Final Experimental Results

With the changes described on the previous page, the students achieved success within a week. On December 3rd, they recorded a snap that achieved a speed near the speed of sound.  They followed up almost immediately with a snap that exceeded the speed of sound by about 20%. This was the definitive result that provided verification of their hypothesis. A clip of this snap follows. 

VFW    QT

As in the other clips, the towel tip is flipping from right to left moving down the page. The greatest speed occurs between the 3rd and 4th (green and blue) and 4th and 5th (blue and red) images. Click here for a full-size frame that can be analyzed.

Having reached their goals, Elizabeth, Nicolas, and Spence didn't stop there. By Christmas vacation and the end of the fall semester, they wrote a paper on their experiment and submitted it to The Physics Teacher journal. The article was published in the September, '93 issue (vol. 31, p. 376).

It's important to take note of what the student experimenters did not prove. They didn't show that the cracking sound was produced when the tip exceeded Mach 1. They did, however, show a weak correlation between the two events. Using a sound trigger instead of a photogate, they investigated a number of snaps. They found that the first image of the towel's tip was always beyond the flip over point. This result would be expected if the cracking sound occurred at the flip over point, since there was a delay of about a millisecond for sound from the event to reach the microphone and actuate the trigger.

There is a technique for finding out if the crack occurs when the tip exceeds the speed of sound. Using shadow photography, it is possible to photograph shock waves produced when an object travels faster than sound. These shock waves give rise to the sound that one perceives as a cracking sound. Bernstein, et. al.¹ used this technique to show that the tip of a cracked bullwhip produces shock waves.

Addendum

The tip of a towel undergoes a tremendous acceleration as it is flipping over. An examination of the frame above shows that the speed of the tip nearly doubles from the 3rd to the 5th image.  That's an increase of about 200 m/s in a ten-thousandth of a second. The average acceleration is 200 m/s ÷ 0.0001 s = 2 x 10⁶ m/s². That's two hundred thousand times greater than the acceleration of an object falling to the ground. How is it possible for an ordinary human to impart such a huge acceleration to a towel or a bullwhip, causing a portion of it to exceed the speed of sound?

There are probably a couple of things that make this possible.   Consider Newton's 2nd law, a = F/m. This says that the acceleration of an object increases directly with the total force acting on it and decreases with the mass of the object. For the towel, the force is provided by the way in which the towel is snapped. Not only must the towel snapper throw the towel forward, but she must also yank back on it at just the right instant, presumably the time when the tip begins its flip. This jerk would greatly increase the tension in the tip. The mass of the object is not well defined, since it's not clear how much of the towel that one should consider to be the tip. However, the mass is small, and the amount of mass in a given length of the towel decreases the closer one gets to the tip. At the very end, there are only fibers.  Such small masses can be given huge accelerations by moderate forces.  A mass of a tenth of a gram, for example, requires a force of 200 N (about 45 pounds) to impart an acceleration of 2 x 10⁶ m/s².  Even a child can produce that large a force for a short period of time.

The tension in the tip of the towel also contributes to fraying. The tension forces between parts of the towel pull it apart. The speed achieved by the tip may also help explain why it stings you if it hits just right. When the fast-moving tip strikes, it is decelerated rapidly by the part of your body that is struck. Your body exerts a large force to slow the towel. By Newton's 3rd law, the towel exerts a force of equal size on you. That force is concentrated over a very small area of your skin and produces a sharp pain. We don't recommend experimenting with this phenomenon.

One more interesting observation about towel snapping is that it seems to be easier to get the towel to crack when it is wet. Perhaps this observation is simply a consequence of the many towel flipping experiments that are done in gym class.  However, there is probably some truth to it. A towel is more massive when wet than dry. The drag force exerted by the air when trying to snap the towel may be more important for a dry towel. Thus, it may be easier to give a wet towel a large acceleration. Moreover, the tip of a wet towel will shed water quickly. As mentioned above, the less massive the tip, the larger its acceleration will be.

Whether or not the previous explanation is correct, the students used a dry towel for most of their experiments. For example, the snap above was done with a dry towel.