Physics 9702/12 — May/June 2022

Background Concept

A “reasonable estimate” question is about order of magnitude: choosing typical everyday values and checking whether a suggested numerical value is in the right ballpark.

For lifting an object vertically, the work done against gravity equals the gain in gravitational potential energy:

$$W = \Delta E_p = mgh$$

where $m$ is the mass, $g \approx 9.8\ \text{m s}^{-2}$ (often $\approx 10\ \text{m s}^{-2}$ for estimates), and $h$ is the vertical height raised.

Understanding the Question

You are given four proposed estimates for different physical quantities (mass, volume, work, power). Only one is realistic for the stated object/situation.

We can quickly test each option using typical sizes:

• grain of sand (tiny mass),
• tennis ball (few cm radius),
• apple lifted by some tens of cm,
• household light bulb (tens of watts).

Approach

Do quick “back-of-the-envelope” calculations:

1. For the work done lifting an apple, use $mgh$ with sensible values.
2. For the others, compare to known typical values (or do a very quick geometric/scale estimate).
3. Pick the one that matches reality to within about a factor of 10.

Step-by-Step Reasoning

Option C (work done lifting an apple):
An apple typically has mass $m \sim 0.1\ \text{kg}$.
The height from waist to head is roughly $h \sim 0.5\ \text{m}$.

$$W \approx mgh \approx 0.1 \times 10 \times 0.5 = 0.5\ \text{J}$$

So the work is of order $1\ \text{J} = 10^{0}\ \text{J}$, which matches option C.

Why the other options are unreasonable (quick checks):

• A: $10^{-3}\ \text{kg} = 1\ \text{g}$, far too massive for a grain of sand (a grain is typically much less than a milligram, i.e. around $10^{-7}$ to $10^{-6}\ \text{kg}$).
• B: $10^{-2}\ \text{m}^3 = 0.01\ \text{m}^3 = 10\ \text{L}$. A tennis ball is only a few cm across; its volume is around $10^{-4}\ \text{m}^3$, not $10^{-2}\ \text{m}^3$.
• D: $10^{4}\ \text{W} = 10\ \text{kW}$. Household light bulbs are typically $10$–$100\ \text{W}$ (even powerful ones are not kilowatts).

Therefore the only reasonable estimate is C.

Key Takeaways

• Use order-of-magnitude thinking: typical values + simple physics.
• For lifting vertically: $W = mgh$.
• Quickly reject options that are many orders of magnitude off everyday experience.

Common Mistakes

• Mixing up prefixes and powers of ten (e.g. thinking $10^{-3}\ \text{kg}$ is “tiny” without realising it is $1\ \text{g}$).
• Treating the height change as $\sim 1\ \text{m}$ or $\sim 0.01\ \text{m}$ without a sensible physical picture.
• Confusing power (W) with energy (J) and judging them on the wrong scale.

Things to Be Careful About

• For estimates, $g=10\ \text{m s}^{-2}$ is acceptable; don’t overcomplicate.
• Convert to familiar units to sanity-check: $0.01\ \text{m}^3 = 10\ \text{L}$, $10^{4}\ \text{W} = 10\ \text{kW}$.
• “Reasonable” usually means within about a factor of $10$ (same order of magnitude), not exact.