Two boys are sliding toward each other on a frictionless, ice-covered parking lot. Jacob, mass 45.0 kg is gliding to the right at 8.00 m/s, and Ethan, mass 31.0 kg, is gliding to the left at 11.0 m/s along the same line. When they meet, they grab each other and hang on. 

(a) What is their velocity immediately thereafter? 

(b) What fraction of their original kinetic energy is mechanical energy after their collision?

(c) That was so much fun that the boys repeat their collision with the same original velocities, this time moving along parallel lines 1.20 m apart. At closest approach, they lock arms and start rotating about their common center of mass. 

(d) Find their angular speed. 

(e) What fraction of their original kinetic energy is still mechanical energy after they link arms? 

(f) Why are the answers to parts (b) and (e) so different?