So I just spent time doing this. We’ve all made this stupid mistake in the game.
Analyzing the scenario here, it seems pretty obvious that the hard hat would kill anybody at that speed.
But it always kind of bothered me. Assuming Ghost Tricks can’t break the Law of Conservation of Energy (sure, they’re supernatural, but they aren’t magic), there’s no way the hard hat could be traveling at the same speed the bullet was. In order for the position swap to fully make sense, one has to assume the objects simply swapped energy values instead of velocity.
Now it’s time for the fun part (read: lots of math). Assuming Inspector Cabanela here was using a standard-issue gun (and that the game took place somewhere in the USA), we can guess the gun was a .40 S&W pistol, which is very common among the police forces of the United States. There are many sorts of bullets available for this type of pistol, but the average bullet weight is around 9.90195 grams, or 0.00990195 kilograms. The average velocity at which this gun fires a bullet hovers around 1181.2222 feet per second, or 360.036515 meters per second. An average hard hat weighs something like 396.893 grams, or 0.396893 kilograms.
The formula relating kinetic energy to velocity and mass is (KE)=(1/2)(M)(V^2), where KE is kinetic energy in Joules, M is mass in kilograms, and V is velocity in meters per second.
Using this formula, we find that the kinetic energy of the bullet leaving the gun was 641.7765318 Joules. Plugging this value back in to find what this energy would do for the hard hat, we discover that the thing was traveling at a whopping 56.86829324 meters per second. Phew.
By the by, that kinetic energy in Joules translates over to 473.35007615012 foot-pounds: almost five times the amount of energy it’d take for a rigid falling object to kill somebody through blunt force trauma in the head.
Now, I don’t take physics, so I don’t know what all this means for the knit cap.
But I do know the structure of the cap is anything but rigid. For the calculations there, we’d need to take that amount of shock absorption and elasticity into account upon impact to see if it’s really as safe as the game portrays it to be.
If anybody’s interested in picking up where I left off here, I’d be fascinated to see the results.