It seems I can't let this BMW motorcycle pulling the tablecloth trick go. Hopefully, you have seen the recent episode of MythBusters where they try to reproduce the commercial (please don't make me re-describe this). Ok,here is my first analysis of this "trick" and my complaint about the MythBusters science-like explanations. Now you are caught up.
After watching this episode, a colleague had a great question:
Interesting. In their last attempt, they had the motorcycle going around 100 mph. It did't work, but some of the stuff did stay on the table. What if they went faster? Could it work?
Suppose I have a long tablecloth with some stuff on it. As I said before, the item that really matters is the last one on the pulling end. This object will have the frictional force on it for the longest time (where the one on the other end has the tablecloth gone relatively quickly). So, here is a diagram showing that last object.
There is more that needs to be known, the coefficients of kinetic friction for both the interaction between the table-objects and tablecloth-objects. I will call these μ1 (tablecloth-object) and μ2 (table-object). Oh, I guess I should explicitly state that I will use the following model for friction:
There is a good chance that this model doesn't really work in this case because of the high speeds involved. Oh well, I will use it anyway. So, what do I want to find? I want to find how far the object moves when the table cloth is pulled out. It will move because of two phases. Part 1 will have the tablecloth being pulled out. This will have a horizontal force (to the left in the picture above) that will make the object increase in speed. After the table cloth passes from under the object, there will be a frictional force from the table that will make the object slow down. If it stops before it gets to the end of the table, it won't fall off.
Part 1: Tablecloth under the object. There are two important things to determine here. How far does it go and how fast is it going at the end (that will be need for part 2 when it stops). Here is a force diagram for the object while the tablecloth is under it:
Since the vertical acceleration is zero, I can get the following expression for the horizontal acceleration:
Oh, but for simplicity, I am going to replace μk with μ1 - ok? And how far does this object move? The first thing I need is the time this force acts on the object and I am going to cheat. If I assume that the object is at rest, then the time the table cloth is under it would be:
This is just your distance for constant speed formula where v is the speed the cloth moves and s is the distance to the end of the cloth. Why isn't this exactly correct? Because the time will actually be a little bit longer. As there is a force on the object, it will speed up and move to the left (same direction as the table cloth) and increase the time it is on the cloth. Why can I cheat? Well, if I want this trick to work, the tablecloth is going to have to move super fast. So fast that the motion of the object will likely have little effect on the time on cloth. Of course, this is an interesting problem - I will have to come back to it. But I have the time (t1). I can now get the distance the object travels and the speed at the moment the object leaves the cloth (assuming it started from rest).
Oh, a couple of more notation things. I will call the right end of the table the x = 0 meters location. Also, I am going to say the velocity of the tablecloth is -v (since it is moving to the left).
Time for a quick check. For the position: as the velocity of the cloth gets bigger, the position x2 is closer to -s - as it should be. Also, the smaller s is, the less the object will be displaced. Ok. That seems ok. A similar thing is true for the final velocity.
Part 2: Sliding on the table. The object has left the tablecloth, but it is still moving to the left. How far will it go? Here is is a force diagram - just to be complete.
Really, the only difference is that the acceleration will have a different value for μ and it will be a positive value. How far will it go? Or rather, where will it end up? Pulling out another kinematic equation, I get:
I think that is it. One thing to look at is the coefficients of friction. If μ1 goes to zero, the thing shouldn't move and this expression agrees with this (there would be no friction to get the thing going). If μ2 is zero then the object would never stop and have an infinite ending position - yep.
What values do I need to use now? Well, first I need the two coefficients of friction. I guess the second thing I will need is an acceptable displacement. Really, this could go two ways - how fast would you have to go to "look" like the fake BMW video and how fast so that the objects don't fall off the table.
To get the coefficients, I will look at the motion of the objects in the MythBusters clip from this angle:
Looking at the motion of one of the dishes on the far left end, I get this:
Note, the table is 24 feet long (that is important for the scale). This gives acceleration of the object at about 3.6 m/s2 which would mean the coefficient of kinetic friction is around 0.37. Just for a check, this is a plot of the position of the tablecloth.
It is not a constant velocity most likely because the tablecloth is sort of springy. Fitting a linear function near the end, you can see it as a speed of around 48 m/s which would be around 107 mph. Ok, good enough for me. Now what about the other coefficient of friction? Here is an object sliding on the table after the tablecloth was not under it.
The above is the motion of an object near the middle of the table. At the end of its motion (while sliding on the table) it has an acceleration of around 1.7 m/s2. This would give a coefficient of kinetic friction a value of about 0.18.
I have my values. How fast would the motorcycle have to be going so that no objects fall off the table? I guess I need one more value. If L = 24 feet = 7.3 meters, then from the video it looks like some of the dishes start about 18 cm from the motorcycle end of the table. I will use x3 = -7.3 meters, and s = 7.12 meters. Solving for v, I get:
Yowzah! That is just a little bit faster than they tried on the show. But, I think there was another problem. When the rope from the motorcycle pulled the tablecloth, it pulled it up some. This made some of the dishes leave the table and become unstable. Perhaps if there was some bar on the end of the table to prevent excess vertical movement of the tablecloth, it would work.
The first question before this answer is: how far would it have to move to still look just right. In the show, Adam was pulling (by hand) a tablecloth from under just one bottle. He would get a total displacement of around 0.01 meters. If the movement of the object on the far left is just 0.01 meters, how fast would you have to go? Using the same ideas as above, I get a velocity of 220 m/s (490 mph). Ok, that is a little fast. What if I relax a little bit and let the object move 0.02 meters? This would require a tablecloth speed of 156 m/s (349 mph).
