A Ball In an Accelerating Elevator
This year’s winter American Association of Physics Teachers meeting was right around the corner from me in New Orleans at the Hyatt Regency Hotel. Probably the best thing about the hotel are the elevators. First, they have a glass wall facing outward. Second, they seem to have fairly high accelerations when starting and stopping.
You know what happens next, right? Physics. Grab a couple of friends and make a video. What I wanted to do was to recreate a video I had seen a long time ago (probably from the last time AAPT was in New Orleans in 1998) where a ball was tossed inside an accelerating elevator.
Here is the video. My partners for this impromptu lab experiment were Duane Deardorff and Eric Ayers – just so you know who to blame if something doesn’t work. Let me point out that this might be the one and only time where a vertical video is ok. Don’t forget about all those that suffer from VVS (Vertical Video Syndrome).
Here is the plan. Measure the acceleration of the ball in the frame of the moving elevator as well as in the stationary frame. Yes, I have talked about this problem before – but I didn’t have awesome video to go with it.
Throwing the Ball in a Stationary Frame
Let me start with the video from outside the elevator – the stationary frame. In this case, I can get a scale for the object. Eric measured the bricks next to the elevator and found that 15 bricks was 113.5 cm. This gives a brick stack (with the mortar) at 0.0757 meters per brick. With this, I can count bricks to get the following scale measurement:
Yes. The bricks are a little bit farther away from the camera than that front part of the elevator. Really, it’s just an approximation. The ball isn’t at that distance anyway, it’s a little behind it.
Here is the vertical position of the ball and the elevator as it accelerates upward from a stationary position (in the stationary frame).
I have fit a parabolic function to the ball data and you can see that the term in front of the t2 term is -4.9 m/s2. This would put the vertical acceleration at twice that with a value of -9.8 m/s2. This make look awesome, and it is. However, note that I just got lucky with the scaling in the video. If you just change the length of that part of the elevator by just a tiny bit, you get different value for the vertical acceleration.
The main point is that in the stationary frame, physics works as you expect it would. There is one force on the ball after you throw it so the force diagram would look like this:
Since there is only the gravitational force on the ball, the ball will have a vertical acceleration of -9.8 m/s2 and everyone is happy.
Ok, while I am at it – what about the acceleration of the elevator? Here is the same plot as before except with a parabolic function fitting the elevator data instead of the ball.
This puts the elevator acceleration around 1.2 m/s2. That’s a little bit larger than I would have guessed. What about the final speed? At the end of this data, it looks like the elevator is going at about a constant speed of 3.68 m/s (8.2 mph).
Throwing a Ball in the Elevator Frame
Now, let’s look at the exact same situation but from a camera inside the accelerating elevator. There is one step first. I need to scale the video. This probably isn’t the best thing to use, but I am going to use the diameter of the ball for the scale. This is a “g-ball” that was borrowed from Arbor Scientific for the experiment. It has a diameter of 10 cm. Done. If we would have planned better, I would have put some type of meter stick in the elevator or something like that. Oh well. I’m sure AAPT will have another meeting in New Orleans at some point in the future.
Since the video is blurry, I used the center of the bottom ball in the motion. Here is the point I used.
With that, I get this plot for the vertical motion.
From the parabolic fit, I get an acceleration of -12.49 m/s2. Even with my crappy scale, this seems much higher than the regular acceleration of -9.8 m/s2. This is actually too high. I will talk about what the acceleration should be in just a bit.
There is something else I can check – the time. How long was the ball in the air in both reference frames? Even though the two frames would disagree on acceleration and the maximum height, the time should be the same (since the elevator isn’t moving near the speed of light). In the stationary frame, the ball was not touching the hand for 0.75 seconds. In the video from inside the elevator, the ball was in the air for also 0.75 seconds. Boom. That’s good. Actually, I was afraid that I had matched up two different throws since we tried this stunt a couple of times. It also helps that both cameras recorded video at 24 frames per second. It’s pretty easy to match up the frames that the ball left and returned to the hand.
What about the case where the ball is thrown while the elevator is stopping. In this case, the elevator is has an acceleration in the downward direction instead of the upward direction. I don’t have a video from outside the elevator, but it is still worth looking at. Oh, and as stated in the demonstration video, this is for the case where the elevator started at a high floor and accelerated down to go to a lower floor.
This has an acceleration (in the frame of the elevator) of -8.22 m/s2. Again, I don’t think this is the best video – the ball even went out of the frame of the camera. However, it is pretty clear that this vertical acceleration is lower than acceleration of a free falling object in a non-accelerating frame (-9.8 m/s2) and much lower than the acceleration in the elevator when it was accelerating upwards.
Physics in an Accelerating Frame
In an accelerating elevator, what should the acceleration be? Well, as I have said before, this is actually a pretty difficult problem (that older post has a lot of details – I think it turned out rather nice).
Here is the main point. When dealing with an accelerating frame, our normal physics rules don’t work. What are the rules? Here are the important ones for this case:
- Forces are an interaction between two objects. In the case of the gravitational force on a ball, it is an interaction between the ball and the Earth.
- Forces change the momentum of an object. If the object is not moving near the speed of light and has a constant mass, you could say that a net force on an object is proportional to the product of acceleration of the object and the mass of the object.
Those are really the only two “rules” we need for this case. Experimentally, we can say that the gravitational force on an object near the surface of the Earth is down and has a magnitude of m*g where g has a value of 9.8 N/kg. If there is only the gravitational force on an object, then the following would be true (in the vertical direction):
And this is why the acceleration of a free falling object is -9.8 m/s2 – only in the accelerating elevator, it isn’t. So, how do we fix this? Well, the simplest way is to cheat. We can make the rules of physics work again if we introduce fake forces. For physics in an accelerating frame, we can still use the old rules if there is an added force that is NOT an interaction between two objects (so, not a real force) that has an expression:
So, to make things agree with the acceleration measurements inside and outside the elevator I can draw the following two force diagrams. This is for the case of an elevator accelerating upwards.
If the elevator has an acceleration in the positive y-direction with a magnitude of ae, the we can write the Newtonian physics as:
Using my measurement for the acceleration of the elevator (accelerating up), this should put the acceleration of the ball as seen from inside the accelerating elevator as -9.8 m/s2 – 1.2 m/s2 = -11.0 m/s2. This isn’t quite the value I determined with video analysis, but it is higher than plain old free fall acceleration.
When the elevator accelerates down (either by starting from rest and accelerating down or moving up and stopping), the elevator acceleration would have a negative value. Let me just assume that the positive and negative acceleration of the elevator have a magnitude of 1.2 m/s2 (which I don’t actually know is true). This would make the second inside the elevator acceleration -9.8 m/s2 + 1.2 m/s2 = -8.6 m/s2. This is a bit closer to the measured value – but again, I don’t actually know the elevator acceleration.
I think I was so excited to make these videos that I didn’t really plan ahead very well. I should have used better cameras both inside and outside the elevator. I have more than one camera that can record at frame rates of at least 60 fps – why didn’t I use those? Who knows. Also, a tripod inside (and outside) would have been quite useful.
Here’s what I should have done. I could get one of those PASCO ball launchers and mount it to some type of vertical measuring stick. That way, I could just plop it on the floor of the elevator and shoot it up. I wouldn’t have to guess at the scale of the video based on the diameter of the ball.
I just realized that in the clip with the elevator starting at a high floor and then accelerating down, I could get the elevator acceleration. If you look at the vents on the wall outside the elevator, you could use those to get the elevator acceleration. Well, you can do that for homework.