1) The center of mass is the point we can focus an extended object or system of objects down to.  If we do so we can apply all of the analysis techniques we have learned so far as if the extended object/system was a point object at the center of mass.

2) Assuming that the car in parts a and b are both initially at rest:
a)conservation of momentum + collision being sticky implies 
v=Mputty Vputty/(Mputty+Mcar)=2.9m/s
b)Conservation of momentum + conservation of K.E. implies 
v=2 Mputty Vputty/(Mputty+Mcar)=5.7m/s.

3)Conservation of energy of the block+block+spring+earth system implies
a)K.E. of both masses=2 M g h-0.5 k h^2=4.384J
b)K.E. of the hanging mass=(2 M g h-0.5 k h^2) (2 M)/(3 M)= 2.92J
c)maximum displacement=4Mg/k=1.18m

4)By conservation of energy of the penny+earth system
a)v=sqrt(2 g h)=77m/s
b)W=change in KE=0.5 m Vf^2-0.5 m Vi^2=-0.77J

5
a) The turning points are the places where the line E=-1J intersects the potential energy graph.  This is at two places, x~0.6mm and x~1.3mm.
b) By definition the KE at a turning point is zero.
c) At the position 1mm U=-2J, the energy of the object is E=-1J so KE=E-U or
v=sqrt(2 (E-U)/m)=3.6m/s
d) An object with E=1J will only have one turning point, so if the object is initially moving in the negative x-direction it will move in the negative x-direction until it reaches the turning point.  It will then reverse directions and move in the positive x-direction, presumably never stopping and going off to +x-infinity.