Understanding the Equation for a Falling Leaf: A Simplified Explanation

[Kaboosh]

I've been online searching for interesting stuff about maths and physics.
I came across to this question in yahoo answers that i found quite interesting.

The first equation would be working out how long it takes to reach the ground, and how fast it's going when it arrives. This is actually going to be a differential equation since the air resistance on the leaf will cause it to reach terminal velocity very quickly, and will have the form:

F = mg - k(dx^2/dt^2).

This is the site where i found it.

http://answers.yahoo.com/question/index?qid=20081026115651AAN2FTv

Thing is, i have not yet reached to the point where i have learned or even heard of terminal velocity and integral calculus.
Could somebody please kinda explain it what these things are and how the equation works.
In other words, it's kinda like ''dumbing it down''.

-Kaboosh

 

[russ_watters]

Welcome to PF.

One of the goals of this site is helping people learn how to learn. When you have a question that is so broad that people spend years of their lives studying it, it is tough to get a short answer to a question like "what is calculus?" But the word and concept are something you can type into google or wiki and get concice (though horribly incomplete) answers. So that should always be your starting point. Here's the wiki on calculus:

Calculus (Latin, calculus, a small stone used for counting) is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern university education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus and join calculus.

Principles:

Limits [how to work around division by zero]
Differential Calculus [how to find the slope of a curve at a single point]
Integral Calculus [how to chop a graph up into little pieces and add them together to get the area under it]

http://en.wikipedia.org/wiki/Calculus

You should try to read through the whole article.

The answer on that yahoo page also gives a definition of "terminal velocity", but in my words, terminal velocity is the speed an object falls at when the aerodynamic drag drag force equals the weight of the object.

I'm not liking that equation, though. It doesn't look quite right to me. In any case, I prefer the first one here in the wiki for terminal velocity: http://en.wikipedia.org/wiki/Terminal_velocity#Derivation_for_terminal_velocity

It starts the same, with the first term "mg" being the force of the weight of the object and the next term being the drag force. Rather than solving the equations of motion (which requires calculus), you can plug a=f/m, s=at and d=st into a spreadsheet to numerically integrate and find the resulting performance of a falling leaf. This method doesn't require more than junior high math.

 

[Kaboosh]

Cheers