Oh and here is the scene if you want to watch it before you move on with reading the post.
I am going to look at wall-e's acceleration about 30 seconds into the embedded video. So first a little bit about fire extinguishers. Did you know that there are several different types of fire extinguishers. and they all target a different kind of fire. Fires are also categorized by their fuel that they are burning. So first about the types of fires and then a little bit about the fire extinguishers mostly because it is something new that I learned while make this blog post and that is what this is all about anyway. So the different types of fires are class A,B,C,D,and K. (I really wonder why K so if you know please post it in the comments.)
Class A fires are fires have wood, paper, cloth, trash, and plastics and their fuel. These are the most common fires. Class B fires are fires in flammable liquids or gasses such as gasoline, petroleum oil, propane and/or butane. Class C fires are fires involving energized electrical equipment such as motors, transformers, and appliances. Remove the power and the Class C fire becomes one of the other classes of fire. Class D fires are fires in combustible metals such as potassium, sodium, aluminum, and magnesium. And then there is the weirdly named class K fires. Class K fires are fires in cooking oils and greases such as animals fats and vegetable fats. Or your worst kitchen fire that you start with the Bacon.
Each fire extinguisher works better with different fires. In fact some fire extinguishers can make a fire worse if you use the wrong one. I don't think I am going to go through all the fire extinguishers here because there are a several different ones but I will say that I think that wall-e is using a dry chemical based fire extinguisher. These are the most widely used so I think that it is a safe bet that he is using this type of one in the clip above.
So now to the physics of his motion while he is floating around using a fire extinguisher to propel himself around. So first I thought it would be go to find Wall-e's mass. I was thinking of several different ways to find that but they all seemed very hard. Like comparing his acceleration to the acceleration of the sodium bicarbonate that is coming out of the back of the fire extinguisher and trying to get his mass that way. But there are several reasons why that would be hard. I don't really know how much sodium bicarbonate comes out of a fire extinguisher and I couldn't find that information on the internet. It apparently depends greatly on how much is left in the fire extinguisher. It was easy to find the density of it which is 2.2 g/cm3 but I still would need the volume to find actual mass. I then thought well why don't I just look up the mass of Wall-e. That was harder to find than I was originally thinking as well when that idea occurred to me. But there was a guy on the internet that said he called Pixar and asked for wall-e's specs and this is what he got here.
She came back a second or two later and said he is 3'4" from the top of his eyes to the ground when he is "standing" normally. She said she had further information if I wanted it (I of course said yes), and she went on to say that he weighs 270 lbs, averages 20 mph with a top speed of 35 mph, and is 7.8 hp.
Now this says that he weighs 270 lbs. So here is going to be my rant on weight and mass. For those that already know that mass and weight are not the same thing you can skip to the end of this paragraph for those that don't keep reading and shame on you. So mass is a measurement of how much stuff something has weight however is the measurement of the force that gravity pulls it down with. So if we want to get the mass of something and we know it's weight on earth we have to take that weight in lbs and divide by 32 ft/s2 which is the acceleration due to gravity in ft/s2 that will give us the mass in slugs. But because no one knows what a slug is we should probably instantly convert that into kg so that we can actually use that mass. So if we do all that it give us wall-e's mass of 122.5 kg.
So now to the physics of his motion while he is floating around using a fire extinguisher to propel himself around. So first I thought it would be go to find Wall-e's mass. I was thinking of several different ways to find that but they all seemed very hard. Like comparing his acceleration to the acceleration of the sodium bicarbonate that is coming out of the back of the fire extinguisher and trying to get his mass that way. But there are several reasons why that would be hard. I don't really know how much sodium bicarbonate comes out of a fire extinguisher and I couldn't find that information on the internet. It apparently depends greatly on how much is left in the fire extinguisher. It was easy to find the density of it which is 2.2 g/cm3 but I still would need the volume to find actual mass. I then thought well why don't I just look up the mass of Wall-e. That was harder to find than I was originally thinking as well when that idea occurred to me. But there was a guy on the internet that said he called Pixar and asked for wall-e's specs and this is what he got here.
She came back a second or two later and said he is 3'4" from the top of his eyes to the ground when he is "standing" normally. She said she had further information if I wanted it (I of course said yes), and she went on to say that he weighs 270 lbs, averages 20 mph with a top speed of 35 mph, and is 7.8 hp.
Now this says that he weighs 270 lbs. So here is going to be my rant on weight and mass. For those that already know that mass and weight are not the same thing you can skip to the end of this paragraph for those that don't keep reading and shame on you. So mass is a measurement of how much stuff something has weight however is the measurement of the force that gravity pulls it down with. So if we want to get the mass of something and we know it's weight on earth we have to take that weight in lbs and divide by 32 ft/s2 which is the acceleration due to gravity in ft/s2 that will give us the mass in slugs. But because no one knows what a slug is we should probably instantly convert that into kg so that we can actually use that mass. So if we do all that it give us wall-e's mass of 122.5 kg.
Now in the clip at about the 30 second mark we see wall-e drifting backward he then decides he needs to go forward and discharges the fire extinguisher backward there by forcing himself forward. Now this deals with the conservation of momentum, also known as Newton's third law. The gas under pressure in the canister is accelerated out of the back of the tube and since forces come in pairs wall-e is accelerated forward. Now that is the introductory physics part but here comes the fun part because rockets are pretty complex things. I mean it is rocket science after all. Now rockets especially rockets that don't have any other forces acting on them such as a rocket that is out in the middle of no where in space, like Wall-e in this clip, are governed by a sweet equation called Tsiolkovsky rocket equation. Which has a great Wikipedia article on it that I linked here. The equation is...
where:
is the initial total mass, including propellant. The mass measurements can be made in any unit form (kg, lb, tonnes, etc). This is because the ratios will still be the same.
is the final total mass without propellant, also known as dry mass.
is the effective exhaust velocity,
is delta-v - the maximum change of velocity of the vehicle (with no external forces acting), and
refers to the natural logarithm function.
We can use this equation to find the mass of the expelled sodium bicarbonate if we have Wall-e's mass and acceleration. So here comes the video analysis. So if we use my favorite video analysis tool tracker then we get the plot of Wall-e's acceleration.
By analyzing the above data we can get Wall-e's acceleration at 8.3 m/s2 which means the fire extinguisher must be providing 1,016.75 N of force. Which also means according to Newton's third law that a force of 1016.75 is pushing the sodium bicarbonate out the back of the fire extinguisher as well. Since we don't know the mass or how much sodium bicarbonate is coming out of the fire extinguisher we have to do some physics tricks in order to make and equation that will help us find the velocity of the sodium bicarbonate. Now you might think you could just use video analysis to find this out well I tried but it was too hard. Plus the math way is fun too. So we know that forces equal mass times acceleration (Newton's second law) but we also know that velocity divided by time is acceleration and mass times velocity is momentum so we can also write it like this...
Which is actually the way that Newton wrote it in the first place. We can then take that and because we don't know the mass we can use the density and times that by the volume and then we get the mass but we also don't know the volume so we replace that by the velocity times the area of the tube time time which gives us and equation that has all the things that we know and the velocity which is what we are looking for. I have always hated typing out in English thing that would take like two lines in math. It is always so much more complicated but I still have not found a good equation writer for free. (maybe you could help by pointing one out to me) Anyway the final equation looks like this.....
Plug in what we know. The density of fire extinguisher exhaust is 0.85 g/ml the area of the hose is 6.28 cm squared and the force is 1016.75 N. Gives us a velocity of 4.36 m/s. Which seems reasonable to me. So fire extinguisher seem like they might make pretty good rockets after all. Too bad they would only propel you for about 13 seconds.
