Two objects moving in opposite directions with the same speed v undergo a totally inelastic collision, and half the initial kinetic energy is lost. Find the ratio of their masses.

Explanation:

We name M the larger mas, and m the smaller mass, so base on this we write the following conservation of momentum equation for the collision:

We can write this in terms of what we are looking for (the quotient of masses :

We use now the information about  Kinetic Energy of the system being reduced in half after the collision:

We can combine this last equation with the previous one to obtain:

where solving for the quotient m/M gives:

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