Physical quantities and estimates

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Every value you write in physics is a number and a unit together.

This section explains why both parts are needed, and how to make a sensible estimate of a quantity when no data are given.

A number and a unit

A physical quantity is a property of an object or a process that can be measured: the mass of a trolley, the current in a lamp, the time for one swing of a pendulum.

Every value of a physical quantity has two parts, a numerical magnitude and a unit.

In $m = 2.4\ \text{kg}$ , the magnitude is 2.4 and the unit is the kilogram.

Neither part means anything alone.

"The length is 3" could be 3 mm or 3 km, a factor of a million apart.

So a final answer with no unit, or with the wrong unit, is a wrong answer, not a small slip: it normally loses the answer mark.

The value is the magnitude multiplied by the unit.

That is why you can treat units like algebra: $2.4\ \text{kg} \times 3.0\ \text{m s}^{-2} = 7.2\ \text{kg m s}^{-2}$ .

It is also why a table column heading or a graph axis label is written as quantity / unit, for example $I$ / mA.

Dividing the quantity by its unit leaves the pure number that goes in the table.

In a practical paper, " $I$ / mA" is the usual form, and " $I$ (mA)" is also accepted; " $I$ mA" or a bare "mA" is not.

Symbols for quantities and for units

The same letter can stand for a quantity and for a unit.

$m$ is mass but m is the metre; $W$ is work done but W is the watt.

Read the context, and write the unit out in full if there is any chance of confusion.

Units are written with a space between them and negative indices: m s⁻¹, kg m⁻³, J kg⁻¹ K⁻¹.

Without the space, "ms⁻¹" would mean "per millisecond".

Making a sensible estimate

An estimate question gives you no data, or only some of it.

You supply sensible values from everyday experience, calculate, and give an answer to one or two significant figures.

The aim is the right order of magnitude, the nearest power of ten, not a precise value.

Estimates appear in multiple-choice items, and an answer out by a factor of a thousand or more is a common slip: a sense of scale is also your best check on any calculated answer.

A good estimate follows four steps:

1. Break the quantity into things you can picture: a mass, a size, a speed, a time.

2. Choose round values for each, and write them down.

3. Calculate with the correct equation.

4. Check the answer against experience. A person with a mass of 7 kg, or a room 30 m high, means a value went wrong.

Table 1.1 gives benchmark values worth knowing.

They are rounded and typical, not exact.

Table 1.1

Quantity

Sensible value

mass of an adult

70 kg

mass of an apple (weight about 1 N)

0.1 kg

mass of a car

1000 kg

height of an adult

1.7 m

height of a room

3 m

walking speed

1.5 m s⁻¹

sprinter's top speed

10 m s⁻¹

car on a motorway

30 m s⁻¹

speed of sound in air

330 m s⁻¹ to 340 m s⁻¹

diameter of an atom

10⁻¹⁰ m

diameter of a nucleus

10⁻¹⁵ m to 10⁻¹⁴ m

wavelength of visible light

400 nm to 700 nm

density of water

1000 kg m⁻³

density of air

1.2 kg m⁻³

atmospheric pressure

1.0 × 10⁵ Pa

room temperature

293 K (20 °C)

power of an electric kettle

2 kW

current in a torch lamp

0.3 A

Figure 1.1 A scale of lengths from 10⁻¹⁵ m to 10⁸ m, one tick per power of ten, marking a nucleus, an atom, a light wavelength, the width of a hair, a person, Mount Everest and the Earth's diameter.

Figure 1.1 puts some of these lengths on one scale.

Each tick is a factor of ten, so the whole range from a nucleus to the Earth spans more than twenty powers of ten.

The idea behind it

An estimate is not a guess.

It is a proper calculation with honest, rounded inputs, and it works because physical quantities spread over enormous ranges.

Figure 1.1 runs across more than twenty powers of ten, and on that scale a factor of two moves you less than a third of one tick.

So an estimate that is twice too big is still a good estimate, while one that is a thousand times too big has gone badly wrong somewhere.

That is why examiners ask only for the right order of magnitude.

Estimates are also more reliable than you might fear, because their errors tend to cancel.

Suppose you multiply three rough values, and each one could be anything from half to twice the true value.

If all three were wrong in the same direction the answer would be out by a factor of 8, but that is unlikely: usually one guess comes out high and another low.

The figure shows the result of trying this a great many times with random guesses.

About two thirds of the answers land within a factor of 2 of the truth, and only about one in twenty-five is out by more than a factor of 4.

Problems of this kind are often called Fermi problems, after the physicist Enrico Fermi, who was famous for them.

Bar chart from a simulation of 200 000 estimates, each the product of three inputs guessed anywhere between half and twice their true value: 67% of answers are within a factor of 2 of the true answer, 88% within a factor of 3 and 96% within a factor of 4; none can be out by more than a factor of 8

When you cannot settle on one value, bracket it.

If you are confident a quantity is more than 10 and less than 1000, take 100: the value halfway between on a powers-of-ten scale, which is $\sqrt{10 \times 1000}$ .

For example, how many times does a heart beat in a lifetime?

Take about 70 beats a minute for about 80 years: $70 \times 60 \times 24 \times 365 \times 80 \approx 3 \times 10^{9}$ .

Neither input is exact, but the answer is certainly a few billion, not a few million or a few trillion.

The same sense of scale protects every calculation you do, not only estimates.

Read each answer back as a statement about the world: "a current of $3 \times 10^{4}$ A in a torch lamp" or "a pressure of 2 Pa under a person's feet" should stop you at once.

At A Level you will estimate things you can never see directly, such as the number of atoms in a coin or the speed of molecules in the air, and the habit you build here is what makes those answers trustworthy.

Key terms

Physical quantity: a quantity that can be measured, consisting of a numerical magnitude and a unit.

Order of magnitude: the power of ten nearest to a value.

Worked example 1: kinetic energy of a sprinter

Estimate the kinetic energy of a sprinter running at top speed.

1. C1: assume a mass $m = 70\ \text{kg}$ and a top speed $v = 10\ \text{m s}^{-1}$ (both written down).

2. C1: $E_\text{K} = \tfrac{1}{2}mv^2 = \tfrac{1}{2} \times 70 \times 10^2$

3. A1: $E_\text{K} = 3500\ \text{J}$ , about $4 \times 10^{3}$ J

Answer: about $4 \times 10^{3}$ J (one significant figure, because the inputs are estimates).

Worked example 2: pressure under a standing person

Estimate the pressure a person exerts on the floor when standing on both feet.

1. C1: weight $W = mg = 70 \times 9.81 = 687\ \text{N}$ , about 700 N.

2. C1: each foot is roughly 0.25 m long and 0.10 m wide, so the contact area is about $2 \times 0.25 \times 0.10 = 0.050\ \text{m}^2$ .

3. A1: $p = \dfrac{F}{A} = \dfrac{700}{0.050} = 1.4 \times 10^{4}\ \text{Pa}$

Answer: about $1 \times 10^{4}$ Pa.

The real contact area is smaller than the outline of the foot, so the true pressure is somewhat higher.

Worked example 3: the mass of air in a classroom

Estimate the mass of air in a classroom, and compare it with the mass of a student.

1. C1: assume the room is 8 m by 6 m by 3 m: volume $V = 8 \times 6 \times 3 = 144\ \text{m}^3$ .

2. C1: density of air $\rho = 1.2\ \text{kg m}^{-3}$ , so $m = \rho V = 1.2 \times 144$

3. A1: $m = 173\ \text{kg}$ , about 170 kg

4. B1: this is more than twice the mass of an adult (70 kg).

Answer: about 170 kg.

A surprising answer is fine if every assumption is sensible and stated.

Exam tip

In an estimate, write down every value you assume before you calculate, and give the answer to one or two significant figures.

The data sheet gives physical constants such as $g$ and $c$ , but never everyday values such as the mass of a person: those you must supply yourself.

In Paper 1, an estimate item is answered by rejecting the options that are wrong by a power of ten.

Common mistakes

Wild orders of magnitude. An adult of 7 kg or 700 kg, a speed of sound of 3000 m s⁻¹, a nucleus as big as an atom. Learn the benchmark values in Table 1.1 and use them to sense-check every answer, not only estimates.

Leaving the unit off a final answer, or giving the right number with the wrong unit (an adult of 70 g). Both lose the answer mark.

Quoting an estimate to four significant figures, as if the guessed inputs were precise.

Writing m s⁻¹ as "ms⁻¹" (per millisecond), or mixing up a quantity symbol and a unit symbol.

Check your understanding

1. Which value is the best estimate of the volume of an adult person? AO2 · L2 [1]

A 0.007 m³
B 0.07 m³
C 0.7 m³
D 7 m³

2. Which row gives sensible estimates of all three quantities? AO1 · L1 [1]

Table 1.2

mass of a family car

speed of sound in air

diameter of an atom

100 kg

340 m s⁻¹

10⁻¹⁰ m

1000 kg

340 m s⁻¹

10⁻¹⁰ m

1000 kg

3000 m s⁻¹

10⁻¹⁰ m

1000 kg

340 m s⁻¹

10⁻¹⁵ m

3. Use Figure 1.1. Approximately how many times larger is the diameter of an atom than the diameter of a nucleus? AO2 · L2 [1]

A 10²
B 10³
C 10⁵
D 10¹⁰

4. A student runs up a flight of stairs, from one floor to the next, in a few seconds. Which value is the best estimate of the useful power output of the student? AO2 · L3 [1]

A 7 W
B 70 W
C 700 W
D 7000 W

5. A student writes "the period of the pendulum is 1.6". State what is missing, and explain why it matters. AO1 · L1 [2]

6. Use Figure 1.1 to estimate the number of orders of magnitude between the diameter of an atom and the diameter of the Earth. AO2 · L2 [1]

7. Estimate the kinetic energy of a car travelling on a motorway. State the values you assume. AO2 · L2 [2]

8. A student estimates the density of the human body using the values in Table 1.3, treating the body as a rectangular block. AO2 · L3 [3]

Table 1.3

Assumed quantity

Value

mass

70 kg

height

1.7 m

width

0.30 m

front-to-back depth

0.020 m

(a) Calculate the density the student obtains.
(b) Identify the unrealistic assumption, explain how you know, and give a better estimate of the density.

Answers

1. B. An adult's mass is about 70 kg and the body's density is close to that of water, 1000 kg m⁻³, so $V = m/\rho = 70/1000 = 0.07$ m³. A is ten times too small (a 7 kg person); C and D are ten and a hundred times too big. [1]

2. B. A car is about 1000 kg, sound travels at about 340 m s⁻¹ in air and an atom is about 10⁻¹⁰ m across. A gives the mass of a large adult, C a speed of sound in a solid, and D the size of a nucleus. [1]

3. C. The atom sits at about 10⁻¹⁰ m and the nucleus at about 10⁻¹⁵ m: $10^{-10}/10^{-15} = 10^{5}$ . A and B under-count the ticks; D counts from the atom to the 10⁰ m mark. [1]

4. C. About 70 kg is lifted through a storey of 3 m in about 3 s: $P = mgh/t = 70 \times 9.81 \times 3 / 3 \approx 700$ W. A and B are far too small (B is about the power of a bright lamp); D would mean climbing a storey in a fraction of a second. [1]

5. B1: the unit (seconds) is missing [1].
B1: without it the number has no meaning: 1.6 s, 1.6 ms and 1.6 min are very different times [1].

6. B1: atom about 10⁻¹⁰ m, Earth about 10⁷ m, so about 17 orders of magnitude (a factor of about 10¹⁷) [1].

7. C1: assume $m = 1000$ kg and $v = 30$ m s⁻¹ [1].
A1: $E_\text{K} = \tfrac{1}{2} \times 1000 \times 30^2 = 4.5 \times 10^{5}$ J, about $5 \times 10^{5}$ J [1].

8. (a) C1: $V = 1.7 \times 0.30 \times 0.020 = 0.0102$ m³ [1]; A1: $\rho = 70/0.0102 = 6.9 \times 10^{3}$ kg m⁻³ [1].
(b) B1: the depth of 0.020 m (2 cm) is unrealistic: the density comes out about seven times that of water, but a body nearly floats, so its density must be close to 1000 kg m⁻³. With a depth of about 0.20 m, $V = 0.102$ m³ and $\rho \approx 700$ kg m⁻³, the right order of magnitude [1].

Physical quantities and estimates

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