 Much has been said about the World War II German 88mm anti-aircraft gun. The 8.8 cm flugabwehrkanone, in various models shortened to flak, could fire high-explosive fragmentation shells to explode in midair, the familiar clouds of exploding anti-aircraft shells present in historical imagery. The gun could also fire deadly armor-piercing anti-tank shells. As the video title suggests, I want to look at what would happen if it were fired on Planet Mars. But really, though, I want to talk about the ballistics of artillery and about world building, when authors write fiction which is not set on Earth. More on that later. In this video, I will be calculating and showing you the trajectories of flak shells under various conditions. I have taken data for this video from this wartime Allied manual for the first-generation flak Model 36. I do not know how widely captured German flak guns were actually used by the Western Allies, but perhaps someone else has made videos on the subject. Artillery and firearms work by setting off propellant in a controlled explosion which shoots around out of the barrel. The high-pressure gases accelerate the projectile down the barrel. You might think that because Mars has no oxygen, can't make a fire, that sort of thing, that firearms simply wouldn't work. Actually, the oxidizer is already inside the propellant, whether it be gunpowder, chordite, and so on. Guns work in space. The greater challenge would be the cold. Mars is far colder than even Arctic conditions experienced in World War II. Assuming that the cold would not pose a problem, however, an artillery gun could certainly be fired. The flak manual tells us that the muzzle velocity of the high-explosive shell is 820 meters per second, or nearly 2,000 miles an hour. The armor-piercing shell, being slightly more massive, leaves about 810 meters per second. For now, I will consider the high-explosive shell. Its muzzle velocity is fixed, meaning that the only things that a gunner can change are the direction the gun is aiming and the elevation angle. It is the elevation which determines the range of the shell, namely how far along it will impact the ground. Once a shell leaves the muzzle, it feels a gravitational force directly towards the center of mass of the planet. If we ignore all other forces and the curvature of the planet, the shell will follow a parabola, an upside-down x-squared curve you might be familiar with from high school. The highest point in the trajectory is called the maximum ordinate. This is where the shell has converted the largest proportion of its kinetic energy into gravitational potential energy, so it has slowed down. The allied manual gives the height of the maximum ordinate and also the distance along the ground underneath the trajectory for different elevation angles. If gravity is the only force acting on the shell, the maximum ordinate is the halfway point of the trajectory and the second half is a mirror image of the first. There are two possible elevations which have the same range, so it is actually possible to fire one shell on a high trajectory, then depress the gun and with careful aiming, fire again to hit the same spot at the same time with the second shell. The force of gravity falls away with the square of the distance from the center of the planet, meaning that the shell feels a weaker force downwards the higher up it is, but this is usually a very small effect. The acceleration due to gravity is lower on Mars than on Earth, it's a much smaller planet, meaning that the range and maximum ordinate of artillery would be much higher. Taking the curvature of both planets into account, but no force other than gravity, the range of flak is about 69km on Earth and 186km on Mars as shown to scale here. The weaker gravity allows the shell to go higher and further on Mars. The other major force that the shell feels in flight is atmospheric drag. This force depends on the density of atmosphere, the square of an object's speed and a ballistic coefficient. The larger the ballistic coefficient, the more aerodynamic the object is. In other words, an object feels stronger drag when it is travelling through thicker air, when it is moving faster, and when its size and shape encounter more air in its path. Because the shell is losing kinetic energy to drag, it will not fly as high or as far compared to the previous case. The maximum ordinate is more than halfway along the trajectory. In other words, the second half of the trajectory is a more pronounced drop because drag has slowed down the shell. It impacts the ground at a steeper angle than it was fired. This might actually be desirable because the shell would be able to drop into a trench from above, or it might make contact with slope tank armour at a more favourable angle. We need to know all of these variables when calculating a trajectory. The speed of the shell is easy, we know what it is at any given time. The density of atmosphere above a planet falls off exponentially. On Earth, atmospheric density drops by half, roughly every 6km up, meaning that it would drop to a quarter at 12km, an eighth at 18km, and so on. Mars has a much thinner atmosphere. At the surface, the density is about 80 times lower than on Earth, so drag on Mars would be much less significant. The ballistic coefficient is a bit more tricky, as the Allied manual does not give a value for it directly, but we can infer it. For example, when the cannon is elevated to 30.5°, the manual states the range, the height, and the distance of the maximum ordnance. We know that if we have the correct ballistic coefficient, the resulting trajectory would connect the dots like so. If we take a relatively low value of 2000, the air resistance is too strong and the shell drops short. If we take a ballistic coefficient of just over 9000, the shell becomes too aerodynamic. The true ballistic coefficient must be somewhere between these two extremes. I will use this value of the ballistic coefficient for the high explosive shell from now on, and further details of how I got it will be given at the end of the video. Note that if you use this to calculate a so-called drag coefficient, which is independent of the dimensions of a given object, then the shell comes out to be half as aerodynamic as a modern jet fighter. On Earth, drag has drastically shortened the maximum range to just under 14.5 km. On Mars, it has made very little difference. The maximum range is 176 km. The combination of weaker gravity and thinner atmosphere lead to a 12-fold greater range on Mars. Whichever planet it's on, a moving shell will experience the Coriolis force due to the pilot's rotation. Imagine Alice and Bob are on a rotating carousel with Bob sitting on the outside and Alice on the inside like so. They both complete one revolution in the same amount of time, but Bob travels further in a wider circle, meaning that his speed is higher. Now imagine that Alice jumps instantaneously to where Bob is sitting. She is effectively moving too slowly and must accelerate forward to match Bob's rotational speed. Alice decides to throw a ball at Bob as it travels outwards within the carousel. That mismatch between the rotational speed it actually has and the speed it needs to have grows larger and larger. The ball therefore looks like it's going slower and slower, as if a force is pushing it back. This is why the Coriolis force is sometimes called a fictitious force, because it merely looks as if an object is being pushed. In the end, the ball will fail to catch up with Bob and will pass behind him. This is the shape of the trajectory it will make relative to the carousel. In summary, you feel an apparent force backwards if you move away from the axis of rotation and an apparent force forwards if you move towards the axis of rotation. The same principle applies to shells on a rotating planet in two particular ways. Let's take the Earth as an example, which is rotating west to east. Liberville and Gabon is nearly at the equator and therefore just about as far away from the Earth's axis of rotation as you can get, while Rome in Italy is due north and, because of the curvature of the Earth, therefore closer to the axis of rotation. These two cities are like Alice and Bob in my previous example. Firing a shell southwards from Rome will cause it to lag behind the Earth's rotation and drift to the west or to the right as you're looking down the barrel. Shooting northwards from Liberville will cause the shell to curve east, also rightwards. The Coriolis force is mirrored in the Southern Hemisphere. Generally speaking, shells flying away from the equator go eastwards, while those flying towards the equator go westwards. This aspect isn't important for the total range of an artillery piece, but it is important for accuracy. Another consequence of the Earth's rotation is the Earthverse effect. As the trajectory takes the shell upwards and forwards, the Earth curves away below. Shells flying to the east are deflected upwards and go further, while shells flying westwards drop short. The north-south Coriolis effect is greater near the poles, while the east-west Earthverse effect is greater at the equator. The total effect on the shell therefore depends strongly on where the artillery piece is located and the direction it's firing. If you thought it was complicated so far, you are right. For this reason, batteries of flak artillery in wartime would have been directed by a primitive fire control computer. So let's go back to comparing Earth and Mars. All of the effects I just mentioned are more pronounced the faster the planet rotates, or alternatively, the shorter the day. Mars has a day 37 minutes longer than Earth, so the Coriolis force is a little bit weaker. However, because shells fly so much further and higher on Mars, the force acts on them for longer. An 88mm flak gun in Rome, firing directly northwards at maximum range, sees its shell go right by just 23m. From the same point on Mars, the deflection is 1.7km over the longer range I mentioned previously. Located at the Earth's equator, a shell fired directly east has its range increased by 17m. At the equator on Mars, a shot to the east increases the range by 2.5km. On both planets, a shot to the west decreases the range by roughly the same corresponding amount. To sum up, shells from the 88mm flak go higher and further on Mars compared to Earth due to a lower gravity and thinner atmosphere. So much for hard facts, but let's talk a little bit about fiction. Fantasy and sci-fi writers put a lot of thought into their setting and history. This is what's called worldbuilding. Writers make up the names and histories of kings or weird aliens, but almost never consider the practical and technical things like ballistics. I wonder if some of the things I've discussed could make an interesting plot point, visual effect, or video game mechanic. Perhaps the protagonist of a story could take a spaceship or magic portal to a world with different properties to Earth. In a modern or futuristic setting, long range artillery and aircraft would be affected by changes in gravity or air density. The history of a fictional world would likely follow our own history in which ballistic calculations drove the development of science and technology. For example, Galileo studied the trajectories of cannons at the inception of modern physics, while one of the first practical electronic computers was used to calculate ballistic trajectories. On a medieval world, with a lower surface gravity than Earth, weapons like bows, crossbows, and even slings might become much more prominent than swords and spears because of their greater range. It would also make acrobatics and dodging projectiles easier. Unlike with Mars, whose atmosphere is too tenuous for us to breathe, even if it contained oxygen, a smaller world need not have a lower surface air density and vice versa. There could be a world with lower gravity than Earth, but also more atmosphere, for example. A rapidly spinning planet would be particularly interesting in a fantasy medieval setting. Imagine this scene. You're an elven archer about to join battle with an orc army. The sun has just risen over the enemy and will set and rise again within an hour. Your regiment engages the orcs to the east with massed volley fire at a greater range than you would normally manage. Orc cavalry closes in, aiming to cut around your left flank, therefore passing to your north. They're in range to take aimed shots now, but the further northward they are from you, the more you have to compensate to the left. This is on top of leading a moving target as usual. They turn for a charge against your regiment's flank. The closer the attack approaches, the shorter the flight time of your arrows, and hence the less you need to compensate. On this planet, an archer needs an eagle eye and a sharp wit to land a hit every time. I hope this video has taught you something and gets you thinking about both real history and about works of fiction. Let me know in the comments if you'd like to discuss or work together on the numerical side of ballistics. Thanks for watching. A few interesting tidbits I didn't have time to mention. Because of the low pressure on Mars and in space, the shells would actually leave the barrel with a higher muzzle velocity than on Earth, because the difference in pressure between the exploding propellant and the outside would be larger by about one atmosphere. I did not bother to account for this because I'm lacking some important technical data, but it would be possible to infer a difference by making some gross assumptions. I didn't mention anything regarding the accuracy of the 88mm flak, some data for which is available in the allied manual. Fired at the same range, the thinner atmosphere on Mars would mean less deflection for the shell, but if fired at the larger maximum range, accuracy would suffer. To arrive at the value of the ballistic coefficient, I used data from all the elevations listed in the allied manual. Let me plot the elevation angle on the x-axis against the range on the y-axis. My aim now is to choose a ballistic coefficient, use it to calculate my own ranges for each of the given angles, and try to match the ones in the manual as best as possible. This is a process known as curve fitting. The best technique from a mathematical and computational standpoint is the Levenberg-Marquardt algorithm, which is just what I used. I couldn't quite match the exact range curves given in the manual, even when I added a linear drag term. The ballistic tables in the manual must have made some additional assumptions, which I just don't know. However, this is the value of the ballistic coefficient for the high-explosive shell which I used.