Video Analysis of an Underwater Bullet

I can just imagine what Destin would say. “Hey, how about I take an AK-47 and shoot it underwater? I can record the motion with a high speed camera. It will be fun.” Yes. That’s exactly what Destin from the awesome Smarter Every Day did along with help from The Slow Mo Guys.

Destin didn’t just make an awesome video. No, he also went on to explain some cool things that you can see when you slow down things like this. In particular, he looks at bubble bounce as well as the difference between water vapor and bullet gas. You’ll have to watch the video to see what I am talking about. Let me just say one more thing about why I love this video. When you take something like this and you look at it through a new lens (in this case, a high speed camera), you just don’t know what you will find. However, all too often you will find something cool. If you look carefully, cool stuff is everywhere.

Video Analysis of an Underwater Bullet

In order to make a model of an underwater bullet, I first need some data. This video is actually very nice for video analysis since it follows some of my guidelines:

• Stationary camera.
• View perpendicular to the motion of the object (mostly).
• Known frame rate (it’s right there in the lower corner of the video).
• Something to scale the video. A meter stick would have been nice, but I can use the AK-47.

Let’s just get straight to the analysis. Really the only thing I will need is the size of the weapon. I’m not an expert, so I will just go with this image that shows the total length of an AK-47 to be 87 cm. I suspect that there are many variations in the riffle, but to me the image matches the gun in the video. Oh, but underwater the shoulder stock has been removed. Based on my estimations from the diagram, the weapon used underwater would have a length of 64 cm.

Now for the video analysis, I will just load the video into Tracker Video Analysis. The only thing I need to do here is to change the frame rate to 18,000 fps. And here is the first plot showing the position of the bullet.

I’m pretty sure that the first region on the graph isn’t the bullet. Instead it is the leading edge of the expanding gas from the gunpowder. I marked it anyway because I didn’t realize this wasn’t the bullet until you could see something that was actually a bullet.

Here is a plot of the velocity of the bullet as a function of time. This is the stuff that will be more useful.

Why do I need the velocity plot? Well, let’s assume that the only force on the bullet in the water is a drag force. Sure, there’s a gravitational force, but this will likely be quite small in comparison to the drag. It also seems obvious that the faster the bullet goes, the greater the drag force. However, is the drag force just like the typical model for air drag with a magnitude proportional to the square of the velocity? I wouldn’t think it would be the same. Anyway, I want a model for the drag force. I have three options.

• Assume that this is just like air drag with a magnitude proportional to the square of the velocity. I could guess on the size and drag coefficient of the bullet and I know the density of water. However, I just don’t think that a high speed bullet in water can be modeled this way. Of course, I could always be wrong about that.
• Assume the drag force has both a term that is proportional to the velocity and a term proportional to the square of the velocity. Then set up a differential equation and solve. With this equation, I could fit the Tracker video data to find the parameters needed. This sounds like a great idea (and what I started out to do), but I couldn’t get it to work.
• Lastly, I could look at the plot of velocity vs. time. From this I can pick different parts of the data. If I pick a small section of data, I can fit a linear function to find the average acceleration. If I do this enough times, I can get a plot of acceleration vs. velocity and use this to get my drag force model.

I will assume the drag force looks like this:

Now I just need to pick some parts of the video analysis data to get velocity and acceleration data. Here is my plot.

I added a linear function to the data – since that’s what it looks like. The slope of this function is -662.8 s^-1. This suggests that the primary drag force is just proportional to the magnitude of the velocity. I can write the acceleration function as:

Now I can check this with a numerical model.

Numerical Model

The nice thing about getting the acceleration as a function of velocity is that I don’t have to worry about the mass or size of the bullet. All of that stuff is already factored into the acceleration function.

Even though it seems like I go over this all the time, here is the key to a numerical model. I can break the motion of the bullet into small time steps. During each step I can assume that the acceleration is constant (even though it isn’t). This will let me calculate the new position and new velocity at the end of the time interval. Let me list the recipe. During each time step, I will do the following.

• Start with a known position and velocity.
• Based on the velocity, calculate the acceleration.
• With this acceleration, calculate the velocity at the end of the time interval assuming that the acceleration is constant.
• Using the velocity, calculate the new position assuming the velocity is constant.
• Repeat.

The constant velocity and constant acceleration assumptions are valid if the time interval is small enough. Although with a smaller time interval, you end up doing more calculations. Wait! I don’t have to do any calculations, I have a computer. Computers rarely complain about being overworked.

Here is a comparison of the velocity from the numerical model with the data from the video analysis.

Not a perfect fit, but good enough for me. Actually, it isn’t. Look at this plot of the position for both the model and the real data.

The main difference is that my numerical model essentially stops but the data from the video shows the bullet at some final constant speed. One fix for this would be to include a gravitational force. Looking back at the video, the gun seems to be shot at about an angle of 17° below horizontal. This means there would be a component of the gravitational force in the direction of the bullet’s motion. However, if I add this in, it still doesn’t look right. In fact it looks just like the plot before.

I can calculate the terminal speed based on the drag and the component of the gravitational force. From my model, this terminal speed would be just 0.014 m/s and the program calculates a final speed of 0.017 m/s – so pretty close. If I look at the data from the underwater video, it looks like the bullet has a final speed of 18 m/s.

I’m really not sure what went wrong. I guess I overestimated the usefulness of my model. One other possibility is that the video shows a changing frame rate and not the constant 18,000 fps like it claims.  Actually, if I change the gravitational field from 9.8 N/kg to 49,000 N/kg – the position data seems to match much closer.  I’m not sure what that works.  Odd.

I was going to see how far you could get the bullet to go by increasing the speed. My guess would be that if you double the speed, it still just goes about the same distance. One way to fix this is to use a slower but more massive bullet. Slower bullets would mean less drag. A higher mass would mean that the drag force as less effect on the speed.

Bubble Bounce

Since I failed with my bullet model, let me leave you with one more plot. Destin talks about these bubble oscillations. So, here is a the radius (perpendicular to the direction of the bullet) of a bubble as a function of time (from video analysis).

At first, I was thinking of this bubble like an oscillating spring. However, it doesn’t do that. Notice that it changes very quickly from collapsing to expanding. This is more like a super nova than a spring. It’s very cool.

A couple more notes. I think I can try to get a better drag model by looking at the other bullets fired from the handguns. That will be on my list of things to do.