# Time of flight for a projectile with stated initial velocity and size at various distances

I've been mooching round the internet looking for an answer to this one and can't find a ready resource, hence the question.

In short, I'm trying to discover the flight time of a shotgun pellet at various distances and not even the manufacturers of cartridges seem able to help and none of them seem able to even state what the coefficient of drag is for their pellets so there are elements of any equation which will based on assumed data.

Initial velocity is a variable as cartridges differ. Pellets will be assumed to be spherical but again differ in size depending on the cartridge so pellet diameter (mm) and weight (grains or grams) will be variables.

Air termperature and therefore density need not be a consideration but decelaration of the projectile must be taken into consideration.

As an example, a typical question would be:

Given an initial velocity of 1500m/s and a spherial pellet size of 2.3mm diameter, what is the flight time in seconds for the projectile to travel 30m, 40m and 50m ?

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• There's really not enough information to make a conclusion. What decelerates the pellet is air resistance/ drag, so that'd have to be part of the model. But the correct drag law isn't obvious, and requires quite a bit of empirical information. (The scaling of drag with speed, the density of air, etc...) These are all physics things, and this isn't really the right forum for that. (Physics Stack Exchange, on the other hand, is ideal.) – Semiclassical Jul 28 '14 at 17:10
• Multiple pellets travel together initially which reduces the apparent drag (like cyclists in the Tour de France); density of the pellet is a big factor. When you say "travel 30 m", do you mean along a curve, or straight (for longer shots, the pellet will rise, then fall, to hit a target level with the gun; this changes the "distance traveled"). You need to be more explicit with your question and assumptions. – Floris Jul 28 '14 at 21:02

A few assumptions and ten minutes with an Excel spreadsheet will get you close to an answer. These are the ones I made:

1. Single pellet (ignore the "cloud" effect of pellets dragging air)
2. Ignore the effect of muzzle blast - the pellet "comes into existence" at the exit of the barrel in perfectly still air
3. The drag coefficient is constant at 0.45 (see below)
4. Vertical velocity can be neglected
5. Pellet is made of lead, $\rho = 11.3 g/cm^3$
6. Air density is $1.225 kg/m^3$
7. Air viscosity is $15.11\cdot 10^{-6} m^2/s$

As for assumption #3, see the following chart (from http://www.uic.edu/classes/me/me211/Lab/lab2.pdf) which shows a relatively flat curve for $C_D$ between a Reynolds number of $10^3$ and $2x10^5$ - roughly the regime we are interested in:

Now we can use the drag equation:

$$F = \frac12 \rho v^2 A C_D$$

to compute the force, and finally integrate Newton's laws to get the deceleration, velocity, position.

Key parameters (note - I am taking your muzzle velocity of 1500 m/s as a given for now, but that is insanely fast for a shotgun pellet: you are probably off by at least a factor 3: most likely you meant feet per second. The mathematics are not all that different)

\begin{align} diameter &: 2.3\ mm\\ mass\ of\ pellet &: 72\ mg\\ area\ of\ pellet &: 4.15\ mm^3\\ Reynolds\ number &: 2.8\cdot 10^5 \end{align}

Now you can write the following for a couple of rows in your excel spreadsheet:

t       velocity       force                   acc        distance               drop
0       =velocity      =0.5*rho*area*B2^2*CD_  =C2/(mass)  0                     =-0.5*9.81*A2^2
0.0005  =B2-D2*(A3-A2) =0.5*rho*area*B3^2*CD_  =C3/(mass)  =E2+(A3-A2)*(B3+B2)/2 =-0.5*9.81*A3^2    =B3


Here the first line sets up the initial conditions (velocity 1500, distance traveled zero), and the next line is the first "time step" in the equation of motion - you compute the force (from the force equation) and the acceleration, then get the new velocity as the old velocity minus the acceleration times the time step. The new distance is the old distance plus the mean velocity times the time step.

You create (in column A) the time steps you want (I chose 0.5 ms - that seems to work well) and then copy the formula from row 3 down as many lines as you want.

You can now plot the velocity as a function of distance - I got the following plot:

Note - I also computed a column "drop", but for the given velocity and distance, the drop of the pellet was really quite small - only 1.5 cm after 50 m. Again, this is because the muzzle velocity you gave was very very high.

If you picked a more reasonable 500 m/s initial velocity (still quite fast), your velocity after 30, 40, 50 m of travel are obviously lower. Here is a little table:

      | init  |  init
after | 1500  |   500
------+-------+---------
30   |  925  |  309 m/s
40   |  768  |  263 m/s
50   |  670  |  224 m/s


I hope this helps.

Here are a few approaches which will give you some solid answers. I don't have access to any pretty math programs, so I've not solved your example problem. You'll be making some assumptions, such as pellet shape, air temperature / density, and terrain. I'll leave it up to you to choose which approach to take, but you can solve it.

Newtownian Mechanics

You may have tried this, but then found that it does not work well, and that it's hard. Newtownian Mechanics will give you only a rough estimation anyways.

If you are dead set on using Newtownian Mechanics, I would use some discrete math. Discrete math is good for this situation, because variables change so much. Basically, instead of figuring out the answer over long distances and times, you figure out the math for a really small distance and time, and repeat that calculation until you've covered your long distance or time. You can do this because, on smaller scales, many forces can be approximated to be constant.

I would calculate how far it travels over a small amount of time, considering the force of drag and gravity constant over that time. If I add up each of these steps, I can figure out approximately where it would go.

My algorithm would look like this:

Choose a time-step (such as .001s), define the current position (s = 0), a current speed (v = 1500), and whatever else I need (such as air density):

• Figure out the projectile's velocity at current time/position step
• Figure out current force of drag on the projectile
• Figure out the new position of the projectile given current velocity (s = v * dt)
• Go to the new position, figure out new speed given the force of drag using the speed at the previous point and the time step (v2 = v1 - F*dt)
• Repeat Until v=0, or some other interesting point in time (such as s = 50m)

That should be easy to code in most programming languages, and it should give you a decent approximation if you choose a small enough time-step. Even better, if you ever do get solid information from manufacturers, you can come back to this program and have your answers! You can also have the program print out a table of positions and times in each of these steps, so you don't have to run it a lot.

Alternatively, using Newtownian Mechanics, you could try to come up with some integrals and hope you can integrate it. However, if you're going to go through that much effort, then you may as well use Lagrangian Mechanics.

Langrangian Mechanics

This is something that Lagrangian Mechanics solves really well. If you're comfortable with calculus, please use this over Newtownian Mechanics. I also suggest using Mathematica or similar software to help with the mathematical leg-work.

To be clear, Langrangian Mechanics works by looking at the forces involved in a system, even if they're not constant, and then figures out how the system should evolve over time given those forces. It's much more math-heavy, but it gives you very accurate results. (As a matter of fact, we've been using Lagrangian Mechanics for ballistics systems for a while now!)

In this particular case, you have the force of gravity on your object, and you have the force of air drag. Also, the angle at which you shoot will matter as well, so make sure your velocities reflects this as you express your forces.

I hope these approaches work for you!