User:Inconspicuum/Physics (A Level)/Exponential Relationships

Graph of x = ae^bt, where b is positive.

Many things are governed by exponential relationships. The exponential relationships which we shall be dealing with are of the following form:

$x=ae^{bt}$,

where t is time, x is a variable, and a and b are constants. e is just a number, albeit a very special number. It is an irrational constant, like π. e is 2.71828182845904523536 to 20 decimal places. However, it is far easier just to find the e (or exp) button on your calculator.

The inverse function of e^t is the natural logarithm, denoted ln t:

$\ln t=\log _{e}t$

Graph of x = e^-bt, showing several different values of b.

Growth and Decay

When b is positive, an exponential function increases rapidly. This represents the growth of certain variables very well. When b is negative, an exponential function decreases, flattening out as it approaches the t axis. This represents the decay of certain variables.

Exponential Relationships in the Real World

An exponential relationship occurs when the rate of change of a variable depends on the value of the variable itself. You should memorise this definition, as well as understand it. Let us consider some examples:

A petri dish with bacteria growing on it.

Population Growth

Consider a Petri dish full of agar jelly (food for bacteria) with a few bacteria on it. These bacteria will reproduce, and so, as time goes by, the number of bacteria on the jelly will increase. However, each bacterium does not care about whether there are other bacteria around or not. It will continue making more bacteria at the same rate. Therefore, as the total number of bacteria increases, their rate of reproduction increases. This is an exponential relationship with a positive value of b.

Of course, this model is flawed since, in reality, the bacteria will eventually have eaten all the agar jelly, and so the relationship will stop being exponential.

Emptying Tank

If you fill a large tank with water, and make a hole in the bottom, at first, the water will flow out very fast. However, as the tank empties, the pressure of the water will decrease, and so the rate of flow will decrease. The rate of change of the amount of water in the tank depends on the amount of water in the tank. This is an exponential relationship with a negative value of b - it is an exponential decay.

Cooling

A hot object cools down faster than a warm object. So, as an object cools, the rate at which temperature 'flows' out of it into its surroundings will decrease. Newton expressed this as an exponential relationship (known as Newton's Law of Cooling):

$T_{t}=T_{env}+(T_{0}-T_{env})e^{-rt}$,

where T_t is the temperature at a time t, T₀ is the temperature at t = 0, T_env is the temperature of the environment around the cooling object, and r is a positive constant. Note that a here is equal to (T₀ - T_env) - but a is still a constant since T₀ and T_env are both constants. The '-' sign in front of the r shows us that this is an exponential decay - the temperature of the object is tending towards the temperature of the environment. The reason we add T_env is merely a result of the fact that we do not want the temperature to decay to 0 (in whatever unit of temperature we happen to be using). Instead, we want it to decay towards the temperature of the environment.

Mathematical Derivation

We have already said that an exponential relationship occurs when the rate of change of a variable depends on the value of the variable itself. If we translate this into algebra, we get the following:

${\frac {dx}{dt}}\ =ax$ , where a is a constant.

By separating the variables:

$dx=axdt$

${\frac {1}{x}}dx=adt$

$\int {\frac {1}{x}}dx=\int adt$

$\ln x=at+c$ (where c is the constant of integration)

$x=e^{at+c}=e^{at}e^{c}$

If we let b = e^c (b is a constant, since e^c is a constant):

$x=be^{at}$