A-level Physics (Advancing Physics)/Exponential Relationships/Worked Solutions

'''1. Simplify Newton's Law of Cooling for the case when I place a warm object in a large tank of water which is on the point of freezing. Measure temperature in °C.'''

Newton's Law of Cooling states that $$T_t = T_{env} + (T_0 - T_{env})e^{-rt}$$

The freezing point of water is 0 °C, so, if we measure T in °C, Tenv = 0:

$$T_t = 0 + (T_0 - 0)e^{-rt}$$

$$T_t = T_0e^{-rt}$$

'''2. What will the temperature of an object at 40 °C be after 30 seconds? (Take r=10−3 s−1.)'''

$$T_t = T_0e^{-rt} = 40 \times e^{-10^{-3} \times 30} = 38.8$$ °C

'''3. A body is found in a library (as per. Agatha Christie) at 8am. The temperature of the library is kept at a constant temperature of 20 °C for 10 minutes. During these 10 minutes, the body cools from 25 °C to 24 °C. The body temperature of a healthy human being is 36.8 °C. At what time was the person murdered?'''

First, we must calculate r:

$$24 = 20 + (25 - 20)e^{-10r}$$

$$4 = 5e^{-10r}$$

$$e^{-10r} = 0.8$$

$$-10r = \ln{0.8}$$

$$r = \frac{\ln{0.8}}{-10} = 0.0223\mbox{ minute}^{-1}$$

Then, calculate t - this is the time between the murder and 8am:

$$25 = 20 + (36.8 - 20)e^{-0.0223t}$$

$$5 =16.8e^{-0.0223t}$$

$$e^{-0.0223t} = \frac{5}{16.8}$$

$$-0.0223t = \ln{\frac{5}{16.8}}$$

$$t = \frac{\ln{\frac{5}{16.8}}}{-0.0223} = 54 \mbox{ minutes}$$

Therefore, the murder occurred at 7:06am.

'4. Suppose for a moment that the number of pages on Wikibooks p'' can be modelled as an exponential relationship. Let the number of pages required on average to attract an editor be a, and the average number of new pages created by an editor each year be z. Derive an equation expressing p in terms of the time in years since Wikibooks was created t.'''

Let n be the number of editors.

$$n = \frac{p}{a}$$

$$\frac{dp}{dt} = nz = z\frac{p}{a}$$

$$dp = z\frac{p}{a} dt$$

$$\int{\frac{1}{p}} dp = \int{\frac{z}{a}} dt$$

$$\ln{p} = \frac{zt}{a} + c$$ (where c is the constant of integration)

$$p = e^{\frac{zt}{a} + c} = e^{\frac{zt}{a}}e^c = ke^{\frac{zt}{a}}$$ (where k is a constant - k = ec)

There must have been a first page, which marked the point where t = 0, so:

$$1 = ke^{\frac{z}{a} \times 0} = ke^0 = k$$

Therefore:

$$p = e^{\frac{zt}{a}}$$

'''5. Wikibooks was created in mid-2003. How many pages should there have been 6 years later? (Take a = 20, z = 10 yr−1.)'''

$$p = e^{\frac{zt}{a}} = e^{\frac{10 \times 6}{20}} = e^3 = 20$$

'''6. The actual number of pages in Wikibooks in mid-2009 was 35,148. What are the problems with this model? What problems may develop, say, by 2103?'''

There are two key problems with this model:


 * We have estimated the values of the constants. These should have been determined statistically.
 * We have assumed that the constants are constant. In reality, as the amount of content on Wikibooks increases, more people think "Wikibooks already contains this content, so I am not going to add anything." This means that both z and a change with time. Our exponential model only applies over small periods of time. Each of these small periods of time has different values for the constants.

In the future, such as 2103, the constants will have changed so radically as to be useless. Question 5 shows how much they change over just 6 years - how much more must they change over a whole century!