Let me continue my exploration of the Bad Piggies world by looking at the balloons. I already know the mass of the some of the stuff, so this will help.
Let's get to work.
Do These Balloons Keep Accelerating?
Here is a wooden box with a pig and two balloons. This is pretty much the simplest setup I can think of.
But while I am at it, let me look at the vertical motion for several balloon cases.
Now I can use the scale of a block with a height of 0.947 meters along with my favorite video analysis tool, Tracker, to get the motion of the objects. Here is the vertical data.
The first thing to notice is that the motion of the wooden box with 1 balloon is just about the same as the motion of a box with 2 balloons. Both seem to move up at a constant speed of 11.6 m/s. That is odd. If two balloons produce twice the lifting force, you would expect the one with two balloons to have different motion. They could be going at a constant speed because of some drag force - but if that were true, they would have different terminal speeds. My guess is that there is a maximum speed limit for balloons. I suspect that a balloon will accelerate until it reaches this 11.6 m/s and then travel at a constant speed. If the lift force from a balloon is significantly high compared to the weight of the box, it will quickly reach this terminal speed.
What about the motion of the box with a pig in it? It seems to accelerate and it doesn't look it even reaches the maximum speed. Here is a function fit for that data.
Now I can compare this fit to the following kinematic equation:
From this, the (0.525 m/s2) term in front of the t2 term has to be the same as the (1/2)ay term. This means that the vertical acceleration would be twice this term, or 1.05 m/s2.
If there is a universal balloon speed-limit of 11.6 m/s, how long would this balloon take to get to that speed? Let me start with the definition of acceleration - I can write it like this:
Since I only looked at the first 4 seconds of video, this wasn't enough time for it to reach this speed. Even though there is more data in the video as the pig goes higher, it won't get to 11 seconds. I will have to make another video. In this case, I will use one pig and one wooden block, but include one more balloon.
Here is the vertical position of a pig in a box with three balloons some time after it was released.
The slope looks constant with a value of 11.4 m/s - close enough to the 11.6 m/s (I need to come up with a better method for scaling videos). So it looks like this maximum speed limit might actually be there.
What About Air Resistance?
I mentioned above that there probably isn't air resistance for the floating balloons. How do I know? Let me start with the assumption that two balloons has more lifting force than just one balloon. I'm not saying it's twice the force, just that it's more than one balloon (I will test this soon). So, here is a force diagram for the two balloons as they move up at a constant speed.
Here is the deal. The two balloons in the diagram on the right have a greater upward force than the one balloon. However, the two blocks are traveling at the same constant velocity. This would mean that the air resistance for both cases would be the same. However, if the air resistance is the same then both cases wouldn't have a net force of zero. Oh sure, the balloons could have significant mass. This could cause a problem but still two balloons would have a greater upward force than one balloon. The only way you could get this to work would be to say that the drag coefficient for two balloons was twice that of one balloon. This might work, but the two balloons don't look like they take up twice the cross sectional area.
The is one other thing against air resistance. If there was a velocity dependent force such as air resistance on the box and balloons as it rises, there wouldn't be a simple quadratic fit with a constant acceleration. As the balloons increased in speed, the drag force would also increase making the acceleration smaller. As it is, the acceleration seems to be fairly constant up to the point where the balloons reach a speed somewhere around 11.5 m/s.
Don't forget, I found that both Angry Birds Space and the regular Angry Birds have a maximum speed. It isn't crazy to think that the balloons would also have a speed limit.
Do Two Balloons Lift Twice As Much?
Here I will start with two objects. The first will be one metal box and one balloon. The second will be two metal boxes with two balloons.
It should be safe to assume that the two metal boxes has a total weight twice that of one metal box. If the two balloons has a force twice one balloon, then these two objects should have the same acceleration. They don't. Here is a video analysis of this case.
I didn't show the parabolic fits for both sets of data, but the one block one balloon had an acceleration of 0.016 m/s2 and the two blocks had an acceleration of 0.012 m/s2. Both of these are in the range "super small" - so, maybe it wouldn't be terrible to say that two balloons has twice the lifting force. There was one odd thing. If you re-run this same case several times, you will find that everyone once and a while the two blocks will move with the same vertical motion. Not sure why.
What Is the Lift Force From One Balloon?
If I stick with the idea that there is no air resistance, I can find the buoyancy force from one balloon. Here is a diagram for one balloon accelerating upwards (but before it reaches the speed limit):
If I just look at the forces in the y-direction, I can write:
The balloon force (FB) can be determined with just the acceleration and the mass of the stuff. I am going to make the crazy assumption that the mass of the balloon is zero - just because. Then, I will measure the acceleration for different payloads to determine the balloon force. Simple enough? Especially since I already know the mass of many of the elements.
Actually, it might help if I write the relationship between mass and acceleration like this:
Here I have a linear relationship between the vertical acceleration and the term (1/m). If I plot ay vs. (1/m) it should be a straight line with the slope having a value of the balloon force. Now, to get the acceleration for different cases I looked at objects that had both a positive and a negative acceleration. In order to get a negative acceleration, I used two balloons. When the object was high enough above the ground, I popped one of the balloons so that the object would accelerate down (and move down) with a negative acceleration. You kind of need to do this since one balloon doesn't lift much.
Now for the data. I only have 5 data points, but it should be enough.
This gives a slope of 8.62 N/wb (remember that wb is the unit of mass in Bad Piggies) with an intercept of -5.32 m/s2. Problem number 1: the intercept isn't what I would expect. I expected it to be around -g, so a value around -9.8. This seems to be half that amount. My best guess is that it is just measurement error. Really, I am stuck on that.
Ok. I have an idea. What if a balloon does two things? When you attach a balloon, it exerts an upward force and it also magically makes the gravitational force on this mass half as much as it was? What if that is true? That would explain the lower value for the y-intercept in my plot. Unfortunately, I can't think of an easy way to test this idea. Oh wait. I just came up with an idea. Check this out.
It is pretty close to actually staying balanced. This is just like the contraption I used to find the mass of stuff in Bad Piggies but with a twist. The balloon pulls up on the right side of the balance and produces a counter clockwise torque. From before, I know the small engine has a mass of 1/2 wb (wooden block) and the sand bag has a mass of 5/2 wb.
If the total torque is zero, this would give the following:
Sadly, this value does not seem to agree with my other method. If I put in a value for g, I get 14.7 N/wb. This isn't exactly twice my other value for the balloon force, but it is close to being twice. I still might be correct about the balloons in that they reduce the mass of the payload when the balloon is floating.
Update: Ciaran in the comments correctly pointed out a mistake above. I made a small algebra mistake when calculating the balloon force. The answer is now corrected above. The value from the balance experiment gives a balloon force of 22.05 N/wb. End Update.
Here is a good example of the problem. If the balloon has a lifting force of (3/2)* (9/4)*g N/wb then if I add an extra balloon AND an extra wooden wheel (which has a weight of (3/2)*g), the two objects should have the same motion. But they don't. Update: and now we see why. My fault.
On the other hand, if I look at the result that says the balloon force is 0.87*g, it shouldn't even be able to lift one wooden block (which has a weight of 1*g). But clearly, one balloon can lift two wooden blocks.
One More Experiment
Help me. I can't stop. Here I am going to use multiple balloons and multiple wooden blocks. Maybe this would be better to show as a video.
Here, there are several different cases where the accelerations should be different. At first, there are 4 balloons with a payload mass of (4 + 5/2) wb's. After that, I pop two balloons so that the stuff falls. It will have the same payload mass but only half the upward balloon force. Next, I drop the sandbag so that the payload mass is only 4 wbs. Here is a plot of the vertical position of the object with quadratic functions fit to the data.
The first thing I noticed that was the last part is messed up. Right after I dropped the sandbag, there are two balloons with 4 boxes and the thing is moving down. The force equation would look like this (in the y-direction):
The acceleration shouldn't depend on the direction of the velocity. However, if you look at the data you can see the best fit comes from separating the downward motion from the upward motion. While the boxes are going down, they have an acceleration of 0.732 m/s2 but after they start moving up, the acceleration drops to just 0.0745 m/s2 - about one tenth the down value. Odd. If I use the last equation to solve for the balloon force, I get two values.
Because of a constant (and large) weight, the difference in acceleration doesn't lead to a huge difference in balloon force. However, looking at the graph of position vs. time, it is clear the down and up have different accelerations. What about the balloon force for the other two parts (going up with 4 balloons and down with 2 balloons and a sandbag)? Using the same idea, I can calculate the force from one balloon based on the acceleration and the mass.
This is crazy.
Fixing Stuff
This analysis is getting out of control. I wanted to go back and collect more data for my graph of acceleration vs. 1/mass since it was giving a different balloon force (about half as much) as the other methods. To do this, I put three wooden boxes with 1 balloon. If you start with 2 balloons, you can get the thing moving up. When I pop one of the balloons, it will accelerate down while moving up and then go down. As I saw before, the acceleration up and down were different - like this:
The acceleration as the object goes up is around -4 m/s2 but on the way down it is around -2 m/s2. My first thought that there were just different physics rules for going up and for going down. However, look at this plot of velocity vs. time.
If the acceleration was constant going up and down (but up would be different than down), you would see two straight lines with different slopes. However, this doesn't look like a straight line. The acceleration is not constant. Maybe there is some type of air resistance. Maybe I was wrong. When I first looked for air resistance, I was looking for a different terminal velocity for objects with different mass. I suspect the reason I didn't find this terminal velocity is that there is also a maximum speed limit of 11.5 m/s (or something like that).
If there is indeed air resistance, then when the object is moving up the air resistance force would be down creating a greater negative acceleration. When the object then goes down, the air resistance would be up making the acceleration a smaller negative number.
Before I try to model this air resistance force, let me just say that I don't think it depends on the shape of the object. These two objects seemed to move side by side and thus have the same air resistance.
So, perhaps the air resistance force just depends on the velocity of the object or maybe it is some constant drag force (like in Angry Birds Space). At this point, I'm just not too sure.
Conclusion
It seems like I didn't get much accomplished. However, let me make some claims.
- There seems to be a speed limit for objects with balloons. The speed limit seems to be somewhere around 11.5 m/s.
- I think my best estimate for the lifting force for one balloon is (3/2 9/4 wb)*g.
- If you have two balloons, it has the same lift as two times the force of one balloon.
- When balloons are lifting objects that are moving, there does appear to be some type of drag force. I am fairly certain that the acceleration when going up and when going down are different for the same object.
- The air drag (or whatever you want to call it), does not seem to depend on shape or orientation of the object. So, it isn't technically air resistance.
Clearly, more data is needed. For homework, measure the up and down acceleration for at least 5 different masses and use this to determine a model for the drag force. Is it possible to find an object going at terminal speed that is lower than the 11.5 m/s speed limit? (if that is indeed the speed limit)
Oh, one more thought. John Burk (@occam98) suggested that maybe the gravitational mass is different than the inertial mass. Gravitational mass is the m in the weight (mg). The inertial mass is the mass in F = ma. In our universe, these two masses seem to be interchangeable. In Bad Piggies, maybe they are different things.
