make reasonable estimates of physical quantities included within the syllabus

Physical Quantities

In physics a physical quantity is a property of a system that can be measured and expressed as a number together with a unit. The ability to make reasonable estimates of the magnitude of a quantity is a valuable skill for A‑Level examinations and for scientific thinking.

1. Base and Derived Quantities

The International System of Units (SI) defines seven base quantities. All other quantities are derived from these.

Base Quantity	Symbol	SI Unit	Unit Symbol
Length	$l$	metre	m
Mass	$m$	kilogram	kg
Time	$t$	second	s
Electric current	$I$	ampere	A
Thermodynamic temperature	$T$	kelvin	K
Amount of substance	$n$	mole	mol
Luminous intensity	$I_{\!v}$	candela	cd

Derived quantities are formed by combining base quantities. For example, speed $v$ is length divided by time ($v = l/t$) with unit metres per second (m s\(^{-1}\)).

2. SI Prefixes – Quick Order‑of‑Magnitude Scaling

Prefixes allow us to write very large or very small numbers compactly. The most common for A‑Level work are listed below.

Prefix	Symbol	Factor
kilo	k	10³
mega	M	10⁶
giga	G	10⁹
milli	m	10⁻³
micro	µ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²

3. Significant Figures and Uncertainty

• When a quantity is measured, only a limited number of digits are reliable – these are the significant figures.
• Rule of thumb for multiplication/division: the result should be reported with the same number of significant figures as the factor with the fewest.
• For addition/subtraction: keep the result to the same decimal place as the term with the least precise decimal place.
• Uncertainty can be expressed as an absolute value (e.g., $5.0 \pm 0.2\,$m) or a relative/percentage value.

4. Reasonable Estimates – Typical A‑Level Quantities

The table below lists common physical quantities that appear in the 9702 syllabus together with typical magnitudes that students should be able to recall or estimate.

Quantity	Symbol	Typical \cdot alue (SI)	Comments / Estimation Tips
Acceleration due to gravity (Earth)	$g$	9.8 m s\(^{-2}\)	Often approximated as $10\,$m s\(^{-2}\) for quick calculations.
Speed of light in vacuum	$c$	3.00 × 10⁸ m s\(^{-1}\)	Exact by definition; useful for order‑of‑magnitude checks.
Elementary charge	$e$	1.60 × 10⁻¹⁹ C	Recall as $1.6\times10^{-19}$ C.
Mass of a proton	$m_p$	1.67 × 10⁻²⁷ kg	Useful for nuclear‑physics estimates.
Planck’s constant	$h$	6.63 × 10⁻³⁴ J s	Rarely needed numerically, but good to know the order.
Permittivity of free space	$\varepsilon_0$	8.85 × 10⁻¹² F m\(^{-1}\)	Often appears in Coulomb’s law.
Permeability of free space	$\mu_0$	4π × 10⁻⁷ N A\(^{-2}\)	Useful for magnetic field calculations.
Typical laboratory voltage	$V$	1 V – 10 V	Battery cells are around 1.5 V; mains supply is 230 V (UK).
Typical current in a circuit	$I$	10⁻³ A – 10 A	Micro‑currents for sensors, amperes for power circuits.
Typical resistance of a copper wire (1 m, 1 mm²)	$R$	0.017 Ω	Use $ρ_{\text{Cu}}≈1.7×10^{-8}$ Ω m and $R=ρL/A$.
Gravitational field strength at Earth's surface	$g$	9.8 N kg\(^{-1}\)	Same numeric value as acceleration due to gravity.
Typical wavelength of visible light	$\lambda$	400 nm – 700 nm	Useful for diffraction and interference estimates.

5. Strategies for Making Reasonable Estimates

1. Fermi‑type reasoning: Break a problem into a product of quantities you can guess, then multiply. Example: estimating the number of piano tuners in a city.
2. Dimensional analysis: Use the dimensions of the required quantity to construct a formula from known constants. Example: estimating the period of a simple pendulum $T\approx2\pi\sqrt{L/g}$.
3. Order‑of‑magnitude comparison: Compare the unknown quantity with a familiar reference value (e.g., “a car travels about $10^2$ m in a few seconds”).
4. Use of standard values: Keep a short “cheat sheet” of the typical values listed above; this speeds up calculations and reduces transcription errors.
5. Check plausibility: After obtaining a result, ask whether the magnitude makes sense (e.g., a speed greater than $c$ is impossible in classical contexts).

6. Example: Estimating the Energy Stored in a 1 m Long, 1 mm² Copper Wire Carrying 5 A

Step‑by‑step estimation using the table values:

1. Resistance $R = \rho L/A$. Use $\rho_{\text{Cu}}≈1.7×10^{-8}$ Ω m, $L=1$ m, $A=1\,$mm² = $1×10^{-6}$ m².
   $$R≈\frac{1.7×10^{-8}\,\text{Ω m}\times1\,\text{m}}{1×10^{-6}\,\text{m}^2}=1.7×10^{-2}\,\text{Ω}$$
2. Power dissipated $P = I^2R = (5\text{ A})^2 × 1.7×10^{-2}\,\text{Ω} ≈ 0.425\,$W.
3. Energy released in 10 s: $E = Pt = 0.425\,$W × 10 s = $4.3\,$J.

The estimate shows that a short copper conductor carrying a few amperes stores only a few joules of heat over a few seconds – a useful sanity check for experimental design.

7. Practice Problems

• Estimate the force exerted by atmospheric pressure on a 10 cm × 10 cm window. (Use $P_{\text{atm}}≈1.0×10^5$ Pa.)
• Using dimensional analysis, find an expression for the terminal speed of a small sphere falling through a viscous fluid (Stokes’ law). Then estimate the speed for a 1 mm radius steel ball in water.
• How many photons per second are emitted by a 60 W incandescent bulb if the average photon wavelength is 600 nm?