Physics (9630) | Estimation of physical quantities

The Skill of Sense-Checking: Estimation of Physical Quantities (Syllabus 3.1.3)

Welcome to one of the most practical and exciting skills in physics: Estimation!
In this short but crucial chapter, you aren't aiming for the perfect, measured answer. Instead, you are learning how to use your knowledge of physics and common sense to find an approximate value for a quantity.

Why is this important? Being able to estimate allows you to:

• Quickly check if a calculator answer is realistic (i.e., spotting when you've accidentally calculated the mass of a pencil as 50 kg!).
• Solve complex problems where you don't have all the exact numbers (like estimating the total mass of air in your school hall).

This skill is all about determining the Order of Magnitude (OOM). Let's dive in!

Section 1: Understanding Order of Magnitude (OOM)

What is an Order of Magnitude?

The Order of Magnitude of a physical quantity is simply the power of 10 closest to its actual value.
It tells you the general scale or size of the quantity.

We typically write any number, \(N\), in standard form as: $$N = a \times 10^b$$ Where \(1 \le a < 10\). The order of magnitude is determined by the exponent \(b\).

The Crucial OOM Rule of Thumb

To find the closest power of 10, you must decide whether to round the pre-factor (\(a\)) up or down.

If \(a\) is less than \(\sqrt{10}\) (which is approximately 3.16), you round the OOM down to \(10^b\).
If \(a\) is equal to or greater than 3.16, you round the OOM up to the next power, \(10^{b+1}\).

Why 3.16? This ensures that the chosen power of 10 is mathematically the closest possible value. It's the midpoint between \(10^b\) (which is \(1 \times 10^b\)) and \(10^{b+1}\) (which is \(10 \times 10^b\)).

Example 1: Mass of a small child

A small child might have a mass of 20 kg.
1. Write in standard form: \(2.0 \times 10^1\) kg.
2. Check the pre-factor \(a = 2.0\). Since \(2.0 < 3.16\), we keep the exponent.
3. OOM: \(10^1\) kg.

Example 2: Mass of a textbook

A textbook might have a mass of 1.5 kg, which is \(1.5 \times 10^0\) kg.
1. Write in standard form: \(1.5 \times 10^0\) kg.
2. Check the pre-factor \(a = 1.5\). Since \(1.5 < 3.16\), we keep the exponent.
3. OOM: \(10^0\) kg (or 1 kg).

Example 3: Speed of a fast runner

A runner might run at 10 m/s. This is \(1.0 \times 10^1\) m/s.
OOM: \(10^1\) m/s.

Example 4: Distance across a city

The distance might be 50,000 m (50 km). This is \(5.0 \times 10^4\) m.
1. Write in standard form: \(5.0 \times 10^4\) m.
2. Check the pre-factor \(a = 5.0\). Since \(5.0 > 3.16\), we round the OOM up.
3. OOM: \(10^5\) m.

Section 2: Estimating Single Quantities (The Physicist's Intuition)

Estimation often relies on knowing or guessing typical values for common quantities. Here are some essential OOM values you should have a feel for:

Key OOM Benchmarks to Memorize:

• Height of an average adult human: Around 1.7 m. OOM: \(10^0\) m.
• Mass of an average adult human: Around 70 kg. OOM: \(10^2\) kg (since \(7.0 > 3.16\), we round up from \(10^1\)).
• Time taken to walk 1 km: Around 1000 s (15-20 minutes). OOM: \(10^3\) s.
• Speed of sound in air: Around 340 m/s. OOM: \(10^2\) m/s (since \(3.4 > 3.16\), we round up from \(10^2\) to \(10^3\)? Wait! \(3.4 \times 10^2\). \(3.4 > 3.16\), so the OOM is \(10^3\) m/s). Let's re-examine: 340 is closer to 100 than 1000. This highlights a subtle difficulty! For simplicity in AS/A Level, unless the rounding is critical, often numbers 1-3 are taken as \(10^b\) and 3-10 are taken as \(10^{b+1}\).

  Stick to the \(\sqrt{10}\) rule: \(3.4 \times 10^2\). Since \(3.4 > 3.16\), we round up to \(10^3\) m/s. (Note: This shows how OOMs can sometimes feel counter-intuitive, but the rule is consistent.)

• Volume of a coffee mug: Around 300 ml or \(3 \times 10^{-4}\) m³. OOM: \(10^{-4}\) m³.

Tip for struggling students: When faced with an object, always compare it to a reference point you know well. If they ask for the mass of a pencil, think: Is it closer to 0.01 kg (10 grams) or 1 kg? It's definitely closer to \(10^{-2}\) kg.

Estimating Dimensions (Length, Area, Volume)

If you need to estimate the volume of an unfamiliar room, don't panic! Break it down into fundamental lengths.

Example: Estimate the volume of a classroom.

1. Height (h): Maybe 3 metres. OOM: \(10^0\) m.
2. Width (w): Maybe 5 metres. OOM: \(10^1\) m.
3. Length (l): Maybe 8 metres. OOM: \(10^1\) m.
4. Volume V = lwh: \(5 \times 8 \times 3 = 120\) m³.

Now find the OOM of the result: \(120 = 1.2 \times 10^2\) m³.
Since \(1.2 < 3.16\), the OOM of the classroom volume is \(\mathbf{10^2}\) m³.

Did you know? Estimation is frequently used in engineering and astronomy. When scientists predict the path of a comet, they first use OOM estimates to ensure their complex numerical models are set up correctly!

Section 3: Estimating Derived Quantities

The specification requires you to use your initial estimates (OOMs) and combine them using physics equations to produce further derived estimates, also to the nearest order of magnitude.

The great thing about OOM calculation is that you often don't have to worry about the specific numerical coefficients (like 1/2 or \(\pi\)) unless they drastically change the overall result (which they usually don't in basic OOM estimates).

The Process for Derived Estimates

When calculating a derived quantity (like Force, Energy, or Power), follow these steps:

Step 1: Write down the relevant physics formula. (e.g., \(E_k = \frac{1}{2} m v^2\))
Step 2: Estimate the OOM for each input quantity. (e.g., OOM of \(m\), OOM of \(v\)).
Step 3: Substitute only the OOMs into the formula.
Step 4: Combine the powers of 10. (Remember: multiplying powers means adding the exponents.)
Step 5: Apply the 3.16 rule to the final result (if necessary).

Example: Estimate the Kinetic Energy (\(E_k\)) of a moving car.

Step 1: Formula: \(E_k = \frac{1}{2} m v^2\)

Step 2: Estimate Inputs (OOM):

1. Mass (\(m\)) of a typical car: 1500 kg. Standard form: \(1.5 \times 10^3\) kg. OOM: \(10^3\) kg.
2. Speed (\(v\)) of a car on a road: 20 m/s. Standard form: \(2.0 \times 10^1\) m/s. OOM: \(10^1\) m/s.

Step 3 & 4: Substitute and Calculate OOM:

We can ignore the constant \(\frac{1}{2}\) because it's less than 3.16.
$$E_k \approx (OOM \, of \, m) \times (OOM \, of \, v)^2$$ $$E_k \approx (10^3) \times (10^1)^2$$ $$E_k \approx 10^3 \times 10^2$$ $$E_k \approx 10^{3+2}$$ $$E_k \approx \mathbf{10^5} \text{ J}$$

Verification (Optional but good practice): If we calculate the actual value: \(E_k = 0.5 \times 1500 \times (20)^2 = 300,000\) J.
\(300,000 = 3.0 \times 10^5\) J.
Since \(3.0 < 3.16\), the OOM is indeed \(\mathbf{10^5}\) J. Our simple OOM calculation was correct!

Common Mistakes to Avoid

1. Forgetting the 3.16 Rule: If you estimate the mass of an aircraft to be 5,000 kg, and incorrectly use \(10^3\) instead of \(10^4\) (since 5 is greater than 3.16), your entire final OOM will be off by a factor of 10.
2. Ignoring Powers (Squaring/Cubing): In the kinetic energy example, it is crucial to remember that the velocity OOM (\(10^1\)) must be squared to become \(10^2\).
3. Mixing Units: Always ensure that your estimates are in the correct fundamental SI units (metres, kilograms, seconds) before finding the order of magnitude. For example, estimate mass in kilograms, not grams.

Section 4: Estimating Constants and Prefixes

When dealing with derived estimates, sometimes you will encounter fundamental constants (like the speed of light, \(c\), or the acceleration due to gravity, \(g\)). You should know the OOM for these.

Essential Constants and Their OOMs

• Acceleration due to Gravity (\(g\)): Approximately \(9.8\) m/s². Standard form: \(9.8 \times 10^0\). OOM: \(\mathbf{10^1}\) m/s² (since \(9.8 > 3.16\)).
• Speed of Light (\(c\)): Approximately \(3.0 \times 10^8\) m/s. Standard form: \(3.0 \times 10^8\). OOM: \(\mathbf{10^8}\) m/s (since \(3.0 < 3.16\)).
• Density of Water: Approximately \(1000\) kg/m³. OOM: \(\mathbf{10^3}\) kg/m³.

Using Prefixes in OOM:

Remember the SI prefixes you learned in section 3.1.1. They are already powers of 10, making OOM conversion simple:

• 1 microsecond (\(\mu\text{s}\)) = \(10^{-6}\) s. OOM is \(\mathbf{10^{-6}}\).
• 500 megawatts (\(\text{MW}\)) = \(5.0 \times 10^2 \times 10^6\) W = \(5.0 \times 10^8\) W. OOM is \(\mathbf{10^9}\) W (since \(5.0 > 3.16\)).

Don't worry if this seems tricky at first! This is a skill built through practice. Every time you calculate a number in physics, take a moment to estimate the answer's OOM first—it’s excellent practice for the exams!