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In a collision between two spheres $A$ and $B$, their velocities are symmetric ($v$ and $-v$, respectively) in the center-of-mass frame of reference. The final speed of $A$ does a $45°$ angle with its initial velocity.
Determine the final speed of the spheres if the collision is elastic, in the frame of reference where the sphere B is initially at rest.
How do I solve this? I am not used to solve these problems with the center-of-mass frame of reference.
The centre of mass frame information is there to give you some information about the relative masses of the two spheres and the initial speed of sphere $A$ in terms of speed $v$.
The rest of the problem can then be done in the reference frame of sphere $B$ which you would have done many times before.
If you decide correctly about the relative masses the problem can be solved fairly easily by writing the conservation of kinetic energy equation from which the shape of the momentum vector addition triangle can be inferred.
Finish the calculation in the CoM frame of reference. That is the frame in which the problem is set.
Then transform from the CoM frame to the frame of reference in which B is initially at rest by adding the reverse of the initial velocity vector of B (ie $-\vec{u_B}=+\vec{v}$) to each of the final velocities in the CoM frame.
The wording of the question does not make clear whether the collision is elastic in the CoM frame or in the frame in which B is initially at rest. Does this matter? No. The amount of KE each sphere has depends on which frame you are measuring it in, but the fact that total (kinetic) energy is conserved in the collision does not depend on which frame of reference you are using. If the collision is elastic in the CoM frame, it is elastic in the rest frame of B also.
