A video on Veritasium’s channel gave a problem which seems to trip up a lot of people! (I got it wrong too)

In the end, it comes down to simple conservation of momentum, and nothing needs to be said about energy. If the sum of all forces adds up to zero, or to gravity’s pull on earth’s surface, then momentum is conserved, or decreases at a rate according to gravity, and so the time when the total momentum is precisely zero will always be the same regardless of where the block hits it. That is, if \(\sum p_{t_0}=a\) (sum of the momenta at time zero is \(a\)), and all the forces obey Newton’s third law except perhaps gravity, that is, \(\sum \dot{p}_{t}=-gM\), then the time when the total momentum is zero is simply \(a/(gM)\), and this is the same, allowing the center of mass of the bullet-block system to travel the same distance every time, regardless of the initial offset of the bullet.

But that’s no fun, what’s more fun is to analyze the dynamics of it! Energy loss! Bullet penetration depth! I made a simple model of the system where I treated the interaction between the bullet and the block as a velocity-proportional friction force, and made an applet and wrote some Mathematica code to boot!

- Applet http://mathandcode.com/block/
- App source code: https://www.khanacademy.org/cs/bullet-block-interactive/1902087509
- Mathematica .nb: http://mathandcode.com/programs/blocklagrangian/lagrangian.nb
- PDF of the notebook for convenience: http://mathandcode.com/programs/blocklagrangian/lagrangian.pdf

We can see that, indeed, when the bullet hits dead center, more energy is lost to heat, but the bullet penetrates further.

Neat dynamics? Check. Neat app? Check. Neat youtube video response? Sorry, my editing software died horribly after a re-install. Have fun with the javascript app!