Let's say two objects are sitting adjacent (in contact) to each other. If we start pushing one of them, we know that both the objects move, remaining in contact to each other. But let's now consider imparting an impulse to one of the objects, instead of a steady force. I expect the two objects to start moving, but this time with different velocities, and thus developing a separation between them.
As a test case, let's say that we push the object which has mass $M$, and the other object has mass $m$; where $M > m$.
I expect the smaller mass to move off with a higher velocity.
But I can't find a mathematical description for it.
Assuming that the force is large enough, we have $F dt$ as the impulse on the combination. On the mass $M$, the impulse is $(F-N) dt$, where $N$ is the contact force between the two masses. Also, $N dt$ is the impulse on the object of mass $m$.
How can I calculate the velocities of the two masses, knowing that both the masses were at rest before the impulse.