DYNAMICS OF PACKAGE CUSHIONING 359 



governs before or after contact, the free-body diagram of Fig. 1.2.2(a) should 

 be used. Then 



X2 = g. (1.2.4) 



Equation (1.2.4) holds (neglecting air resistance) from the instant the 

 package starts to fall until the instant it strikes the floor and from it we 

 can find the package velocity at the instant of first contact. Integrating 

 (1.2.4) with respect to time, we find 



Xi^gt^A, (1.2.5) 



where Xi is the velocity {dx^/dt) and A \s o, constant of integration whose 

 value is found from the initial condition that when / = (the instant of 

 release) ;V2 = 0. Thus yl = and 



X'i = gL (1.2.6) 



Integrating again, 



x^ = hgt''+B. (1.2.7) 



The value of the integration constant B is found from the initial condition 

 that .T2 = — // (the height of drop) when / = 0. Hence B = —h and 



X2 = ig/2_/;. (12.8) 



At the instant of contact, X2 = and, from (1.2.8), the time at first contact 

 is given by /q = 2/?/g. Substituting this value of / in (1.2.5) we find, for the 

 velocity at first contact, 



[xo]x,=o = \/2p. (1.2.9) 



We now have the initial conditions for finding the values of the integration 

 constants in the solution of equation (1.2.3), which we proceed to obtain. 



First multiply both sides of (1.2.3) by dxi/dt and write ^2 = t ( -j^ ]: 



dx2 d fdxi\ dx2 _ dx2 /1 o in\ 



or 



, _, dx2 dx2 



+ ^-57 """^^^ 



Multiplying by dt and integrating once: 



to + / " i' dx2 = j ' niog dx2 + C, (1.2. 11; 



2^2 Xi 



