Canoes, Carrots and Capacitors

If you stop paddling a canoe, its speed will decrease over time in a unique way: if it takes 3 seconds (for example) for its speed to become half the original, then in 3 more seconds it will be half of the half, etc. With a little playing around with the numbers, you will find that v = v_o (2)^-t/3 where v_o is the initial speed and v is the speed at time t (check this). There is nothing special about the number 2; we could find the time it takes to get to 1/5 of its original V and then our equation would have 5 to the -t/t power, where t is some positive time constant.

There is something special about the number e. You do not need to know calculus for understanding this. If you have not played with e before, first let's see what it is. If your calculator does not have e^x on it, find someone whose calculator does, and enter a 1, then e^x .(In some calculators, you push 1 after e^x.) This will tell the calculator to find e¹, which is e. Here is what is special about it: ce^x (where c is any constant) is the only function whose value is equal to its slope at all points. You could check this out by using the calculator to get enough values of e^x to plot a decent graph, then determine some slopes. Let's check it out another way. The slope is the change of e^x divided by the change of x. Since e^x is a curve, make these changes very small. For example, the slope at x=2 is approximately (e^2.01-e^1.99 )/.02. Do this on a calculator and compare to e². Repeat, using a D x of .002 and notice that the error gets smaller. It is also worth noting that ce^-x is the only function equal to the opposite of its slope.

Because of the property of e noted above, when a variable (like t) is in an exponent in an equation, people usually use e. Instead of v = v_o 2^-t/t we have v = v_oe^-t/t where t is the time it takes for the velocity to get to 1/e of its initial value. (1/e is about 0.37.)

The physical condition needed to make our velocity equation valid is that the drag force is proportional to velocity. We write F = -kv where k is a positive constant and the minus sign means that F is opposite in direction to v. So -kv = ma. But acceleration a is the slope of the v Vs t graph, which should lead us to speculate that v = v_oe^-t/t is reasonable. If you learn calculus, you will find out how to derive this and get t = m/k. At this point let's just verify it. Note that from the f=ma equation, we get -kv = ma, or -v = t a. Choose the easiest quantities to deal with, v_o =1 m/s and t = 1 s. Then at a time of 2 s, for example, the acceleration is approximately (e^-2.01 -e^{-1.99 )}/.02. (Use the calculator.) Compare this to -v, which is -e^-2.

The velocity of the canoe after a long time will be called v_f. If the canoe is on still water with no wind, v goes to zero eventually, so v_f = 0. However, if the canoe is on a stream, the ultimate velocity is the stream's velocity. It turns out that in the general case, v = v_f + (v_o - v_f )e^-t/t and t = m/k, same as before. Note that if you have v_f = 0, this equation reduces to the previous one.

If you take a bag of carrots out of the refrigerator, its ultimate temperature (T_f ) will be the the room temp. The rate of change of T is proportional to T_f -T. Now the rate of change of T is to acceleration as T is to v.Thus we can write T = T_f + (T_o-T_f )e^-t/t , where T_o = initial temp and t = a positive time constant which needs to be determined experimentally.

When a capacitor is discharged through a resistor, the rate of discharge (current) is proportional to the charge, since the voltage is proportional to the charge. So charge q follows the same law as velocity and temperature. Voltage then must obey that law also: with any network of resistors and batteries, V = V_f .+ (V_o - V_f )e^-t/t . In this case the time constant t is RC, where R is the resistance as seen by the capacitor when the batteries are replaced by their internal resistances. An Ohm×farad is a second.

We see that when the rate of change of a quantity (u) is proportional to u_f -u, then u = u_f + (u_o-u_f )e^-t/t , where u_o is the initial value of u and t is the inverse of the constant of proportionality. Check t to be sure it is in time units. Here are a couple of graphs of u vs t, with t on the horizontal axis:

Other cases where this applies (under the right conditions) include current in an inductor, magnetization or demagnetization, radioactive decay, and perhaps the decay of knowledge after completion of a course.

RABBITS, $, THE BOMB, COMPUTER SPEED, AND FADS

Positive exponentials abound in nature, but for limited times. If rabbits are supplied with food and protection from enemies, the rate of increase of n is proportional to n: dn/dt = ct, where c is a positive constant. This leads to n = n_oe^ct. Eventually one would run out of money for food or space to house them.

Other examples include money in the bank (yes, exponential growth does not always mean fast growth), the beginning of a bomb blast, computer speed and the beginning of a fad. Eventually you remove the money, or you die and someone else removes it. Any bomb blast is self-limiting due to running out of fuel. New computer speeds are currently increasing exponentially, but eventually will be limited by the laws of nature, so the rate of increase will taper off. Fads loose their appeal when they are old. So the initial exponential growth of these things must cease. No quantity in the real world can increase exponentially forever. (But do not believe statements like this which tell how nature must behave. This one seems safe, but….)

You probably want to hit your browser's back button.

My main pages:
Mechanics
Fluids, heat, electricity and magnetism
Vibrations and waves
Quantum
or look up this kind of baloney alphabetically in my index.

Comments, questions: fredrick.gram at tri-c.edu (but remove “at” and spaces and insert @)