"What would Galileo do ?" ("WWGD"): Let's assume that the bullet in a rifle barrel is accelerated by the gas pressure behind it, and let's assume that the difference in gas pressure when the bullet starts moving and the gas pressure when the bullet exits the barrel to be approximately the same -- i.e., the barrel is far too short for there to be any significant pressure drop. (This is a terrible assumption for longer guns -- e.g., naval guns -- but this case provides a simple starting point.) So, without any rifling in the barrel at all, the bullet would be subject to *uniform acceleration* from a *constant* force -- a situation Galileo had studied quite extensively in his work on inclined planes. So the bullet would gain speed quadratically w.r.t. time -- a fact which could be "graphed" by throwing a bullet horizontally out of the Tower of Pisa and watch its parabolic trajectory as its X coordinate is changing with its constant horizontal velocity while its Z coordinate is changing quadratically due to the constant acceleration of gravity. Galileo could also illustrate this on an inclined plane by rolling a spherical bullet horizontally while gravity simultaneously accelerates the bullet *down* the inclined plane. (To minimize angular momentum effects, use a styrofoam ball with a lead sphere at its center.) --- So far, we haven't dealt with rifling. So what is a reasonable way to model rifling? I contend that rifling should *minimize* the torque on the bullet, else the frictional forces will consume a large portion of the energy. I.e., the torque on the bullet should be *constant*. But Galileo has also studied this case: we simply *roll* the bullet down an inclined plane; the energy of the bullet at the bottom of the inclined plane will consist of rotational energy + translational (kinetic) energy, and will equal the potential energy at the top of the inclined plane. The fact that the bullet is facing orthogonal to its direction of motion is insignificant; its speed and rotational energy are still correct. The only problem with Galileo's "inclined plane" model of a rifle barrel is that the rotational energy and the translational energy are *too closely linked*. Our rifle engineer wants another degree of freedom to adjust the ratio of the two energies. So Galileo considers a fireworks rocket which accelerates its payload to a maximum height -- say 500 feet -- at which point it ejects a bullet with a little tiny rocket on the side which acts to merely rotate the bullet faster and faster. The bullet is now accelerating downward under the force of gravity, but its angular momentum is being increased by this tiny rocket motor exerting a constant torque. But the torque can be varied independently of the force of gravity, so torque is no longer strictly proportional to g. Finally, Galileo says, "Aha! Let's simply *unroll* the rifle barrel (many times, of course); ideally, the angle of the rifling on this flat planar plot should be the ratio of the instantaneous translational velocity of the bullet to the instantaneous rotational velocity of the bullet. Since both are accelerating under a constant force, these will be *straight lines*, and hence have come from a helix of *constant pitch*." So -- at least for the case of constant translational and constant rotational acceleration -- the optimum rifling should be a *fixed pitch* twist. If the chamber pressure builds up quickly (i.e., infinitesimally) to a particular value, and then remains *constant* while the bullet is accelerating down the barrel, then we're done, because the force delivered to the bullet is constant all the way down the barrel. Of course, any pressure remaining after the bullet exits the barrel is wasted, so ideally, the powder will burn the entire time the bullet is in the barrel, but finish at the same time the bullet exits. Furthermore, the powder burn will actually have to *speed up* to make up for the expansion of the gas volume as the bullet travels down the barrel. But through clever choice of the shape of the burn surface -- analogous to that used in solid fuel rockets -- our modern Galileo can approximate a constant force and therefore profit from a fixed pitch rifling strategy. At 03:53 PM 11/3/2018, Henry Baker wrote:

I was watching an old WWII movie, which showed the rifled insides of a cannon barrel.

A similar view (of a smaller gun) is found at the beginning of one (or more!) of the James Bond movies.

This rifling all seems to be of a *fixed* pitch -- i.e., a helix of constant twist angle.

But if you think about it, a bullet starts at velocity zero and angular momentum zero, so *ideally*, this rifling should start out relatively shallow, with slightly more twist angle as the bullet travels further down the barrel.

On the other hand, the bullet is also travelling *faster* at the end of the barrel, than at the breech of the barrel, so the twist could actually be less at the end, since it may already have achieved the correct angular momentum.

So what's the ideal rifling -- by "ideal" here I will choose that rifling that maximizes the energy transferred to the bullet, while still achieving a given angular momentum ? We have to transfer a certain amount of energy from linear motion to rotating motion, and I presume that there is a certain cost associated with this transfer.

Obviously, the ratio of the moment of inertia to the mass of the bullet should enter into the calculation at some point. In order to maximize the power transfer, there needs to be a kind of *impedance match* of the bullet to the rifling pitch.

https://en.wikipedia.org/wiki/Rifling

"In some cases, rifling will have twist rates that increase down the length of the barrel, called a *gain twist* or *progressive twist*".