Using Bad Physics To Explain Video Game Physics
by Eric Hall
June 7, 2014
As a little departure from serious debunking, I felt it necessary to correct this egregious error in an explanation of video game physics. While watching a video sent by a friend, the video posted below came up as a suggestion. Being a physics instructor, I always love finding new and fun ways to talk about the basics of physics. However, the basics became very jumbled in the video, as you will see.
If you choose not to watch the video, here's a quick synopsis: a narrator explains using various iterations of Mario to determine his height, thereby determining the size of the Bullet Bill characters. This is important because in order to figure out how the Bullet Bill will affect Mario we need to know the mass of both objects. They then go on to calculate the force each different-sized the Bullet Bill will place on Mario. This is where their calculation derails.
The video tries to apply Newton's Second Law to determine the force applied to Mario. This would be a good place to start. Usually in a collision, it is better to use a modified version of Newton's Second Law known as the impulse-momentum theory. It is really just a rearrangement of Newton's Second Law, and it will generally lead to the answer one is looking for more quickly. Let's start with the video approach.
The video starts with the idea of the impulse-momentum theory, though they don't state it or show the equation. They get it right in the sense that a fall from a significant height kills you because of the fast deceleration at the end, not the fall itself. The video posted an image of various bullet shapes and made mention of the coefficient of drag. They then explain a little bit about what acceleration is, and then say because the Bullet Bill doesn't change velocity its acceleration is zero. They then apply Newton's Second Law in the following way:
A typical application for this equation might be pushing an object across a level floor or table. If the acceleration of the object is zero, then the force applied to the object would be equal to the frictional force, which is the normal force (m*g, since it is not accelerating in the y-direction) multiplied by the coefficient of friction.
What the video concludes from this equation is the force applied from the equation is the force Mario experiences from the Bullet Bill. However, this is in error in two ways. First, the frictional force as presented here is not the same as the drag force. The drag force is the force experienced as an object travels through a fluid; thus using the drag coefficient as a kinetic friction coefficient is incorrect. Kinetic friction is what is experienced as two objects slide past one another, like a book sliding across a table. If we ignore the misapplication of the coefficient for a moment, what they then calculated is the force required to keep the Bullet Bill moving at a constant velocity (thus the zero acceleration on the left side of the equation), and not the force applied to Mario.
What the video needed to do was first examine their assumptions more carefully. We make these assumptions all the time in physics, and it just depends on the scale and the situation which assumptions we make. For example, if I push a person standing on roller skates and measure their acceleration, I can fairly easily calculate the force I applied to cause that acceleration. It makes sense in this case to ignore friction and air resistance, because they would have a small effect on the results in that short amount of time and at those speeds. Thus ignoring those factors are part of the assumption set I am making. One could also argue I am assuming no relativistic effects, because the relative velocities are far below the speed of light. These are all things in my assumption set.
A big assumption to make in this video would be to ignore pretty much all physical laws except for at the time of impact. In the original NES version of Super Mario Brothers, the Bullet Bills do not fall in a parabolic arc — they simply stay on a straight horizontal path and don't appear to experience gravity or air resistance. Because of this constant velocity situation, we can assume either the physical laws don't apply, or we could simply assume there is a force being applied to the Bullet Bills to counteract gravity and air resistance, thus making the net force on them zero. I will go with the latter because it is at least plausible, and provides an explanation for how the later games had Bullet Bills that were able to track Mario vertically (by changing the vertical force applied to the bullet — perhaps Star Trek-like thrusters on board?). Either way, we can say the Bullet Bills are moving at a constant velocity when they impact Mario.
Using the calculations for the scale of things from the video, they estimate in the original Super Mario NES game that the Bullet Bills are moving about 12 m/s. Using the video estimate of mass for the Bullet Bill (I will round up to 900 kg) and assuming mini-Mario at .65 meters high and about 30 kg, we can calculate Mario's velocity after the impact. However, because the Bullet Bill is so much more massive than Mario, there will be very little change in its velocity. When you hit a bug with your windshield, you technically are slowed a tiny bit. While this isn't quite bug/car ratios, it isn't unreasonable to make this estimation since I estimated the Bullet Bill speed anyway and it avoids me punishing you with more equations!
So if Mario accelerates from 0 m/s to 12 m/s in a very short impact time as indicated by the video analysis of the game, let's call that impact 0.1 seconds. Here I will look at an equation to figure out the force Mario would experience:
Thus I calculate a force of about 3600 Newtons. This is a little higher than the video estimated, and is in the range where damage could happen.
The video then goes on to calculate forces for bigger and bigger versions of Bullet Bills, but the size of Bullet Bill only matters in the sense it will be slowed less by the impact. I assumed the smallest Bullet Bill wouldn't be slowed, but calculating the slow down based on my estimates, after the impact the Bullet Bill would still be going 11.6 m/s. The larger the Bullet Bill, the closer to 12 m/s it will stay. Thus the forces caused by the large Bullet Bills wouldn't increase dramatically, but would asymptotically approach 3600 Newtons.
There are a few other assumptions made here. The area over which the force is applied will make a difference too. This is calculating the pressure applied. If I apply 3600 Newtons to a small area of the body, it would crush that body part. Applied over the entire body, it might be uncomfortable, but I don't think it would cause permanent damage.
Keep in mind, I am doing this calculation on a Friday and on summer vacation. Feel free to submit any corrections that might need to be made. However, I do believe conceptually, this is a correct assessment of what should have been done in the video. And don't stand in front of a Bullet Bill!
by Eric Hall
@Skeptoid Media, a 501(c)(3) nonprofit