Kevin Zhou Physics Olympiad Handouts

Problem Solving II: Experimental Methods

For more about the theory of uncertainty analysis, see this handout, or for a more introductory
take, this handout and this comic. For practical tips for real experiments, especially at the IPhO,
see chapter 7 of Physics Olympiad: Basic to Advanced Exercises. For some entertaining general
discussion, see chapters I-5 and I-6 of the Feynman lectures. There is a total of 84 points.

1 Basic Probability
Idea 1
If a quantity X has the probability distribution p(x), that means
Z b
the probability that a ≤ X ≤ b is p(x) dx.
a

In particular, the total probability has to sum to one, so

Z ∞
p(x) dx = 1.
−∞

Using the probability distribution, we can calculate expectation values, i.e. averages. For
example, the expectation value of X, also called the mean, is
Z ∞
hXi = xp(x) dx
−∞

while the expectation value of an arbitrary function of X is

Z ∞
hf (X)i = f (x)p(x) dx.
−∞

One especially important quantity is the variance of X, defined as

var X = hX 2 i − hXi2 .
√
The standard deviation is defined by σX = var X. It describes how “spread out” the
distribution of X is, and it will play an important role in uncertainty analysis.

[1] Problem 1. Suppose that x is a length. What are the dimensions of p(x), hXi, var X, and σ?

Example 1

Trains arrive at a train station every 10 minutes. If I arrive at a random time, and X is the
number of minutes I have to wait, what is the standard deviation of X?

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Solution
We see that X can be anywhere between 0 and 10, with all possibilities equally likely, so
(
1/(10 min) 0 ≤ x ≤ 10,
p(x) =
0 otherwise

where the denominator guarantees the total probability is 1. For the rest of this example,
we’ll suppress the units. We have
Z ∞ Z 10
x
hXi = xp(x) dx = dx = 5
−∞ 0 10

which makes sense, as I should have to wait half the maximum time on average, and
Z ∞ Z 10 2
x 100
hX 2 i = x2 p(x) dx = dx = .
−∞ 0 10 3

Then the standard deviation is

p 5
σX = hX 2 i − hXi2 = √ min.
3

[3] Problem 2. Consider an exponentially distributed quantity,

(
ae−ax x ≥ 0,
p(x) =
0 otherwise.

Verify that the total probability is 1, and compute the mean and standard deviation. To perform
the integrals, you will have to integrate by parts.

[2] Problem 3. The purpose of subtracting hXi2 in the variance is to make sure it doesn’t change
when a constant is added to x, since shifting something left or right on the number line shouldn’t
change its spread. Verify that for any constant c, var X = var(X + c).

[3] Problem 4. We say X is normally distributed if

2
p(x) ∝ e−a(x−b) .

For simplicity, let’s shift X so that it’s centered about x = 0, so

2
p(x) ∝ e−ax .

You may use the result given in P1,

Z ∞ √
2
e−x dx = π.
−∞

Find the constant of proportionality in p(x), the mean, and the standard deviation.

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[2] Problem 5. If two random variables X1 and X2 are independent, then

hX1 X2 i = hX1 ihX2 i.

Use this result to show that

var(X1 + X2 ) = var(X1 ) + var(X2 )

which implies that the standard deviation “adds in quadrature”,

q
σX1 +X2 = σX 2 + σ2 .
1 X2

This is an important result we’ll use many times below.

2 Uncertainty Propagation
In this section, we’ll establish the fundamental results needed to compute uncertainties.
Idea 2
When a physical quantity is measured in an experiment and reported as x±∆x, it is uncertain
what the true value of the quantity is. If the quantity has a probability distribution p(x),
then the reported uncertainty ∆x is essentially the standard deviation of p(x).

Remark
In practice, you’ll have to use intuition and experience to assign uncertainties for real mea-
surements. For example, if you’re using a clock that times only to the nearest second, you
might take ∆t = 0.5 s. If you’re using a good ruler, which has millimeter markings, you might
take ∆x = 0.5 mm, though you can actually do a bit better if you look carefully. Of course,
the ultimate test is the results: if you assigned the uncertainties right, your final uncertainty
should encompass the true result most (but not all) of the time.

[2] Problem 6. Suppose x has uncertainty ∆x and y has uncertainty ∆y, where x and y are indepen-
dent. Explain why the uncertainty of x + y is
p
∆(x + y) = (∆x)2 + (∆y)2 .

This is called “addition in quadrature”. What is the uncertainty of x − y? How about x + x?

Remark
Note how this differs from “high school” uncertainty analysis. In school, you might be told
to show uncertainty using significant figures, and when adding two things, to keep only the
figures that are significant in both of them. That corresponds to

∆(x + y) = max(∆x, ∆y)

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which is an underestimate. Or, you might be told that the uncertainty needs to encapsulate
all the possible values, which implies that

∆(x + y) = ∆x + ∆y

which is an overestimate, since the uncertainties could cancel.

Example 2: F = ma 2016 25

Three students make measurements of the length of a 1.50 m rod. Each student reports an
uncertainty estimate representing an independent random error applicable to the measure-
ment.

• Alice performs a single measurement using a 2.0 m tape measure, to within 2 mm.

• Bob performs two measurements using a wooden meter stick, each to within 2 mm, which
he adds together.

• Christina performs two measurements using a machinist’s meter rule, each to within
1 mm, which she adds together.

Rank the measurements in order of their uncertainty.

Solution
√
The uncertainty in Alice’s measurement is 2 mm. The √ uncertainty in Bob’s is 2 2 mm by
quadrature, while the uncertainty in Christina’s is 2 mm by quadrature. So the lowest
uncertainty is Christina’s, followed by Alice’s, followed by Bob’s.

[2] Problem 7. Given N independent measurements of the same quantity with the same uncertainty, √
xi ± ∆x, find the uncertainty of their sum. Hence show the uncertainty of their average is ∆x/ N .
This result is extremely important, since repeating trials is one of the main ways to reduce
uncertainty. But it’s important to remember that the results derived above hold only for independent
measurements. For example, taking a single measurement, then averaging that single number with
itself 100 times certainly wouldn’t reduce the uncertainty at all!

Idea 3
If x has uncertainty ∆x, and f (x) can be approximated by its tangent line, f (x0 ) ≈ f (x) +
(x0 − x)f 0 (x) within the region x ± ∆x, then f (x) has approximate uncertainty f 0 (x) ∆x.

√
[2] Problem 8. If x has uncertainty ∆x, find the uncertainties of x2 , x, 1/x, 1/x4 , log x, and ex .

[2] Problem 9. The tangent line approximation doesn’t always make sense. For example, suppose that
x is measured to be zero, up to uncertainty ∆x. Show that the above results for the uncertainties
√
of x2 and x give nonsensical results. What would be a more reasonable uncertainty to report?

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[2] Problem 10. Consider two quantities with independent uncertainties, x ± ∆x and y ± ∆y.

(a) Show that the uncertainty of xy is

s 2  2
∆x ∆y
∆(xy) = xy + .
x y

To do this, start by writing xy as exp(log x + log y).

(b) If we set x = y, then we find

s 
∆x 2 √

2 2
∆(x ) = x 2 = 2x∆x.
x

On the other hand, in a previous problem we found ∆(x2 ) = 2x∆x. Which result is correct?

(c) Find the uncertainty of x/y.

[2] Problem 11. A student launches a projectile with speed v = 5 ± 0.1 m/s in gravitational accelera-
tion g = 9.81 ± 0.01 m/s2 . The resulting range is d = 1.5 ± 0.02 m. Given that the launch angle was
less than 45◦ , find the launch angle, with uncertainty, assuming all uncertainties are independent.

[2] Problem 12. Two physical quantities are related by y = xex .

(a) If x is measured to be 1.0 ± 0.1, find the resulting value of y, with uncertainty.

(b) If y is measured to be 2.0 ± 0.1, find the resulting value of x, with uncertainty.

Idea 4
For practical computations, it is often useful to use relative uncertainties. The relative
uncertainty of x is ∆x/x, and can be expressed as a percentage.

[1] Problem 13. Some basic relative uncertainty results.

(a) Show that the relative uncertainty of the product or quotient of two quantities with independent
uncertainties is the square root of the sum of the squares of their relative uncertainties.

(b) Show that averaging

√ N independent trials as in problem 7 reduces the relative uncertainty by
a factor of N .

Remark
There are many situations where the rules above can’t be used. For example, consider the
uncertainty of x + y 2 /x, where x and y have independent uncertainties. You can calculate the
uncertainty of either term with the standard rules, but you can’t calculate the uncertainty
of their sum, because the terms are not independent (both contain x).

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In these cases, you can use the multivariable equivalent of the tangent line approximation,
∂f ∂f
f (x0 , y 0 ) ≈ f (x, y) + (x0 − x) + (y 0 − y) .
∂x ∂y
Adding the two contributions to the uncertainty in quadrature gives
s 2  2
∂f ∂f
∆f = ∆x + ∆y .
∂x ∂y

This is the general rule that includes the rules you derived above as special cases. However, it
shouldn’t be necessary in Olympiad problems. If you run into such situations in an experiment,
often one of the uncertainties is much smaller, and can be neglected entirely.

Remark
As you saw in problem 9, the tangent line approximation can sometimes fail. The proper way
to handle situations like these would be to find the full probability distribution of the desired
quantity, rather than just describing it crudely with its standard deviation. However, this
can’t be done analytically except in the simplest of cases. So when professional physicists run
into situations like these, which are quite common, they often just numerically compute a few
million or billion values, starting with randomly drawn inputs each time, and use that to infer
the probability distribution. This technique is called Monte Carlo. It’s very powerful, but
certainly not needed for Olympiads! On Olympiads, you should just fall back to something
reasonable, such as taking the minimum and maximum possible values.

3 Using Uncertainties
Example 3: F = ma 2022 B21

Amora and Bronko are given a long, thin rectangle of sheet metal. (It has been machined very
precisely, so they can assume it is perfectly rectangular.) Using calipers, Amora measures
the width of the rectangle as 1 cm with 1% uncertainty. Using a tape measure, Bronko
independently measures its length as 100 cm with 0.1% uncertainty. What are the relative
uncertainties they should report for the area and the perimeter of the rectangle?

Solution
To compute the area, we multiply the two measurements, which means we add the relative
uncertainties in quadrature,
∆A p
= (1%)2 + (0.1%)2 ≈ 1%.
A
Note that in this case, the relative uncertainty of Bronko’s measurement is negligible; the rel-
ative uncertainty of the area is approximately the relative uncertainty of Alice’s measurement.

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Computing the perimeter involves adding the measurements, which means the absolute
uncertainties are added in quadrature instead. These are 0.01 cm and 0.01 cm for Alice and
Bronko’s measurements, respectively, so the absolute uncertainty of Alice’s measurement
is negligible. Thus, the relative uncertainty of the perimeter is approximately the relative
uncertainty of Bronko’s measurement, 0.1%.

In simple Olympiad experiments, often only one uncertainty will really matter. This can
dramatically simplify calculations, but it might take a little thought to tell which one.

[3] Problem 14. 01^‚ Solve F = ma 2018 problems A12, A25, B19, and B25, and F = ma 2019
problems A16, B18, and B25. Make sure to strictly adhere to the total time. Since these are
F = ma problems, you don’t have to produce a writeup. If you find these questions difficult to
finish in the allotted time, go back and review the earlier material!

[2] Problem 15. Suppose the goal of an experiment is to measure the ratio T1 /T2 of the durations of
two physical processes, where T1 is about 15 seconds, and T2 is about 3 seconds. Also suppose your
stopwatch is only accurate to the nearest second. You have two minutes to perform measurements.
Assume each measurement is independent.

(a) Using your instinct, figure out whether it’s better to spend more total time measuring T1 ,
more total time measuring T2 , or an equal amount of time on both.

(b) To confirm this, qualitatively sketch the relative uncertainty of T1 /T2 as a function of the
fraction of time x spent measuring T1 , using explicit numeric examples if necessary.

Calculations of this sort are common when doing Olympiad experimental physics. You should be
able to do them instinctively, getting the ballpark right answer without explicit calculation.

[3] Problem 16. In the preliminary problem set, you measured g using a pendulum. If you didn’t do
uncertainty analysis for it, as we covered above, then you should go back and estimate uncertainties
more precisely. In this problem you’ll do a different experiment: you will estimate g by finding the
time needed for an object to roll down a ramp, with everything again made of household materials.

(a) Before starting, think about what the dominant sources of uncertainty will be, and how you
can design the experiment to minimize them. In particular, do you think the result will be
more or less precise than your pendulum experiment?

(b) Perform the experiment, taking at least ten independent measurements, and report the data
and results with uncertainty.

[3] Problem 17. [A] Consider N independent measurements of the same quantity, with results xi ±∆xi .
They can be combined into a single result by taking a weighted average. What is the optimal weighted
average, which minimizes the uncertainty?

All of the examples above involve combining continuous quantities, so we’ll close this section with
some applications to “counting” experiments, which work slightly differently.

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Remark
In this problem set, we have given rules for calculating the mean and standard deviation
of derived quantities. But in general, probability distributions can have all kinds of weird
features, which aren’t captured by those two numbers. The reason we focus on them anyway
is because of the central limit theorem, which roughly states that if we have many independent
random variables, the distribution of the sum will approach a normal distribution. As you
saw in problem 4, normal distributions are characterized entirely by their mean and standard
deviation, so we don’t lose any information by reporting only those two quantities.

Example 4

A fair coin is tossed 1000 times, and the number of heads is counted. If this process is
repeated many times, what is the standard deviation of the number of heads?

Solution
Consider one trial of 1000 tosses. The number of heads is X = X1 + X2 + . . . + X1000 , where
(
1 heads on toss i
Xi = .
0 tails on toss i

Of course, the mean of each of these variables is hXi i = 0.5, so that the mean of X is 500. In
addition, the Xi are independent of each other, so the variances add. The variance of each
one of them is
1 1 1
var Xi = hXi2 i − hXi i2 = − = .
2 4 4
Thus, the standard deviation of the number of heads is
√ p
var X = 1000/4 ≈ 16.

So getting 520 heads would not be surprising, but if you got 550, you might be justified in
suspecting the coin isn’t fair. (Also, the number of heads is very close to normally distributed,
by the central limit theorem mentioned above.) To check whether you understand p this, you
can redo it with a general probability p of getting heads, where you should get 1000 p(1 − p).

[3] Problem 18. At any moment, a Geiger counter can click, indicating that it has detected a particle
of radiation. Suppose that there is an independent probability α dt of clicking at each infinitesimal
time interval dt. Let the number of clicks observed in a total time T be X.

(a) Find the expected value and standard deviation of X, and thereby compute its relative
uncertainty. (Hint: split the total time into many tiny time intervals, and let Xi be the
P
number of clicks in interval i, so X = i Xi .)

(b) Using a Geiger counter on a sample, you hear 197 clicks in 5 minutes of operation. Estimate
the activity α of the sample (i.e. the expected clicks per second), with uncertainty. If you
measure for longer, how does the uncertainty reduce over time?

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(c) Now suppose that for a different sample, N = 0 after 5 minutes. Estimate the activity α of
the sample (i.e. the expected clicks per second), with a reasonable uncertainty. If you measure
for longer, and continue to hear no clicks, how does the uncertainty reduce over time?

[4] Problem 19. [A] This problem extends problem 18 to derive some canonical results.

(a) Let λ = αT . Find the probability p(X = k) of hearing exactly k clicks in terms of λ and k.

(b) To check your result, show that the sum of the p(X = k) is equal to one.

(c) ? In the limit λ  1, show that the probabilities p(X = k) approach that of a normal
distribution with the mean and standard deviation calculated in problem 18, thereby providing
an example of the central limit theorem at work. This is a rather involved calculation, which
will use many of the techniques from P1. It will also require Stirling’s approximation,
√  n n
n! ≈ 2πn
e
for n  1, which will be important in T2. (Hint: because the relative uncertainty falls as λ
increases, start by writing k = λ(1 + δ) for |δ|  1, and expand in powers of δ. Be careful not

01mƒ
to drop too many terms, as δ is small, but λδ isn’t.)

[3] Problem 20. IPhO 2023, problem 1, parts A, B, and D.3. A short derivation of the key features
of Brownian motion. It requires only the ideas of this problem set, and some basic mechanics.

4 Graphing
Idea 5
All graphical data analysis for the USAPhO and IPhO can be performed by drawing a line
and measuring its slope and intercept. This is a bit artificial, but it’s necessary because of
the limited calculation equipment you have during these exams. Despite this, drawing lines
can be surprisingly powerful.

Example 5

The activity of a radioactive substance obeys A(t) = A0 e−t/τ . Using measurements of t and
A(t), plot a line to find A0 and τ .

Solution
To handle exponential relationships, take the logarithm of both sides for

log A(t) = log A0 − t/τ.

Then a plot of log A(t) vs. t has slope −1/τ and y-intercept log A0 .

[1] Problem 21. For a power law y = αxn where y and x are measured, what line can be plotted to
find α and n?

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[2] Problem 22. The rate R of electron emission from a solid in an electric field E is
R = βe−E/E0
for some constants β and E0 . The particular form is because the effect is due to quantum tunneling,
and you will derive it in X2.
(a) If E and R are measured, what line can be plotted to find β and E0 ?
(b) Your answer for part (a) should have formally incorrect dimensions, by the standards of P1.
This often happens when one takes logarithms. What’s going on? If the dimensions are wrong,
how can the result be right?
(c) Suppose both β and E0 have 1% uncertainty. For small E, which is more important for the
uncertainty of R? What about for large E? Around where is the crossover point?

Example 6

Suppose that y and x are related nonlinearly, as

y = bx + ax2 .

For example, this could model the force due to a non-Hookean spring. Using measurements
of x and y, plot a line to find a and b.

Solution
If we divide by x, we find
y
= ax + b.
x
Therefore, we can plot y/x versus x, which gives a line with slope a and intercept b. More
generally, we can plot a line whenever we can rearrange a given relation into the form

(known) = (unknown)(known) + (unknown)

where all four terms can be arbitrarily complicated. In this way, it is possible to turn a lot
of very nonlinear relations into lines.

[3] Problem 23. Some more examples of finding lines to plot.

(a) Suppose that you are given points (x, y) that lie on a circle centered at (a, 0) with radius r.
What line can be plotted to find a and r?
(b) Consider an Atwood’s machine with masses m and M > m. The acceleration of the machine
is measured as a function of M . However, since the pulley has mass, it slows the acceleration
of the Atwood’s machine, so that
M −m
a= g.
M + m + δm
Find a line that can be plotted to find g and δm, assuming m, M , and a are known. This is
an example of how plotting a line can separate out a systematic error, i.e. the value of δm,
which would be impossible if only one value of M were used.

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(c) Suppose an object is undergoing simple harmonic motion with amplitude A and angular
frequency ω. Given measurements of the position x and velocity v, what line can be plotted
to find A and ω?

Remark
When performing data analysis in practice, you should neatly organize your work. Always
make a data table that explicitly shows what you’re calculating, and make a neat graph with
a ruler and graph paper. Set the axis scale so that the graphed data points cover almost
the entire page, and let the x-axis include x = 0 if you need to find a y-intercept. The
computation of the slope should be explicitly shown. For each line you should use at least
about five points; you don’t have to use them all. If you have a calculator that can find best-fit
slopes for you, don’t use it, as these features are generally not allowed on real Olympiads.

[3] Problem 24. 01W USAPhO 2012, problem A2. (This one requires basic thermodynamics.)

[3] Problem 25. 01W USAPhO 2011, problem A2.

[3] Problem 26. 01c‚ INPhO 2018, problem 7. (This one requires basic fluid dynamics.)

Idea 6
Historically, uncertainty analysis has only appeared on the F = ma, and data analysis has
only appeared on the USAPhO, but the two appear together in the IPhO.

To perform uncertainty analysis for best fit lines, plot the uncertainties of the data points
as error bars. Then draw the steepest and shallowest lines that still pass through most of
the error bars. These will give you the bounds on your slope and intercept. We’ll see some
examples of this procedure in later problem sets. It isn’t the most mathematically rigorous
method, but it gives decent results.

5 Estimation
Estimation is a useful skill for checking the answers to real-world problems.
Example 7

Estimate the circumference of the Earth.

Solution
If you know that the United States is 3,000 miles wide, and there is a time zone difference of
three hours between California and New York, then a reasonable estimate is 24,000 miles.
Or, if you know the factoid that light can go about seven times around the Earth in a second,
then a reasonable estimate is (3/7) × 108 m ≈ 4 × 107 m.

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Let’s check these results are compatible. There are about 5 miles in 8 kilometers, a fact
you can get by remembering how your car’s speedometer looks, or by noting that 3 feet are
about 1 meter. Then 4 × 104 km ≈ (5/8) × 4 × 104 mi = 2.5 × 104 mi, so the two results are
compatible. There are probably at least a hundred more ways to perform this estimation.

Example 8

Estimate the density of air, and compare this to the density of water.

Solution
We can directly use the ideal gas law, P V = nRT . The density is ρ = µn/V where µ is the
mass of one mole of air, so
µP
ρ= .
RT
Atmospheric pressure is about 105 Pa, typical temperatures are about 300 K, and air is mostly
N2 , which has a molar mass of µ = 28 g/mol, so

(0.028)(105 ) kg kg
ρ= ≈ 1 3.
(8.3)(300) m3 m

The density of water is, almost by definition,

kg
ρw ≈ 103 .
m3
Most liquids and solids have densities within an order of magnitude of this, since in all
cases the atoms are packed close together. Evidently, air molecules are about a factor of
(103 )1/3 = 10 times further apart than typical water molecules.

Example 9

Estimate how much useful power you can produce in a short burst.

Solution
This is a bit tricky to test, because most exercises just burn energy against air resistance or
friction, which is hard to estimate. However, a task that directly performs work is useful. I
weigh about 75 kg and can run up a 3 m high staircase in around 3 s, so

P = mgv = (75)(10)(3/3) W ≈ 750 W.

This is a typical max power output, while typical steady state power outputs are several
times smaller, and the corresponding numbers for elite athletes are several times larger.

For the below questions, feel free to look up specific numbers if you’re stuck. In all cases, an answer
to the nearest order of magnitude is good enough.

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[3] Problem 27. Some questions about light energy.

(a) Estimate the number of photons emitted per second by a standard light bulb. (The energy of
a photon is E = hf , and the frequency of a photon is related to the wavelength by c = f λ.)

(b) The Sun supplies power of intensity 1400 W/m2 to the Earth. The nearest star is about 4
light years away. Assuming this star is similar to the Sun, about how many of its photons
enter your eye per second?

[2] Problem 28. Estimate the radius of the largest asteroid you could jump off of, and never return.

[4] Problem 29. Some questions about energy.

(a) Estimate the digestible energy content of a stick of butter. (A calorie is about 4000 J, and is
also the energy needed to raise the temperature of a kilogram of water by 1 K.)

(b) Estimate the rate at which your body burns energy when at rest.

(c) Estimate the rate at which a human being radiates energy. (The Stefan–Boltzmann law states
that the radiation power per unit area from a blackbody is σT 4 , where σ = 5.7×10−8 W/m2 K4 .)
Is radiation a significant source of energy loss for a human being, or is it negligible?

(d) A human being develops hypothermia, with their core body temperature dropping by 5 ◦ F.
Neglecting any heat transfer with the environment, estimate the number of calories required
to raise their temperature back to normal.

Now let’s verify the energy content of the butter microscopically. This will be a very rough estimate,
so expect answers to be only within two orders of magnitude.

(e) A chemical bond typically involves two electrons, and a characteristic atomic separation
distance of one angstrom, r ∼ 10−10 m. Estimate the binding energy of one chemical bond.

(f) The fats in butter are digested by inputting energy to break the bonds in the molecules, then
harvesting energy by combining the atoms into CO2 and H2 O, which have somewhat more
stable bonds.

Estimate the energy content of a kilogram of butter. How close is this to the true result?

[2] Problem 30 (Povey). When human beings lose weight, most of it is by exhalation of carbon.
About 20% of the air in the atmosphere is oxygen. When we breathe in and then out, about 25%
of the oxygen is converted to carbon dioxide.

(a) Estimate the mass of air contained in a single breath.

(b) Estimate the amount of weight we lose every day by breathing alone.

[2] Problem 31 (Insight). How long a line can you write with a pencil?

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6 Experimental Technique
At both the national and international Olympiad level, it’s important to have practical know-how
in order to make experiments work. It’s very hard to train this with only theoretical problems.
However, the Australian Physics Olympiad has some useful problems in this direction, since it has

01^‚
a strong emphasis on real-world physics.

[3] Problem 32. AuPhO 2010, problem 12.

[3] Problem 33. 01^‚ AuPhO 2012, problem 14.

[3] Problem 34.

answer sheets.
01Y‚ AuPhO 2016, problem 14. You will need to print out pages 8 and 9 of the

You can look at other AuPhO questions for further practice, but as you can see here, many AuPhO
questions are confusing, misleading, or even wrong, which is an unfortunate consequence of the
innovative nature of the contest. I’ll only assign you the best ones.
Recently, there have been several “simulation” based experiments, from EuPhOs, APhOs, and
IPhOs hosted during the pandemic. However, they tend to require substantially more background
to approach, so I’ll sprinkle them throughout later problem sets.

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