Thinking Outside the box 255 Crushing a layer of boxes To ensure that a layer of boxes collapses uniformly, we build it out of n identical boxes. The total amount of work required to crush such a layer is k(l+ w)d T=n Once the structure starts to collapse, we want the rider to maintain a roughly constant average deceleration g over each layer. It follows that the layer should do total work m(g +h, so nkd(l+ w) h m(g+g)h Define A= nlw to be the cross-sectional area of a layer of boxes rearrangil (1) produces adk how B (g+9)l The constant B gives a necessary relationship among the dimensions of the box if we wish to maintain constant deceleration throughout the collision Finally, we would like to minimize the total amount of material, subj the above constraint. To do so, consider the eficiency of a layer with a given composition of boxes to be the ratio of amount of work done to amount of material used. If the motorcyclist peaks at a height ho, we must do work mgho to stop the motorcycle. We minimize the total material needed by maximizing the efficiency of each layer The amount of material in a box is roughly proportional to its surface area, S=2(hl+ lw+ wh). Thus the amount of material used by the layer is pro- portional to nS=2n(hl+ lw wh). It follows that the efficiency E of a layer composed of boxes of dimensions h×l×wis nkd(l+w) kd(l+ w) Eo nS 2nh(hl+lw+wh) 2h(hl +lw+ hw) We maximize E for each layer, subject to the constraint(2). The calculation tre easier if we minimize 1/E. Neglecting constant factors, we minimize f(h,,)=;(h+h+lu) subject to the constraint holy B where B is the constant defined in(2). However, as long as we are obeying this constraint(that each layer does the same total work), we can write hlu BThinking Outside the Box 255 Crushing a Layer of Boxes To ensure that a layer of boxes collapses uniformly, we build it out of n identical boxes. The total amount of work required to crush such a layer is WT = n k(l + w)d h . Once the structure starts to collapse, we want the rider to maintain a roughly constant average deceleration g over each layer. It follows that the layer should do total work m(g + g )h, so WT = nkd(l + w) h = m(g + g )h . (1) Define A = nlw to be the cross-sectional area of a layer of boxes; rearranging (1) produces B ≡ Adk m(g + g ) = h2lw l + w . (2) The constant B gives a necessary relationship among the dimensions of the box if we wish to maintain constant deceleration throughout the collision. Finally, we would like to minimize the total amount of material, subject to the above constraint. To do so, consider the efficiency of a layer with a given composition of boxes to be the ratio of amount of work done to amount of material used. If the motorcyclist peaks at a height h0, we must do work mgh0 to stop the motorcycle. We minimize the total material needed by maximizing the efficiency of each layer. The amount of material in a box is roughly proportional to its surface area, S = 2(hl + lw + wh). Thus the amount of material used by the layer is pro￾portional to nS = 2n(hl + lw + wh). It follows that the efficiency E of a layer composed of boxes of dimensions h × l × w is E ∝ WT nS = nkd(l + w) 2nh(hl + lw + wh) = kd(l + w) 2h(hl + lw + hw) . We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E. Neglecting constant factors, we minimize f(h, l, w) = h l + w (hl + hw + lw) subject to the constraint h2lw l + w = B, where B is the constant defined in (2). However, as long as we are obeying this constraint (that each layer does the same total work), we can write f(h, l, w) = h2 + hlw l + w = h2 + B h ,