A lunar lander is making its descent to Moon Base I. Thelander descends slowly under the retro-thrust of its descentengine. The engine is cut off when the lander is 5.0{\rm m} above the surface and has a downward speedof 0.80 {\rm m/s} . With the engine off, the lander is in freefall.
What is the speed of the lander just before it touches thesurface? The acceleration due to gravity on the moon is 1.6\;{\rm m}/{\rm s}^{2}

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a vertical plane with constant angular velocity \omega. write down the lagrangian for a bead of mass m threaded on the rod, using r as your generalized coordinate, where r, \phi are the polar coordinates of the bead. (notice that \phi is not an independent variable since it is fixed by the rotation of the rod tobe \phi = \omega * t) solve lagrange's equation for r(t). what happens if the bead is initially at rest at the origin? if it is released from any point ro > 0, show that r(t) eventually grows exponentially. explain your results in terms of the centrifugal force m* \omega ^2 *r.