2012 Semifinal Exam Part A 10 Question A4 Two masses m separated by a distance l are given initial velocities vo as shown in the diagram. The masses interact only through universal gravitation. a.Under what conditions will the masses eventually collide? b.Under what conditions will the masses follow circular orbits of diameter 1? c.Under what conditions will the masses follow closed orbits? d.What is the minimum distance achieved between the masses along their path? Solution a.In order for the masses to collide,the total angular momentum of the system must be zero, which only occurs if vo =0. b.In this case,the masses undergo uniform circular motion with radius and speed vo,so that Gm2 mvo2 12 Gm 072 c.The masses follow closed orbits if they do not have enough energy to escape,i.e.if the total energy of the system is negative.The total energy of the system is 1 02、Gm2 2·2m so that the condition required is Gm2 <0 Gm %7>1 Copyright C2012 American Association of Physics Teachers2012 Semifinal Exam Part A 10 Question A4 Two masses m separated by a distance l are given initial velocities v0 as shown in the diagram. The masses interact only through universal gravitation. l v0 v0 a. Under what conditions will the masses eventually collide? b. Under what conditions will the masses follow circular orbits of diameter l? c. Under what conditions will the masses follow closed orbits? d. What is the minimum distance achieved between the masses along their path? Solution a. In order for the masses to collide, the total angular momentum of the system must be zero, which only occurs if v0 = 0. b. In this case, the masses undergo uniform circular motion with radius l 2 and speed v0, so that Gm2 l 2 = mv0 2 l 2 Gm v0 2l = 2 c. The masses follow closed orbits if they do not have enough energy to escape, i.e. if the total energy of the system is negative. The total energy of the system is 2 · 1 2 mv0 2 − Gm2 l so that the condition required is mv0 2 − Gm2 l < 0 Gm v0 2l > 1 Copyright c 2012 American Association of Physics Teachers