Physics 9702/13 — October/November 2013

Background Concept

A “realistic estimate” question tests whether you have a feel for typical sizes (orders of magnitude) of common physical quantities. You do not need exact values; you check whether each option is roughly in the right range.

Useful reference ideas:

• Kinetic energy:

$$E_k = \frac{1}{2}mv^2$$

• Temperature: the kelvin scale is related to Celsius by

$$T / \text{K} = \theta / ^\circ\text{C} + 273$$

• Volume: $1\ \text{m}^3 = 1000\ \text{L}$, so $0.03\ \text{m}^3 = 30\ \text{L}$.

Understanding the Question

You are given four statements (A–D), each claiming a typical value for a real-world situation (bus kinetic energy, light power, oven temperature, tyre air volume). Only one is intended to be a sensible everyday magnitude.

Approach

For each option:

1. Recall or estimate a typical value (mass, speed, typical appliance power, typical oven temperature, typical tyre volume).
2. If needed, do a quick calculation (especially for kinetic energy) or unit conversion (kelvin to Celsius, cubic metres to litres).
3. Decide whether the stated value is about right or clearly wrong by a large factor.

Step-by-Step Reasoning

A: Kinetic energy of a bus
A bus has a large mass, typically around $10\ \text{tonnes} = 1\times 10^{4}\ \text{kg}$. Expressway speeds are of order $20$ to $30\ \text{m s}^{-1}$.

Using $v = 25\ \text{m s}^{-1}$:

$$E_k = \frac{1}{2}mv^2 \approx \frac{1}{2}(1\times 10^{4})(25^2) = 0.5\times 10^{4}\times 625 \approx 3\times 10^{6}\ \text{J}$$

So the given $30000\ \text{J} = 3\times 10^{4}\ \text{J}$ is about $100$ times too small. Not realistic.

B: Power of a domestic light
A typical domestic light bulb is usually tens of watts (e.g. older incandescent bulbs about $40$–$100\ \text{W}$; modern LEDs are much lower). $300\ \text{W}$ is unusually high for a normal “domestic light”. So it is not the best realistic estimate.

C: Temperature of a hot oven
Convert to Celsius:

$$300\ \text{K} \approx 300 - 273 = 27\ ^\circ\text{C}$$

That is room temperature, not a hot oven (which is typically a few hundred degrees Celsius, e.g. $200\ ^\circ\text{C} \approx 473\ \text{K}$). Not realistic.

D: Volume of air in a car tyre
Convert to litres:

$$0.03\ \text{m}^3 = 0.03\times 1000\ \text{L} = 30\ \text{L}$$

A tyre contains air of order a few tens of litres, so $0.03\ \text{m}^3$ is plausible. This is the realistic estimate.

Key Takeaways

• Use order-of-magnitude reasoning to spot values that are wrong by factors of $10$ or more.
• For vehicles, kinetic energy is usually very large (MJ range), because $E_k \propto v^2$ and mass is large.
• Always do quick unit sense-checks (K to °C, m$^3$ to L).

Common Mistakes

• Treating $300\ \text{K}$ as “hot” without converting to Celsius.
• Underestimating vehicle kinetic energy by forgetting the large mass or the $v^2$ dependence.
• Not recognising that $0.03\ \text{m}^3$ corresponds to $30\ \text{L}$.

Things to Be Careful About

• Realistic estimate questions are about typical values; an uncommon special case (e.g. an unusually powerful lamp) is usually not what is intended.
• Keep track of powers of ten: $10^{4}\ \text{kg}$ and $v^2$ can quickly push energies into $10^{6}\ \text{J}$.
• Remember the fixed offset between kelvin and Celsius (about $273$).