# Gravity-powered cannonballs for mountain dwellers

My mountain dwellers have endless patience. They spend their spare time fashioning spherical rock cannonballs of approximately equal size. This is in case of attack by enemies. Around their mountain they have built many wooden ramps down which the balls can roll. Enemies won't be able to fire back because of the height advantage.

My question is a technical one, hence the hard-science tag.

What factors do I have to take into account to get the maximum range? I presume that the balls should leave at 45 degrees, but how do I ensure they leave with maximum velocity for a given height of mountain? How important is the shape of the slope?

• $E = mgh$, ain't it so? Then $v = \sqrt{2gh}$. No place for the shape of the ramp to come in. (And be careful with that upturn. It eats the energy of the cannonball, so that the effective height is the difference between the height of the cliff and the height of the upturn.) (And please note that solid cannon balls need to be sent at a shallow angle so that they skip over the ground like a stone skips on water. Shooting solid cannon balls as in the figure is a pure waste of ammunition, because they will bury themselves in the dirt and not kill anything.) Commented Mar 15, 2023 at 23:26
• @AlexP - I'm wondering if there is an equivalent of a brachistochrone but for maximum exit velocity. Also if, as you say, I need a flatter trajectory, I must start with some upward velocity or I'm just rolling stuff at them across rough ground. How can I find a best solution? Commented Mar 15, 2023 at 23:37
• Strictly speaking, you only need a flat trajectory if you're hitting troops, if you're reducing a fortification it won't skip in either case so you might as well go for distance. (Getting your enemy to build a fort within shooting range of your mountain is left as an exercise to the reader.) Commented Mar 16, 2023 at 0:01
• Anyway, the real question is how do you aim? Is your mountain entirely ringed by these ramps? How close together can you build them? Commented Mar 16, 2023 at 0:02
• "Enemies won't be able to fire back because of the height advantage" - obviously not using gravity alone, but cannon have been known to shoot uphill and as @Cadence implies, unless there are an infinite variety of ramps then a cannon can be positioned faster than a new ramp can be built. More importantly, I'm not seeing the worldbuilding aspect here, this looks like something that could be asked on PhysicsSE. Commented Mar 16, 2023 at 0:57

Some quick googling gives a vertical terminal velocity of a rigid spherical object of ~ 70 Mph (Source ) - bearing in mind that is straight down. But that site also has a calculator, so...

Using a typical Cannonball of 4.1 Inches or 10.4 cm, and being made of Steel the above gives us a terminal velocity of 128 m/s

Using this: Ballistic Calculator and given a value of 128 M/s (70 Mph), a 45 degree ramp and a 10 metre height - we get a range of approximately 1680 metres.

However, the real answer is going to be less than that due to not being at terminal velocity and the Ramp wasting a lot of the Energy.

• The cannonballs are supposed to be stone. Commented Mar 15, 2023 at 23:45
• That ballistic calculator does not use drag. And this is bad as here we have enough velocity to require calculation of drag influence, yet this provides the topmost theoretically reachable distance. I'd rather make the ramp be located further up the slope, to both protect it from plain reach and having fire arc higher to gain some advantage should the mopuntain be steep enough. Commented Mar 16, 2023 at 7:34

Being gravity a conservative force, we know that the maximum velocity you can get dropping from a given height is $$v=\sqrt{2gh}$$.

That maximum velocity will be reduced by a part due to losses. What are these losses?

• air drag: making an analogy with cycling, air drag becomes significant above 12 km/h and grows with the square of the velocity
• rolling friction: this is due to the deformation of the ball and the ground while they are in contact. It doesn't depend on the velocity.
• rotational energy of the ball itself. Part of the potential energy of the ball will go into rolling the ball. The larger and heavier the ball, the more energy will be going into its rolling.

The above points can be summarized in some design advice:

1. no need to go much faster than your exit velocity: the last, going up, part of your launch device shall be as short as possible to avoid wasting energy in drag at higher speeds.
2. to reduce the rolling friction, select a material for the slope that has the lowest possible rolling friction coefficient.
3. to properly direct the ball, you might want to shape the slope with a V groove to better control the rolling direction. In turn, you can also wrap the ball into two belts in correspondence of the contact points with the V for optimizing the rolling friction.
4. again to reduce losses, avoid sharp turns: the larger the acceleration during the turn, the larger the deformation in the ball and thus the loss of energy
5. the ball will be rolling with a "top spin", which will tend to lower its trajectory with respect to a non-rolling ball. This is a consequence of the Coandă effect. You probably want to work the surface of your ball with dimples to look like a golf ball.

it was found that dimpling the ball provided even more control of the ball's trajectory, flight, and spin. 6. to prevent wasting too much energy into rolling, the ball shall not be too big

I think 1, 4 and 6 give a sweet spot for the ball size and ramp profile, depending on the actual materials you are using. But that's part of a more advanced modeling.

45 degrees is only good if you have no drag. With drag present, the optimal angle is lower. Reference: artillery tables based on farthest point of reach. An overview of what is in them could be read here, however a picture there showed general distance travelled to actually peak at 45 degrees ("phi" curve). The pic referenced: Angles are in degrees on the left.

You here have a gravity-based "artillery" with top speed attainable at about 105 m/s (used this calc with 10-cm round stone orb with 2500 kg/m^3 density), but you won't be able to convert all of it into "muzzle velocity" regardless of your ramp's construction. First, if the ramp would have a lowermost point, which it must have in order to curve the already energized ramp-ball to change speed without getting broken, then the terminal velocity could only have been reached right there (or rather a little before that point, as friction and drag would start eating more energy than the gravity would give to it while it still descends the ramp at low angles), this in turn means that the best speed would be attained at a slightly negative angle of the ramp, taking away distance even if the ramp's exit point would be pretty high. Second, as the answer by L.Dutch pointed, the ball would be launched with significant energy in its momentum of spin, and this rotation of the ball would pull it downwards, lessening firing distance even further. Third, your mountain is only as high as it stands, so you only have as much initial potential energy to convert, and it goes two ways, acceleration and rotation, the latter effectively going against you (so maybe make a sliding ramp instead if feasible?). And final from what I can see, that the balls of yours are made of stone, a rolling stone could well break before reaching terminal velocity due to impurities, cracks and/or uneven tension generated by rolling.

Now. The energy stored in rotation for a uniform sphere of mass m and radius r, rotating at speed w is 0.4*m*r^2*w^2. Since for a sphere that's rolling down the slope r*w=v, where v is linear speed of its mass centre along the slope, the total energy conversion from potential splits the energy of fall into 1 part of speed and 0.8 parts of spin, making the height required be 1.8 times larger just in order to reach the terminal speed, less drag and friction. So for a mountain 1 kilometer high the maximum speed the ball can reach by free-rolling would be sqrt(2*g*h/1.8) or 104 m/s, about equal to terminal velocity. This in turn means that any ramp length above 1.0 km high (and some margin I can't calculate as it depends on drag and friction) would lose more energy than provide by height difference.

And finally, you need you ramp to be able to be turned both by elevation angle and by rotation angle, otherwise your balls would fly predictably long, meaning that your enemies would know where your balls would land and thus avoid going there. The elevation angle can be adjusted with relative ease, as the lowermost profile is somewhat straight relative to the ball's axis of spin, thus adjusting this would only result in changing the force that the ball applies to the ramp while curving upwards. The change of azimuthal angle would involve that ball to pitch, losing energy to alter its already high spinning momentum, thus it would be best applied at the top, yet turning the entire 1.5-km long ramp would be too hard to accurately implement, and any smaller alterations would put a great strain over the part of ramp in contact with an off-spinning stone ball, probably resulting in damage to the ramp and inaccuracy of the ball, as it might not fully reach the new course until going off the ramp one way or the other.

In short, such a ramp can gain your mountain dwellers a hardly controllable range of fire of about 1300 meters at best, but since it's wood and stone, it might not be sturdy enough to actually launch enough stones at any incoming invaders before they reach the ramp, and once there, it would provide a great shelter for them from the avalanche above that would otherwise deliver quite some damage. But if that happens, a smaller removable ramp with negative release angle could be used to direct the same stones into destructing the ramp together with whoever hides underneath (compare ski trampoline with its slope).