One of the main things that separate the examples in physics
class from reality
. Be they projectile motion
problems, there never seems to be any air resistance taken into consideration
. This makes us wonder how, exactly, some of these problems come about.
For example, the problem at hand would state that a parachutist drops from an airplane at an initial vertical velocity of 0. Fair enough, but then it goes on to say the parachutist free-falls for a certain amount of time—without air resistance. Presumably, this makes it easier for the problem solver to just go ahead and plug the acceleration due to gravity into the necessary equations. Then, of course, the parachute opens, and (because of air resistance) it slows the parachutist at a given deceleration.
"Why wouldn't he have air resistance on the free-fall part of the way down?" we asked our esteemed physics teacher.
"It makes it easier. Free fall is free fall. If we had to factor in air resistance for the free fall, it would become a whole mess."
"But we like the mess."
And that was that. But we began to wonder: if there was no air resistance, then there must be no air; there were no molecules to collide with any objects and thus cause friction. There would not only be a lack of effect on behalf of the parachute (it would just trail behind the parachutist without inflating), but also by the wings of the airplane. What would Bernoulli effect mean if the pressure on both sides of the wing was null? We didn't even get into how the parachutist would breathe, much less keep from exploding, on the way down.
That said, I believe it's time to define air resistance in mathematical terms. As a problem in my old calculus textbook suggests, the force due to air resistance on an object is directly proportional to the square of the velocity of the object through the air (relative to the air, I should say—this is assuming all the air in the immediate area of the object is moving in the same direction and speed at any given instant). Unfortunately, that constant of proportionality, as far as I know, can only be found by experimentation. This explains the phenomenon that skydivers experience that I've heard called "the Superman effect," where their acceleration is null, i.e. the force due to air resistance (up) is equal to the force due to gravity (down).
Because of this, displacement equations, and by extension, velocity equations, of free fall with air resistance factored in often contain e or some other number raised to a power that contains t or whatever the time variable happens to be.
Any suggestions, comments, or complaints can be directed to "/msg pmdboi". Fourth in a series of MathRants by PMDBoi.