"}];
Solve[fuel[t]==jet[t],t]
{{t -> 0.896805}, {t -> 0.896805}}
Using this model, the fuel plane will meet the jet at t = .897 hours.  The quadratic is not a good model after this point because the fuel plane must hold the position of the jet for 30 minutes, not just for an instant.  For this 30 minute interval, the position of the fuel plane will be determined by a linear function with constant slope of 300 mph.   A point on this line is the point (time,  f(time) ).  The coordinates of this point can be substituted into the equation y = mx + b to find the y-intercept of the linear function.
time=t/.sol[[2]];
sol2=Solve[fuel[time]==300 * time + yint,yint]
{{yint -> 200.}}
fuelwjet[x_]=300 x + yint/.sol2[[1]]
200. + 300 x
This linear function will model the necessary movement of the plane in the 30 minutes following the time of contact.  The best model for the jet will be a piecewise-defined function which uses the quadratic function before the contact time and then the linear function after the contact.
fuel2[t_]:=If[ t<=time, fuel[t],
			If[ time< t ,  fuelwjet[t] ]  ,  ];
Plot[{fuel2[x],jet[x]},{x,0,5},
			AxesLabel->{"Time","Distance North of Base"}];
One more refinement is now necessary for this piecewise model of the position of the fuel plane.  After the refueling has been completed at time .897 + .5 hours, the fuel plane must return to base with a total flight time of 3 hours.  Thus, the position of the fuel plane must be 0 when t = 3.

The fuel plane's departure from the jet should happen gradually, so one constraint is that the position and velocity at the time of "break away" must be the same as the position and velocity of the linear position function that the plane had been following.  (The plane's position must be continuous and differentiable; i.e., the plane's position and velocity must be continuous functions.)

The three constraints allow for three unknowns.  While the time of contact was an unknown when developing the model for the first subinterval, the time of break away is known in terms of contact time.  Thus, the fuel plane position can again be modeled by a function which uses three parameters.  This is the quadratic f[t] which was used previously.
f[t]
             2
c + b t + a t
The parameters a, b, and c of the quadratic which satisfy the stated conditions can be found by simultaneously solving the system of three equations.
sol3=Solve[{ f[time+.5]==fuelwjet[time+.5],
			  f'[time+.5]==fuelwjet'[time+.5],
			  f[3]==0},{a,b,c}]//N
{{a -> -427.977, b -> 1495.6, c -> -635.01}}
Applying these values of the parameters to the quadratic model f[t] gives the model for the fuel plane position in the final time subinterval.
fuelhome[t_]=f[t]/.sol3[[1]]
                              2
-635.01 + 1495.6 t - 427.977 t
The position can now be defined in each of the three distinct subintervals of time (time 0 to the time of contact,  time of contact to the time of break-away, and time of break-away to the time of landing) by altering the previous piecewise function to include this third definition.
fuel3[t_]:=If[ t<=time, fuel[t],
			If[ time< t <=time+.5,  fuelwjet[t],
			If[ time + .5 < t <=3, fuelhome[t], ]  ]]
Plot[{fuel3[t],jet[t]},{t,0,3},
			AxesLabel->{"Time","Distance North of Base"}];
We now have a function which meets the necessary conditions for the flight of the fuel plane.  After a successful rendezvous, the fuel plane returns to the base right on time.
ISSUES IN SOLUTION
A computer algebra system is quite useful for the solution of this problem because it involves solving a system of four equations simultaneously.
It might be necessary to review with students the notion of parameters and constraints.  A hint could even be provided that the students should begin by looking for a second degree polynomial to model the first portion of the flight of the fuel plane.  
This is not the only possible answer.  In constructing the first portion of the model, the condition that the return to base be completed in 3 hours is not essential in finding a parabola that has the appropriate initial position and meets the condition of matching the velocity of each when the positions meet.   It is reasonable to use this as a constraint in order to guarantee that the meeting will occur well before the 3 hour flight time is exhausted.  However, students might choose to use a different fourth constraint in order to find the values of the four parameters.  In that case, they may find other functions that complete the rendezvous.
Some calculus problems have students complete an abstract exercise of finding the parameter(s) such that a piecewise function is continuous and differentiable at the "connecting point" of the definition subintervals.  This problem provides a practical motivation for having this type of connection for the pieces of the graph.