Safe landings 231 of boxes. Another challenge is that cardboard boxes probably have a cost proportional to the surface area of cardboard used, not to the number of boxes Appendix: On Buckling We derive(2) Consider one wall of a cardboard box with external forces acting on both ends. Considering the effects of shearing and inertial dynamical acceleration the displacement of the box wall from its equilibrium(unstressed) position obeys the equation of motion F where a= w/ge is the mass per unit length of the box wall [Thorne and Blandford 2002] Table al Nomenclature Property Symbol Units Horiz. displacement from equilibrium(unstressed) location n(z, t)m xural rigidity of the box wa D Young s modulus of the box wall Length of the box wall in z-direction Width of the box wall Thickness of the box wall Force applied to each end of the box wall Weight per unit length of the box wall W N/ Acceleration due to gravity Buckling force We seek solutions for which the ends of the box wall remain fixed. This is a good assumption for our model, since we require the box catcher to be held together so that the boxes remain horizontally stationary with respect to other. Thus, we set the boundary conditions to be n(0, t)=n(e, t) the separation ansatz n(z, t)=s(z)T(t), we get the linear ordinary differential equations &'t d22 AnS and 2 T, where Kn is the separation constant. The normal-mode solutions are thus mn(a, t)=AsinSafe Landings 231 of boxes. Another challenge is that cardboard boxes probably have a cost proportional to the surface area of cardboard used, not to the number of boxes. Appendix: On Buckling We derive (2), FB = Y π2 12 wτ 3 2 . (2) Consider one wall of a cardboard box with external forces acting on both ends. Considering the effects of shearing and inertial dynamical acceleration, the displacement of the box wall from its equilibrium (unstressed) position obeys the equation of motion −D ∂4η ∂z4 − F ∂2η ∂z2 = Λ ∂2η ∂t2 , where Λ ≡ W/ge is the mass per unit length of the box wall [Thorne and Blandford 2002]. Table A1. Nomenclature. Property Symbol Units Horiz. displacement from equilibrium (unstressed) location η (z, t) m Flexural rigidity of the box wall D J · m Young’s modulus of the box wall Y Pa Length of the box wall in z-direction m Width of the box wall w m Thickness of the box wall τ m Force applied to each end of the box wall F N Weight per unit length of the box wall W N/m Acceleration due to gravity ge m/s2 Buckling force FB N We seek solutions for which the ends of the box wall remain fixed. This is a good assumption for our model, since we require the box catcher to be held together so that the boxes remain horizontally stationary with respect to one other. Thus, we set the boundary conditions to be η(0, t) = η(, t)=0. Using the separation ansatz η(z, t) = ζ(z)T(t), we get the linear ordinary differential equations −D d4ζ dz4 − F d2ζ dz2 = κnζ and Λd2T dt2 = κnT, where κn is the separation constant. The normal-mode solutions are thus ηn(z, t) = A sin nπ  z e−iωnt ,