xkcd what if?

First of all, if you haven't been reading xkcd, I recommend you start reading it soon! It's an awesome webcomic, perfect for people who actually bother to research on the multitude of references inside, ranging from movies to quantum physics!


Next of all (after you've convinced yourself of the worth of reading the main strip of comics), do check out the sideline series of articles by the same author (Randall Munroe) "what if?".


Today's article will randomly make comments on the 44th what if? issue: High Throw. Here's the link: http://what-if.xkcd.com/44/ . I'm not of the opinion that what if? is substandard. In fact, I do think it's awesome and enjoy reading it. But that shouldn't stop me from commenting about it and trying my very best to poke loopholes... right? :)


Quote: "A timing error of half a millisecond in either direction is enough to cause the ball to miss the strike zone... To put that in perspective, it takes about five milliseconds for the fastest nerve impulse to travel the length of the arm."


Comment: This might have something to do with systematic errors being cancelled out: since the time to throw the baseball is determined beforehand by the pitcher, possibly in the brain (and if it's reflexive enough maybe part of it is generated in the spine), there is a fixed amount of delay between giving the signal to swing the arm around and the signal to release the ball in order to hit the strike zone, and this is possibly learned through trial and error, since nobody is THAT good when they were first introduced to baseball. An example of a similar phenomenon can be observed when timing the timespan between two of the same events with no warning (eg. 2 lightning flashes), where the reaction times tend to cancel each other out almost completely (In my experience, my reaction time varies less than 0.03 seconds when doing the "drop a ruler to find out reaction time" test, whereas my reaction time is close to 0.2 seconds. Note that this is still out of the 0.5 millisecond range mentioned by about 3 times. Maybe that's why I don't play baseball.)


Quote:"But we could also sidestep the whole problem by using a device like this one:

It could be a springboard, a greased chute, or even a dangling sling—..."


Comment: This is actually unlikely to be efficient. The first problem is that the entry angle is unlikely to be sufficiently exact for it to just slide up as depicted in the picture. It's likely to bounce around a bit, which tends to lose energy quite well. The second difficulty to overcome is, as in most physics problems, friction. No I'm not talking about air resistance, that has already been (somehow) accounted for in the what if? comic. I'm talking about friction with the deflector. Firstly, let's talk about an instantaneous deflector (eg. a stiff, hard 45-degrees-to-horizontal board). This will cause a great big bounce, which loses energy according to the coefficient of restitution (COR). "Generally, the COR is thought to be independent of collision speed...(except when collision speeds are in the range of 1cm/second)". The highest permissible COR for a tennis ball, according to the international Table Tennis Federation, is about 0.92, on a standard steel block (Wikipedia). This means that about 8% of the energy is lost in one bounce, or about 8% of the height is lost (with an uncertainty of a factor of 2, considering air resistance). This is close to half a giraffe of height. For a baseball, this is likely to be much more, although to be fair, colliding with a board at 45 degrees will probably lessen the impact.


The second kind of energy loss through friction with the velocity converter is the friction that comes when the ball slides along the (for brevity) chute. Usually, when dealing with such dynamic (or kinetic) friction, we can use a formula:

"The coefficient of friction (COF), often symbolized by the Greek letter µ, is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together." -- Wikipedia, yes again.

Lubricated steel on steel has a coefficient of 0.16, again from Wikipedia (what a glaring testimonial to the unreliability of this article!). So, where does the normal force come from? It turns out that there has to be normal force acting on the ball, since it changes its velocity, and hence this means it accelerates, and using the standard F=ma, we can see that the Kinetic friction is Fric= ma = m(v^2/r), where r is the radius of the chute (Yes that's the formula for centripetal acceleration). So how much energy is lost? We shall make the simplifying estimate that velocity is somewhat constant throughout the ride (else it would be a bad idea anyway). We get:

Work done (against friction) = 2(pi)m*v^2

Well, not quite. After all, can't the ball roll? (The coefficient for rolling friction tends to be a lot lower, which is why cars are even remotely efficient) Turns out that even if it rolls, it won't go as high anyway, because the kinetic energy will be converted into rotational kinetic energy, which doesn't really cause the ball to go higher. (Note that this is in particular because it is a ball; projectiles such as bullets are typically given spin so that they stabilise in an aerodynamic position, a phenomenon known as the gyroscopic effect)


As an aside, note that r is not quite the radius of the chute (I lied.). r is actually the difference between the chute and the ball (Details are left as an exercise to the reader, but rest assured everything tends to cancel out.). There is an interesting special case, which will probably lead to the discussion of a bounce, which I shall not discuss due to its theoretical complexity (i.e. r=0.). And I've been using ".)." too often (Right.).


Yet another point he neglected to discuss (probably too boring) was whether we could thin out the air resistance, for example by climbing to the top of Mount Everest, where the air is thinner (minus some power to account for altitude sickness), or possibly from a hovering helicopter even higher up (Baseball probably would instantly fall downwards from the downwash of the helicopter's rotors or just plainly crash into it).


And one last thing: I don't get the caption for the last comic drawing.