260 The UMAP Journal 24.3(2003) the harmonic mean of its other two dimensions the ratio of which can be specified arbitrarily. Finally, for very large jumps, we may need to revise our calculation of g. While a rider may be comfortable with a deceleration of 5g for a small fraction of a second, it is less reasonable to assume the same comfort level if the deceleration is to last several seconds. However, we shouldn t be too concerned about this, since for jumps that are high enough for it to be an issue, our model is likely to break down in other ways Weaknesses of the model o The primary weakness of our model is its dependence on the adiustable arameter d, the distance through which the work of crushing the box is done We assume that over the box sizes that we are concerned with, d is roughly independent of the dimensions of the box; verifying the truth of this assertion would require experimentation. However, so long as d varies at most linearly with h(which is reasonable, since fo box we must have d< h), our method still works; we can still find a uniform optimal box size for the pile. If d depends substantially on the amount of force applied(or the velocity of the motorcycle), this will no longer be the case: the optimal box size will vary with position in the stack of boxes. However, we think this hypothesis is unlikely Our assumption that each layer collapses in a reasonably uniform manner is also a weakness in the model, at least for some parameter values. If the mo- torcycle hits the structure with too much velocity, or the desired cross-sectional rea of the structure is too large, it may not be possible to layer enough flats on top of the structure to ensure uniform collapse, especially if restricted to commercially available cardboard sizes Finally, it is unlikely that we could find easily a supply of boxes whose faces are as large as 1 m x 3m, to create the flats we want. While we could custom-order such flats, this would likely drive the price of construction up substantially. The other alternatives are to use several smaller flats in the place of each large one, or to unfold large cardboard boxes to make the flats. Doing this could weaken the structure, but this problem could likely be circumvented by varying the positions of the weak spots in each layer of flats(and possibly by slightly increasing the safety factor in the number of flats used Conclusion le have designed a landing platform out of cardboard boxes for a stunt motorcyclist who will jump over an elephant. Our model of this platform pre- dicts that we can minimize the material used by using boxes with dimensions 6inx28in×28in.260 The UMAP Journal 24.3 (2003) the harmonic mean of its other two dimensions, the ratio of which can be specified arbitrarily. Finally, for very large jumps, we may need to revise our calculation of g . While a rider may be comfortable with a deceleration of 5g for a small fraction of a second, it is less reasonable to assume the same comfort level if the deceleration is to last several seconds. However, we shouldn’t be too concerned about this, since for jumps that are high enough for it to be an issue, our model is likely to break down in other ways. Weaknesses of the Model The primary weakness of our model is its dependence on the adjustable parameter d, the distance through which the work of crushing the box is done. We assume that over the box sizes that we are concerned with, d is roughly independent of the dimensions of the box; verifying the truth of this assertion would require experimentation. However, so long as d varies at most linearly with h (which is reasonable, since for any box we must have d<h), our method still works; we can still find a uniform optimal box size for the pile. If d depends substantially on the amount of force applied (or the velocity of the motorcycle), this will no longer be the case: the optimal box size will vary with position in the stack of boxes. However, we think this hypothesis is unlikely. Our assumption that each layer collapses in a reasonably uniform manner is also a weakness in the model, at least for some parameter values. If the mo￾torcycle hits the structure with too much velocity, or the desired cross-sectional area of the structure is too large, it may not be possible to layer enough flats on top of the structure to ensure uniform collapse, especially if restricted to commercially available cardboard sizes. Finally, it is unlikely that we could find easily a supply of boxes whose faces are as large as 1 m × 3 m, to create the flats we want. While we could custom-order such flats, this would likely drive the price of construction up substantially. The other alternatives are to use several smaller flats in the place of each large one, or to unfold large cardboard boxes to make the flats. Doing this could weaken the structure, but this problem could likely be circumvented by varying the positions of the weak spots in each layer of flats (and possibly by slightly increasing the safety factor in the number of flats used). Conclusion We have designed a landing platform out of cardboard boxes for a stunt motorcyclist who will jump over an elephant. Our model of this platform pre￾dicts that we can minimize the material used by using boxes with dimensions 6 in × 28 in × 28 in.