Understanding Airsoft BB Trajectory

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Originally posted by Savage Haggis (2012.10.07 - 2129) - For Jsae... (Don't say I never gave you nothing...)

Quote:

Physics is Your Friend (or: the application of simple two-dimensional kinematics in understanding airsoft BB trajectory)

By Supergeek (http://www.airsoftretreat.com/reviews/showproduct.php?product=59&title=retreat-challenge-3a-physics-is-your-friend&cat=51)

Most people have no conception of how things move. There seems to be a quasi-understanding involving a hazy grasp of momentum and something about velocity. Hopefully, this article will clear things up a bit and offer a basic understanding of the motion of objects through space.

Way back in the day, Aristotle thought that things move through space because of an inherent property of impetus. It wasn't until the enlightenment and the 17th century that the true nature of motion was conceived by Galileo and formulated by Newton. The most basic concept is this: things are moved by force - every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it (law 1). What is force? That's the second law - essentially f=ma (force = mass * acceleration). This is the most basic equation in physics. Force is solely dependent on the mass of an object, and how much it is accelerating. Acceleration is change in velocity - it can be positive (going faster) or negative (going slower). Velocity itself has nothing to do with force.

Think about it this way - you're sitting in a car that's driving at 60 miles per hour. You don't feel anything, because the velocity is constant. But then, let's say that the car crashes into something. That's going from 60 to 0 in... let's say .1 seconds for simplicity. Converting 60 mph to meters/sec in metric (everything in physics is metric) and dividing by .1s, we get 268.2 m/s^2. That's the acceleration. Say you weigh 150 pounds. That's about 68 kilograms. That means that a force of over 18,000N is applied to you. That's quite a bit of force. So, in short - you feel it, whereas you don't feel anything when the car is just going steady.

So now that we know about force, we can talk about motion. Kinematics is the study of motion. We can simplify the path a BB takes by ignoring air resistance (which is a real pain to calculate) making two dimensional - we don't care about sideways motion. All we care about is the BB going forward and the BB going down to the ground. Another key principle - in this simplified two dimensional view, the two dimensions of motion are independent. What does this mean? It means that something will fall at the same speed no matter how fast it's going forward. Stand on a cliff over an ocean. Shoot a bullet perfectly straight over the water and drop a bullet at the same time the first bullet leaves the barrel. Negating the curvature of earth, any affects of rifling, and terminal velocity, both bullets will hit the ocean at the same time. So, while the first bullet is flying out at very high speed, it's also dropping at the same rate of acceleration that the second bullet is because of gravity. Near the earth, this acceleration is 9.81 m/s^2. Which means that every second, you're going 9.81m/s faster.

Take this variation of a classic example problem - you're on a hill and directly in front of you, hanging onto a tree and perfectly level with you, is another player that you want to hit. You take aim directly at him and shoot a single BB from your gun. But he sees you and at the same time the BB comes out of the barrel, he lets go and starts falling. We're ignoring hop-up here. Assuming that the BB has enough range to reach your target, do you miss or do you hit him?

Well, the answer is that you do. Because The BB is falling at the same acceleration that the other guy is. After one second, they're both falling at 9.81m/s. After two seconds, both are falling at 19.62m/s, and so on. It doesn't matter how fast the BB is going as long as it gets there before it hits the ground.

This is important because then we can calculate how long it takes for a BB to fall to the ground. Using that time, we can then calculate how far it'll go before it falls. Kinematics is governed by a basic equations:

x=v0 * t + 1/2 * a * t^2

v = v0 + a * t

x = ½ (v0 + v)* t

v^2 = v0^2 +2 * a * x

It's all very neat and pretty, because you can integrate and derive your way through all of them and they all work perfectly... but I'm sure you guys don't want me to get into that. For this, we'll first just need x = v0*t + a*t^2. Here, x is the displacement (in linear motion, same as the distance), v0 is the initial velocity, a is the acceleration, and t is time. Since we need to figure out how long it takes for a BB to hit the ground, we'll have to mess around with the equation so that we get t= something. First, since right when the BB is fired, it's only going straight forward and not down at all, we can plug 0m/s into initial velocity (v0). This means that the entire v0*t block is zeroed out and we don't have to worry about it anymore. So, we're left with x = a*t^2, which turns into t = sqrt(x/a) with a little work. X in this case is how far the BB has to fall. Let's say it's 1.5m for an average person. A is the acceleration due to gravity, which I've said before is 9.81m/s2. So, t = sqrt (1.5/9.81) = .4 seconds.

Now, all we have to do is take that equation (x=v0*t + a*t^2) again and this time, plug in the time. On the left side of the equation is the initial velocity. Let's say we're shooting a 400FPS upgraded AEG. That's around 121m/s. On the right side of the equation is the acceleration that occurs as the BB flies through the air. There aren't any forces pushing it along anymore, just air resistance to slow it down. It's fairly complex to figure for air resistance, so we'll just leave it blank for now and assume that the BB has a constant velocity. That means that the right side is now zeroed out and we're left with x = v0*t = 121*.4 = 48.4m. About 150 ft.

Now, obviously this isn't too accurate - we left out air resistance and hop-up, and we are assuming ideal conditions. 150ft is how far a BB would go if fired from 1.5m off of the ground by a gun with no hop-up in a vacuum. So, it's not really a very good prediction of range. However, the calculations that I've described are useful in comparing the ranges of different guns, and when coupled with the kinetic energy equation Ke=(1/2)mv2, different BB weights.

Plugging in mass (in kilograms) and velocity into that equation gives you the kinetic energy at the muzzle. I guess a long discussion into kinetic and potential energy is unnecessary - all that is needed is knowing that with it, you can find a pretty good approximation of the velocity of various BBs. Let's take that 400 FPS AEG again. V is 121m/s, and the BB weight is .2g, which is 0.0002 kilograms. So, Ke=(1/2)(.0002)(121)2 = 1.4641J. Remember how many of airsoft guns have power rated in joules? Well, now you know what it means. You can reverse the equation by doing v = sqrt((2*Ke)/m)). That way, you can replace the mass with a different BB weight (.00025kg for instance) and figure out how fast the gun will shoot with that BB. Then, go back to kinematics and figure out how much range you sacrifice for stability.