When I was a kid, we didn’t have Nerf guns. If you wanted to shoot someone, you had to tell them when you hit them.

“I hit you!”

“No, you didn’t.”

“I totally did hit you.”

“Nope. I dodged it at the last instant.”

The Nerf guns avoid this problem. Oh, and check this out. A belt-fed automatic shooting Nerf gun. What better way to train and prepare a kid for the zombie apocalypse than a Nerf machine gun?

And now for some physics. What is the launch speed of this gun? How consistent is it? What about air resistance?

Just one shot

This is actually pretty straight forward. Take the gun, shoot it. Video record it. Use Tracker Video for analysis. Here is one of the videos I created. If you want to use it, feel free. I put a meter stick on the fence for scale.

Here is the data for the first shot. In the x-direction, I have:

Linear is good. A linear function suggests that there is negligible air resistance. If the air resistance was a significant factor, the slope of the line (the x-velocity) would decrease as time increased. This seems like a fairly constant slope of 8.73 m/s.

Here is a plot of the y-position:

The good thing is that the acceleration looks fairly constant. The bad thing is that it is a bit too large (or larger than expected) with a value of around -10.2 m/s². Odd. I am pretty sure my video is scaled correctly. Also, I am pretty sure the axis is set so that y is in the vertical direction. The only thing I can think of is that there is some sort of aerodynamical thingy happening. Either that, or I video recorded this in the middle of a gravity wave that LIGO failed to detect.

So, for this shot I will say that it was shot horizontally and has an initial velocity of 8.73 m/s.

A whole bunch of shots

What happens if I find the acceleration and the initial velocity for two belts worth of darts? Here is what you get. This is the distribution of the x-velocities (which I am assuming to the be the initial velocity).

And this gives an average of 10.4 m/s with a standard deviation of 1.5 m/s. So, is it a normal distribution? Hard to say with so few points, but here is a plot of the same histogram normalized so that the total area is 1 along with a curve of a normal distribution with the same average and standard deviation.

Sure, I could do some statistical tests, but I am going to stick with the eyeball. Seems normal enough.

Ok, but what about the variation in the gun angle? Is that what could cause the variation in the launch speeds (since I am really just measuring v-x)? Good question. So, I went back and looked at the video. It seems that the barrel of the Nerf gun goes from a minimum of -1.1 degrees below the horizontal to 2.6 degrees above. Suppose I have an actual launch speed of 10.4 m/s, but I shoot it 2.6 degrees above the horizontal. What would the x-velocity be for this case?

This is small compared to the standard deviation of the x-velocities. So, I am going to say it didn’t matter. Really, if you look at the video closely, you will see these Nerf bullets are not very stable in flight. Maybe the gun barrel needs to be rifled.

Acceleration Data

Here is a histogram of the acceleration data.

This gives an average vertical acceleration of -9.60 m/s² with a standard deviation of 0.72 m/s². So, overall the acceleration wasn’t too bad. After looking at more motions of bullets, I am going to go with the aerodynamic instabilities as a major contributor to the variation in the accelerations. I guess what I really need to do is to toss a whole bunch of balls and find the accelerations. I am pretty sure that plain balls would have a fairly constant vertical acceleration. I guess that will be a post for another day.