What Happens When You Put a Book on an Air Bag?
- 8:41 am |
Here is a cool video from the New York Physics Teachers Alliance meeting recently (video by Jeffery Yap and Michael Belling).
You might ask: Why would you do this? The answer, of course, is: Why wouldn’t you do this? See, I answered a question with another question. That is an example of the Socratic dialogue.
No really, this video demonstration does show just how much explosive energy there is in an air bag. These are serious things and can be fairly dangerous if you don’t know what you are doing. That is why it is so nice to have a YouTube version of it.
Trajectory of the Book
The YouTube version of the video includes some useful information. The bike rack is 77 cm tall and the frame rate is 420 frames per second. That should be enough to get a nice plot using Tracker Video. Here is a plot showing the vertical position of the first part of the motion.
Can I find the acceleration? That won’t be too easy from this plot. Why? Well, look at the vertical speed — oh, I didn’t include that. Sorry. Well, from a linear fit to this equation (which is fairly linear in this time frame) the vertical speed is around 14.5 m/s (32 mph). However, the time the book is in the frame is just around 0.16 seconds. If everything is scaled correctly, this would correspond to a decrease in vertical velocity of just (-9.8 m/s2)(0.16 s) = 1.56 m/s. That isn’t enough of a change from the 14.5 m/s to measure the acceleration.
When I have a video like this, I get a little uneasy. Finding the vertical acceleration is like my safety net. It makes me feel like I am on the right track. So, let me look at another object — the airbag itself. Yes, it probably has some air resistance that will affect its motion. However, I can at least get a ballpark figure for the vertical acceleration since it is moving much slower. Here is a plot of its vertical motion.
From the fitting function, I get a vertical acceleration of 2*(-5.57 m/s2) = 11.14 m/s2. Sure, this isn’t too close to the expected value of g but it is in the same ballpark. Close enough that I assume the video frame rate is correct. I am happy to proceed.
How High Did It Go?
Since you can’t see the whole motion of the book in the air, I will have to estimate its maximum height. First, let me assume that the air resistance is small enough to ignore. With that, I will use the following:
- Initial vertical velocity of 14.5 m/s
- Vertical acceleration of -9.8 m/s2
Instead of just plugging into your standard kinematic equations, let me take a slightly different approach (really the same as the kinematic equations though). First, what happens at the highest point of the book’s trajectory? At the highest point, the vertical velocity is zero. So, how long would it take to get this high? Here I can use the definition of average acceleration:
Now that I have the time, I can use the definition of average velocity. Note that in this case the average velocity during this time is half of the initial velocity since the final velocity is zero m/s. So, from the average velocity:
That seems like a reasonable height; 10.7 meters is about 35 feet. Well, there is one thing to check. If there is no air resistance, the time it takes for the book to get to the highest point is the same as the time to get back to the ground. This means that with this initial vertical velocity of 14.5 seconds, it should be in the air for 2*1.48 seconds = 2.96 seconds.
From the video, the book left the airbag around 0.167 seconds and hit the ground at about 2.65 seconds. This is a “hang time” of 2.48 seconds. Close. Does this say anything about the air resistance force on the book? Yes. It says there probably is some measurable air resistance. With air resistance, the book won’t go quite as high as without air resistance (since the air is pushing down on the book while it is going up). Even though the book is going slower also on the way down (because now the air resistance is pushing up on the book), it doesn’t have as far to fall. Honestly, I am just guessing here. I probably should run some simulations to see if this is always true.
Instead of overanalyzing this video, let me suggest some things you could do with it. First, notice that the book is about 2 kg (from the video info). Consider these your homework questions.
- Estimate the air resistance force on the book right after it leaves the air bag. What would be the acceleration of this book initially? Some estimates required.
- How much energy was stored in the air bag?
- If you sat on the airbag and let it explode, how high would you go? DON’T actually do this.
- Estimate the acceleration of the book during the explosion.
There. That should keep you busy for some time.