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Let's say Santa got himself a new toy: a present delivery cannon. With this new cannon, he wants to deliver presents to children by shooting their presents from the cannon into their chimneys. Let's assume the cannon has 100% accuracy with a "present protection" feature, where presents shot from it can travel the world and go into chimneys safely with 0 damage to the present or its surroundings (because Christmas magic?).

Santa now wants to just sit at home shooting out the presents. He only has 1 cannon though, so he can only shoot and deliver 1 at a time.

What kind of force and speed would the cannon need to deliver all the presents in 24 hours?

Edit: The present should safely land in the chimneys without causing any damage. I do not intend to make Santa a terrorist.

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    $\begingroup$ Every child in the world: "I just want santa." $\endgroup$ – user6760 Dec 26 '19 at 5:19
  • $\begingroup$ The present rate is going to depend on cannon load speed and aiming speed more than fire force (although fire speed will be relevant). $\endgroup$ – Arcanist Lupus Dec 26 '19 at 5:48
  • $\begingroup$ The firing force should be enough to get a present nearly to the other side of the world. "Automatic reloading" can also be assumed to simplify the problem. Seeing how the "it's midnight" line "scans" across the surface of the Earth, I think aiming speed won't be a huge issue if the cannon just changes the target position, and thereby the cannon's firing angle, as it changes its target. From some quick thinking, I'd say Power Control and Firing Speed (probably beyond an AK-47's automatic) are the 2 biggest factors. $\endgroup$ – John Zhau Dec 26 '19 at 5:53
  • $\begingroup$ This is one of the few instances where it might be easier to build a teleporter $\endgroup$ – nzaman Dec 26 '19 at 12:36
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    $\begingroup$ Hard-Science Tag on a Question with Santa and Magic? $\endgroup$ – Julian Egner Jan 3 at 9:58
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This approach does not scale

Every aspect of this cannon requires magic to work. There are just too many presents to deliver. Of course, this is the problem with how Santa operated before the cannon, and you've already said the cannon is magic. But the cannon is so magical that things like 'force' and 'speed' don't really make sense when applied to it.

Number of deliveries

There are about two billion Christians in the world. Very roughly a tenth of that number will be children receiving presents. The number of presents each child receives will of course not be constant (some children want one big ticket item, others want lots of smaller items) but let's assume the cannon's magical delivery protection allows Big Red to fire the cannon once and stack all presents for a given address into a single shot.

Let us further assume everyone lives in a place with a chimney. How many deliveries per person do we need? One per family is a reasonable approximation. We will assume a nuclear family of 4 people, 2 of whom need presents, all serviced by one delivery.

Yes, there are orphanages, hospitals, statistical outlier families with eight kids, etc. but there aren't a lot of them. They will roughly cancel single parent homes, families with one child, etc.

This means each delivery services, on average, two children. Our population estimate of two hundred million happy clients therefore works out to 100M deliveries.

Optimal delivery strategy

Santa wants his cannon to adjust its aim as little as possible. This poses a similar problem as seek time for modern disc-based data storage. Probably the optimal strategy is to deliver presents in strips running from one pole to the other, north to south back up to north, etc, in line with the rotation of the earth. That way he doesn't have to worry about wild divergences in time zones.

Roughly, the rate should work out to approximately one time zone per hour. Regardless of where you are in the world, you can expect your presents to arrive at roughly the same (local) time (say between 2 and 3 AM) because as the cannon delivery pipeline finishes one time zone at 3 AM local time, it rolls over into the next time zone, where it is one hour earlier.

So that means (1/24)th of the total number of deliveries need to be made in one hour (3600 seconds), giving us a final answer of approximately 1160 deliveries per second.

It just doesn't work

Firing a cannon 1160 times per second is going to destroy the cannon. Adjusting its aim 1160 times per second isn't much better. Loading the cannon in less than a millisecond is just silly. You may as well just give up and say "It's a magic cannon, it works very well, thank you very much."

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    $\begingroup$ Compared to flying to the roof, grabbing your sack of presents, climbing into the chimney, placing the presents, working on your diabetes by eating all the milk and cookies, climbing back out and then moving to the next house 1160 times a second this canon is a massive improvement on Santa's MO. $\endgroup$ – Demigan Dec 26 '19 at 9:17
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    $\begingroup$ Santa should look into MIRVs. If the presents can self-guide the last part of their descent, he could service whole clusters of houses at once. $\endgroup$ – Cadence Dec 26 '19 at 13:12
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    $\begingroup$ I highly doubt that 2 billion Christians map to just 100M families with children. $\endgroup$ – Alexander Dec 26 '19 at 17:45
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    $\begingroup$ True, the new-fangled cannon does involve a hefty dose of Christmas Magic, just like the old-fashioned approach. But I wouldn't really think the cannon would be a massive improvement. It's just as much of a hassle, just a different kind of hassle. For one thing, he'll still need the sleigh to deal with mistakes anyway. Yes, of course he makes mistakes, why else would he be making a list an' checking it twice? Plus it loses the whole personal touch, which I imagine Santa considers a perk of the job anyway. $\endgroup$ – Ton Day Dec 26 '19 at 21:21
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What you envision won't work and will make Santa the most destructive terrorist of all times.

Reason for this is that to reach the far end of the world with a ballistic trajectory you will need to give your projectile a large initial velocity, in the order of few km/s. For the conservative properties of the gravitational field, that same velocity will be present at the moment of delivery.

Santa has a "present protection" feature, but no receiver protection is mentioned. This means that all the kinetic energy of the present will be dissipated by the receiver, with catstrofic consequences.

Think of what would mean to stop a cannon ball just to get a raw idea.

After your edit: still cannot work on a science based way. Science requires conservation of energy, but if you don't want to damage neither the gift nor the receiver you are violating the conservation of energy.

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  • $\begingroup$ I've updated my question to account for this... I did not mean to make Santa a terrorist... $\endgroup$ – John Zhau Dec 26 '19 at 7:29
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    $\begingroup$ @JohnZhau, Santa moving from house to house is one of the most destructive forces in the universe. $\endgroup$ – Separatrix Dec 26 '19 at 8:01
  • $\begingroup$ Oh right... But still, may as well make him no longer a terrorist? $\endgroup$ – John Zhau Dec 26 '19 at 8:02
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First, let us define the problem: Santa wants to deliver presents ballistically, launching them from a cannon to descend down the recipient's chimney, yet not leave a crater at each present's intended destination. Santa's presents - at least those that are traditionally deliverable - share one characteristic: they are human portable, meaning that they have a maximum mass of around 100kg, and linear dimensions that at most are around 3m maximum length, 2m in the next largest dimension, and perhaps 1m at most in the smallest dimension, though most will be considerably smaller. Any bigger, and Santa couldn't lift it to get it out of his old sack. The old guy is probably far more muscular than his suit makes him look, given that he can sling potentially large, heavy presents around with such speed and ease for 24+ hours at a stretch, so something that only a very strong human could grasp and lift alone would seem to be the upper limit to a traditionally deliverable present.

Additionally, while presents in Santa's presence are immune to the extreme speeds and acceleration involved in delivering so many presents in one night by sleigh, simple experience will show that many presents are not immune when away from Santa's presence, as a launched present must be. Some are very fragile, so we may assume a relatively low minimum g-tolerance of about 4g, the maximum that may be expected in a modern airliner.

So... TL,DR: A present has a maximum mass of 100kg, maximum dimensions of 3m × 2m × 1m for a total of 6 cubic meters maximum, and a g-tolerance which may be as low as 4g.

In order to propel a projectile as far as possible with the minimum launch velocity, over a small scale, a launch angle of 45° is optimal, but over a global scale, where the curvature of the world becomes a significant factor, such a launch angle is no longer optimal. Using a software package such as Systems Tool Kit (available as an online or downloaded trial at http://licensing.agi.com/stk/), experimentation shows that the optimal launch angle to achieve a minimum launch velocity sufficient to propel a projectile to the far side of the world is around 30° to the horizontal, and requires a launch velocity on the order of 5000 meters per second.

However, a g-tolerance of 4g is incompatible with any cannon with a practical barrel length that is capable of propelling a projectile to a velocity of up to 5 kilometers per second. How can this problem be overcome? The answer lies with the Gyrojet range of guns: Gyrojet guns appear to be traditional firearms, however they do not fire traditional cartridges, but instead launch miniature rockets. While Jules Verne's Columbiad may have turned any real human crew into paste on firing, rockets have proved to be suitable for use with g-sensitive payloads such as humans. The Gyrojet sysyem launches its miniature rockets at a low velocity and acceleration, and they continue to gain velocity after launch, making it an eminently suitable system, if scaled up appropriately. So, Santa's "Present Cannon" may in fact be more accurately described as a cannon-shaped rocket launcher.

The logistics of packing, loading and firing all these Present Projectiles would be no less impressive than delivering all those presents by sleigh. If we assume a hundred million presents and on average 3 children per household (necessitating only one delivery for all of the children in that household), and Santa's cannon fires from the moment of dusk on the earliest Christmas Eve timezone as the first recipients go to sleep until just before dawn on the latest Christmas Day timezone, a period of around 34 hours, that works out to be 273 deliveries per second. Obviously, then, one launch must contain on the order of 300 deliveries if they are launched one per second with brief pauses to realign the cannon. This means that each launch - if we consider that the average weight and volume of a present may be on the order of 3.3kg and 0.033 cubic metre - must contain a payload on the order of 1000 kg and 10 cubic meters.

I leave it as an exercise for the OP to design a tube-launched rocket capable of lifting 1000 kg and 10 cubic meters of payload at 4g maximum acceleration and propelling it to a velocity of 5 kilometers per second.

So... Santa's "Present Cannon" would in actuality be a relatively thin-walled tube (compared to a real cannon firing similar sized projectiles) with a bore of around 3 to 6 metres diameter, potentially a hundred or more metres long, and with a complex autoloading and aiming mechanism at the base.

For the final part of our design of the projectile, we need to consider the delivery of the presents. Within each rocket would be a delivery pod, somewhat similar in purpose to the multiple independent reentry vehicles of a modern nuclear armed missile, that being to deliver the payload of one rocket to multiple targets. However, presents are not typically g-tolerant, and Santa wants to deliver intact presents, not impact craters. So, the challenge is to deliver the presents at a velocity significantly less than the potential 5 kilometers per second. Fortunately, the relatively shallow launch angle means that the angle of descent will be similarly shallow. While the cannon may be directed far more vertically, and the projectiles pitch over toward their target after leaving the atmosphere, in order to minimise the amount of air it must pass through, a shallow angle of descent increases the distance the projectile passes through the air. This is useful, since the delivery vehicles can use aerobraking in order to shed speed. By using a drag-inducing device such as a ballute, the speed of a supersonic object can be reduced in high atmosphere without requiring an ablative reentry shield. Then, once the delivery vehicle is subsonic, it can deploy a parafoil with which it can maneuver to dock with its target chimney.

Next comes the target: the recipient's chimney. As Santa has traditionally come down the chimney to deliver any presents, even in dwellings which do not normally include such a suitable architectural feature at any time other than Christmas Eve after the residents are asleep for the night, we may safely assume that the magic of Christmas ensures that each dwelling will posses a chimney large enough to allow Santa and the largest possible traditionally deliverable present to descend down it's length. Obviously, this is larger than any chimney found in a dwelling these days, so it forms a standardized target for the delivery vehicle to dock with.

Finally, presents delivered, the virtual chimney disappears, along with the delivery vehicle still attached to it.

Of course, this system would share another problem with nuclear missiles, that of interference between delivery vehicles. This can be minimised by targeting the most distant unserviced target on any given bearing so that subsequent launches result in the projectiles not passing through the descent path of an earlier missile or Delivery vehicle.

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  • $\begingroup$ "I leave it as an exercise for the OP to..." I smell a physics major. $\endgroup$ – John Zhau Jan 4 at 2:10

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