Question

From the end of a string (whose other end is firmly attached to the ceiling) we attach a mass m1.Similarly, we attach a mass m2 from the end of a different string. Both strings have equal lengths and the masses are barely in contact when they are hanging freely. We pull out both masses so that they form angles θ1and θ2, respectively, with their equilibrium (vertical) positions. We then release the masses such that there is an elastic, heads-on collision when they reach their lowest point. If, after this collision, the masses return to their initial positions given by those angles θ1and θ2, find the ratio of the two masses, m1/m2.

Answer 1

This elastic collision is exactly equivalent to the 1D head on collision as is given below
And in general after the collisions the masses m1 and m2 will have different velocities. But, if we impose the condition that the masses will return back to their initial position in the pendulumic motion, so, from energy conservation, immediately after the collision the masses will have the equal but opposite velocity of the masses immediately before the collision.
   Now from the collision kinematics, we know the standard result, that if two masses m1 and m2 moves towards each other with velocites u1 and u2 respectively and collide head on and moves back, and immediately after the collision the masses have velocities u'1 and u'2, then the kinematics relation is
   
Now if the masses moves back to their initial positions then the velocity immediately after the collision should be
   
So, by putting these conditions, we get
   
   
  
     
And again from the energy conservation of the pendulumic motion, the velocities immediately before collision is

   
So, we get
   
And so,