We received several interesting responses to the Santa’s Reindeer problem. The best one was by Dave Montesano of IMAC Motion Control Corporation. What Dave did was recognized that the harness between and reindeer should be treated as a two force member, exerting only an equal and opposite force vector on each end and carrying no moments. He then analyzed a Free Body Diagram with N reindeers in front and 8-N reindeers and a sleigh behind. Analyzing the ratio of X and Y force vector components behind different reindeer would tell the shape of the system.
From Dave:
Since the information states that the sleigh will not experience a change in speed or altitude, the sum of the forces on the sled should equal zero in both the X and Y direction. Also, since each reindeer contribute equally to lift and thrust, and drag is equal for each reindeer, the forces at each reindeer should equal to zero in X and Y. Therefore, we can derive an equation to determine the angle Theta at Reindeer “N” with respect to the ground as:
Tan (Theta) = Tension in Y / Tension in X
Another thing to note is that because the sleigh is in straight and level flight (and assumed to not be accelerating also) and all reindeer thrust and lift equally, the net thrust of a reindeer is 1/8 the drag of the sleigh and the net lift of a reindeer is 1/8 the weight of the sleigh.
So, the tension vector after any reindeer N can be calculated as the N times the net force of the reindeer in front of it, or
- TY = N * 1/8 WS where WS is the Weight of the Sleigh and TY is the Y component of the tension vector and
- TX = N * 1/8 DS where DS is the Drag of the Sleigh and Tx is the Y component of the tension vector
Solving as Dave recommends for TY/ Tx
TY/ TX = WS/ DS
reveals that the tension vector and therefore the angle between reindeer is independent of reindeer position. Therefore the reindeer are in a straight line and the correct answer is C.
Bruce Polson points out a couple of issues with the problem statement:
1. Don't Reindeer shed antlers during winter?
You got me
2. I would reason it would depend on the rigidity of the hitch. A rigid hitch would be necessary for braking, and a straight hitch would be required for parking on the rooftop. Therefore, it needs to work in a straight line somehow.
That’s a good point. However, because this is a flying machine and weight reduction is important, even a rigid hitch should have some flexibility like an airplane wing. So answers A, B, and C are still available. Your point about having to land on a rooftop invites some further optimization. In order slow down and land, the front reindeers would have to push down while the back reindeer lift extra to keep this sleigh under control. Because the reindeer in the rear have to lift extra, it makes sense to sort the reindeer by their lift capacity with the highest lift reindeer in the rear. Since all of Santa’s reindeer are very capable, this sort puts the highest thrust reindeer in front. In this configuration, the correct answer would Concave Down.