Does something similar to a gravitation impulse exist?

Question

Let there be a block of mass $M$ sitting on top of a table. there is a hole in the table right under the center of mass of the block.

A bullet of mass $m$ is shot through this hole and it hits the block and passes through it, losing some velocity.

The impulse imparted in the block due to the bullet passing through it is $I = Fdt = \Delta P = m\Delta v = M\Delta V$. Now there is some velocity $\Delta V$ imparted into the Block due to this impulse.

Now I wanna find the minimum mass of the block such that it doesn't leave contact with the surface.

I have 2 thoughts of how the situation could theoretically proceed :

1. velocity is imparted into the block regardless due to conservation of momentum and the block will leave the table for any mass of the block, albeit it might be for a negligible duration as the mass gets higher.

2. something like an impulse imparted by the gravitation force $I_g = Mgdt$. Now we can equate the impulses to get $Mgdt = Fdt \implies Mg = F$. And from here we can find the critical value of $M$.

Are any of these thoughts right or is there a different ways to go about this problem?

Answer 1

Unfortunately, impulse is not the correct way to approach this problem. You need to work in forces and momentum. If you try to think in just impulses, you will find that it is impossible to have a block that doesn't move. No matter how big you make the block, it will have a positive upward velocity after the interaction. This means it will leave the table. Or at least it looks like it does.

What's missing from the equation is a time period over which the impulse is being delivered. To consider that time period meaningfully, we should be talking in terms of continuous momentum transfers, not impulses. Your bullet is applying a force over a period of time, while the bullet deforms. As long as that force is lower than the force of gravity, the block will not move upward.

While you can technically use the concepts of impulse in any situation, we usually use them in scenarios where the momentum transfer is fast enough that we don't really care how the transfer happens. We just care that it does happen. In your case, you really do care about how the transfer happens, so the impulse equations simply don't give you the whole story.

So what does the impulse story tell you? It tells you that if you transfer momentum fast enough (over a short enough period of time), you can generate arbitrarily high forces and overcome stabilizing forces. When designing devices, we often have to subject them to "shock and vibe," which is testing where we intentionally expose it to movement to see if anything gives. A shock, in particular, is very high frequency. It can occur over microseconds. As such, a shock can often take a joint which is stable, using static friction, and temporarily make it unstable, with kinematic friction (because that small delta-V can result in very high forces that start the parts moving with respect to one another). This can cause a part to fail if one was dependent on the high coefficient of friction that comes from static friction.