What minimum speed must the rocket have just before impact in order to save the explorer's life?

Answer: Assuming that I understand the geometry correctly, the combine package-rocket will move off the cliff with only a horizontal velocity component. The package will then fall under gravity traversing the height of the cliff (h) in a time T given by h = 0.5*g*T^2 However, the speed of the package-rocket system must be sufficient to cross the river in that time v2 = L/T Conservation of momentum says that m1*v1 = (m1 + m2) * v2 where m1 is the mass of the rocket, v1 is the speed of the rocket, m2 is the mass of the package, and v2 is the speed of the package-rocket system. Expressing v2 in terms of v1 v2 = m1*v1 / (m1 + m2) and then expressing the time in terms of v1 T = (m1 + m2) * L / (m1*v1) substituting T in the first expression h = 0.5*g * (m1 + m2) ^2*L^2 / (m1*v1) ^2 solving for v1, the speed before impact is given by v1 = sqrt (0.5*g/h) * (m1 + m2) * L/m1