# Maximum velocity of a mass on a box

I need some help with the following question. A square with mass $$M$$ is at rest on a frictionless surface and a point-like particle with mass $$\frac{M}{2}$$ hits it's top corner with velocity $$v_0$$. What is the maximum speed of that particle?

So for that part I tried to calculate the initial $$\omega$$ with respect to the bottom right corner, but couldn't carry the calculations. What is the answer to this question? Also, there is a follow-up question that states that a second mass $$\frac{M}{4}$$ falls on the square such that the CoM speed is constant, does this mean that it is necessarily not rolling? Thank you!

• Does the particle bounce off, or embed itself to the block? Jan 17 at 22:52

So there is an equal and opposite impulse $$J$$ acting on the block due to the particle, and also two reaction impulses $$A$$ and $$B$$ acting from the ground to the corners of the block
Using conservation of linear momentum you can find $$J$$, which is then used to find $$A$$ and $$B$$ using the fact that $$M$$ should not tip over.