Category Archives: General

Projectile Motion in Portal 2 – part 2

So, is projectile motion in Portal 2 accurate?

Short answer: Yes! And in the next post, I’ll cover how Portal 2 handles terminal velocity.

Long answer:

Projectile Motion in Portal 2

To analyze projectile motion, we need a way to measure the position of a falling object over time. In a perfect world, this data would get exported directly from Source. You can find it in the game with console command physics_debug_entity. You would think that this would make getting position data easy, but as far as I know there are no tools available to directly export position data from the console to an external file, and I don’t have the time to manually record position data for a falling cube frame-by-frame. Incidentally, this post took me so long because I’ve been learning scripting for Portal 2 to export position data automatically (still a work in progress). In the meantime, I used Tracker. Tracker is a fantastic program that analyzes the motion of objects in videos. I like using Tracker because it gives me a chance to collect data in a way that better replicates the process physicists (and physics students) go through to collect data and deal with error in the real world. After all, the universe doesn’t have a physics_debug_entity command.

In the video below, we can see Tracker track the position of six falling contraption cubes. Notice how the graph on the right in the video replicates exactly what you’d expect for projectile motion.

However, we now need a way to convert between the units Tracker uses and panels, our favorite unit for distance in Portal 2. There is a viewpoint difference between objects at the top of the fall and objects at the bottom of the fall as a result of perspective. By using the measuring tool built in to Tracker, we measure that each panel is equivalent to 32.4 +/- 0.5 units.

zoomed in cube in tracker

Tracker treats contraption cubes like point objects despite the fact that they aren’t. The point which Tracker uses to track these objects may shift, which gives us another source for uncertainty. Each contraption cube has a size of 15 units, according to Tracker, giving us a reasonable maximum bound for uncertainty of 15 units (I would argue uncertainty of 7.5 units, but that would assume each tracking point begins directly in the middle of the cube, which may not be the case).

The error from the tracker shift on a cube is significantly larger than that from perspective. I feel safe saying that the extreme maximum bound for uncertainty is 15 units.

With that in mind, let’s look at data from multiple falling objects.

projectile motion overview

At a cursory glance, figure 1 shows that all objects initially follow projectile motion (the solid red line). Though smaller objects reach terminal velocity relatively early in the fall (which will be covered next), for the most part we can see that contraption cubes accelerate correctly until they reach terminal velocity. Let’s zoom in on the initial moment of a falling object and see just how closely the measured positions compare to predicted positions.

freefallaccuracy

Here we can see the 85 kg contraption cube’s position over the first two seconds of falling. Using Gnuplot, we can fit a curve to the data to determine exactly the rate at which the cube is acceleration. We want to fit an equation of the form

yfinal = yinitial – 0.5gt2,

which draws from equation 1. Gnuplot’s parabolic curve fit found that g = 152.1 +/- 0.8 units/s2, which is equivalent to 4.694 +/- 0.02 panels/s2. According to the game, gravity should be accelerating objects at 4.6875 panels/s2, which is within the bounds of our calculated value for g!

As far as we can tell, Portal 2 accurately depicts gravity!

To be continued with terminal velocity tomorrow…

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Projectile Motion in Portal 2 – part 1

I’ve based most of my laboratories, lessons and videos off the fact that the Source engine is frighteningly accurate in its depiction of the way objects move under the influence of gravity. In fact, my very first Portal 2 video was about projectile motion. In the video, I was able to accurately predict where I would land based on the laws of physics we know and love in the real world. That alone might be good enough reason for some people to assume the laws of physics in the game are accurate enough. Apparently, I’m not one of those people.

In today’s post, I’m giving an overview of projectile motion. Following this, I’ll present an in-depth analysis of how Portal 2 handles projectile motion and whether or not air resistance is actually a factor for falling objects.

The Physics of Projectile Motion

The vertical path of objects falling solely under the influence of gravity follows the equation

[equation 1]      Δy = vyt + (1/2)gt2 ,

where Δy is the displacement of the fall, vy is the initial vertical velocity, g is the strength of gravity (g = – 9.8 m/s2 in the real world), and t is the time elapsed. If we assume that our object simply started falling without being launched upward or downward, we can let vy = 0, which simplifies our equation to

[equation 2]     Δy = (1/2)gt2 .

At this point, we have a simple equation to determine the vertical distance an object falls based only on how long it has been falling (though it is important to note that equation 2 is only for ideal situations because it ignores air resistance, which has significant effects in the real world). Equation 2 does not include a reference to mass, which indicates that ideally mass has no bearing on the way an object falls, as first proposed by Galileo in the 17th century. This was famously tested on the Moon during Apollo 15 when astronaut David Scott simultaneously dropped a hammer and a feather and watched them hit the ground at the same time. If you were to scour a physics textbook for opportunities to employ equation 2, you’d find problems involving marbles rolling off the edges of tables, packages dropping out of airplanes, and people dropping garbage out of second floor windows. In Portal 2, I’ve found the simplest way of replicating this type of problem is with a cube sitting atop a hard light bridge that suddenly disappears.

hard light bridge

In this situation, the object is free to fall. If the Source engine follows the laws of physics correctly, the object should accelerate at a constant rate as it falls and cover more distance per unit time.

[figure 1: freefall t vs y]sample parabolic motion

Do not be fooled into thinking that this falling object is falling in a parabola by the parabolic shape of figure 1. The figure above only describes motion in one direction, downward. Notice that it’s simply showing a downward change in position that changes more as time goes on. In other words, the object gets faster over time. If figure 1 looks familiar, it should. It’s half of an upside down parabola that most math students create when they graph y = x2 .

[figure 2: parabola]

parabola

Projectile motion is parabolic in nature. As such, projectile motion is often referred to as parabolic motion because launched objects follow graceful parabolic arcs.

blob parabola

In the real world, objects rarely follow perfect parabolas. Friction, in some form or other, will impede or alter an object’s path. The faster an object travels, the more friction is a factor. Anything built to be fast, whether that’s an airplane, racecar or bullet train, has a sleek profile to minimize friction’s effects. Any child who has stuck her hand out of the window of a fast moving car can distinguish between a non-aerodynamic surface (where her palm is held perpendicular to the wind) and an aerodynamic surface (when her palm is held parallel to the wind).

As an object falls, it feels a constant force of gravity but an increased force of friction. Eventually the two forces equal each other and an object reaches what is called terminal velocity, or the maximum velocity attainable in free fall by an object of a particular shape and size (generally denser objects fall faster). For a human, terminal velocity is reached around 60 m/s near the surface of the Earth.

So, in studying falling objects in Portal 2, we have two metrics we can use to determine accuracy: gravity and terminal velocity.

To be continued…

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wait… what is going on here?

I’m currently hard at work analyzing the depiction of projectile motion in Portal 2. What started as a quick, one minute video has turned into a veritable research paper on the accuracy of projectile motion’s representation in the game. Keep an eye out for some hardcore physics nerdiness in the next few days.

In the meantime, here’s an interesting discussion on Reddit about the physics of a rod stuck between two moving portals.

The rod should oppose the motion of the portals in this diagram until it breaks. Increasing the distance between the portals would require stretching the rod, in which case the applied force would in some way be proportional to the tensile strength of whatever material makes up the rod. This begs a few questions, of course. What about the air between two moving portals such as these? Would moving portals create suction? Would a moving portal have to also apply a force to oppose air resistance? Would vertical portal movement (in this diagram) be possible? In a vacuum with no rod present, would you still have to apply a force to change the separation between the portals?

I recently read Chad Orzel’s fun book, “How to Teach Relativity to Your Dog,” which has put me in the relativistic mindset. I can’t help but think that a situation involving moving wormholes would experience significant effects from general relativity in ways that may not be immediately apparent (or allow this kind of situation in the first place). We’re looking at a moving spacetime connection between two distance places, which hurts my brain.

Anyone have any thoughts? I’d love to hear.

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student projects part 1

At the LEAD Academy, our exams must come in three parts: content knowledge (usually in the form of multiple choice), ACT style questions (in whatever form makes sense for your particular class) and an essay. I decided to get creative with my essay and asked my students to build then summarily describe a level they built inside the Puzzle Maker. They had almost two weeks to work on their projects. I’m pretty happy with their results. For the next few posts, I’ll be spotlighting and describing their levels.

This first level, like many that will follow, lacks coherence or logical design. It looks more like a hodgepodge of random elements, which is pretty common. However, it does show two physics concepts, projectile motion and conservation of energy, and I did not grade on principles of level design. This student wrote an excellent essay about their level and how it relates to physics so they earned an A for this portion of their exam.

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student puzzle

In about 5 minutes, one of my students figured out a clever way to make an in-game timer using linked buttons and cube droppers.

Interestingly, the student who made it has poor communication skills. I had trouble following his explanation of what he had in mind, yet he clearly knew what he was talking about. The Puzzle Maker gives students like him the opportunity to show what they know in a totally unique way, that is, for some students, more natural than traditional communication.

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