1.1A Physics Toolkit
Mathematics and Physics — American System Physics Revision Notes & Topical Questions
10 min read 7 parts Lesson 3 of 5 1 quiz
Lesson summary
Use SI base and derived units and metric prefixes correctly.Speedspeed = distance ÷ time, v = d/t; distance in m, time in s, speed in m/s.Densitydensity = mass ÷ volume, ρ = m/V; kg/m3 (or g/cm3; 1 g/cm3 = 1000 kg/m3).Converting unitsmultiply by a conversion factor equal to 1, e.g.
- Area units square the conversion
- Multiplying powers of ten
- Dimensional check example
- Dimensional analysis
- The scientific method is cyclic
- A car travels at 90 km/h. Express this speed in m/s
Part 1 of 7Key ideas
1Speed
Speed speed = distance ÷ time, v = d/t; distance in m, time in s, speed in m/s.
2Density
Density density = mass ÷ volume, ρ = m/V; kg/m3 (or g/cm3; 1 g/cm3 = 1000 kg/m3).
3Converting units
Converting units multiply by a conversion factor equal to 1, e.g. 72 km/h × (1000 m/1 km) × (1 h/3600 s) = 20 m/s.
4
To convert km/h to m/s divide by 3.6; to convert m/s to km/h multiply by 3.6.
5Area units square the conversion
1 m2 = 104 cm2; volume units cube it: 1 m3 = 106 cm3 = 1000 L.
6Multiplying powers of ten
10a × 10b = 10a+b; dividing: 10a ÷ 10b = 10a−b.
7Dimensional check example
Dimensional check example d = vt → (m/s)(s) = m ✔.
8
Numbers that are counted or defined (e.g. 1 km = 1000 m exactly) have unlimited significant figures and do not limit the answer.
Part 2 of 7Learning objectives
1
Use SI base and derived units and metric prefixes correctly.
2
Convert units using conversion factors and dimensional analysis.
3
Write numbers in scientific notation and apply significant-figure rules.
4
Rearrange and solve equations symbolically before substituting numbers.
5
Distinguish between a model, a scientific law and a scientific theory.
Part 3 of 7Essential knowledge
Essential knowledge
Physics seeks a small number of rules that explain a huge range of phenomena, from falling apples to orbiting planets; these rules are usually written as equations.
Magnitude × unit
Every physical quantity is a magnitude × unit. A number with no unit (e.g. "the length is 3") is meaningless and loses marks.
The SI system has seven base units. For mechanics you need three: metre (m), kilogram (kg) and second (s). Other units, such as the newton and joule, are derived from these.
Always convert to base SI units before substituting into equations: grams to kilograms (÷1000), centimetres to metres (÷100), minutes to seconds (×60), hours to seconds (×3600).
Scientific notation makes very large and small numbers manageable, e.g. the speed of light 3.00 × 108 m/s, the mass of an electron 9.11 × 10−31 kg.
Dimensional analysis
Dimensional analysis units on both sides of a correct equation must match. If you calculate a speed and your units come out as m·s, the equation was rearranged wrongly.
Conversion factors are fractions equal to 1; arrange them so the unwanted unit cancels, e.g. 5.0 km × (1000 m / 1 km) = 5.0 × 103 m.
Significant figures reflect how precisely a value was measured. A calculated result can never be more precise than the least precise measurement used.
When rounding, look at the digit after the last one kept: 5 or more rounds up, less than 5 leaves it unchanged. Keep extra digits in intermediate steps and round only the final answer.
The scientific method is cyclic
The scientific method is cyclic observe, ask a question, form a testable hypothesis, experiment (changing one independent variable while controlling others), analyse data, and draw a conclusion that may lead to new questions.
Law
A law (e.g. the law of conservation of energy) describes what happens; a theory (e.g. the kinetic theory of matter) explains why. Theories are never "proven" — they are supported by evidence and can be revised.
Models (such as treating a car as a single point) simplify reality so problems can be solved; good models make predictions that match measurement.
Part 4 of 7Process & sequence
1
Read the problem and list every known quantity with its symbol and unit; identify the unknown.
2
Convert all values into SI base units using conversion factors (e.g. km → m, g → kg, min → s).
3
Choose the equation that links the knowns to the unknown and rearrange it algebraically before substituting numbers.
4
Carry out a dimensional analysis check: substitute the units alone and confirm they simplify to the unit of the unknown.
5
Substitute values, calculate, and round the answer to the correct number of significant figures (the fewest in the data for × and ÷).
6
State the answer with its unit, and judge whether its size is reasonable for the situation.
Converting units with dimensional analysiswrite these steps in this order; each step is a marking point.
1
Write the given quantity with its unit, e.g. 72 km/h.
2
Multiply by a conversion factor with the unwanted unit on the bottom: × (1000 m / 1 km).
3
Multiply by a second factor to replace the time unit: × (1 h / 3600 s).
4
Cancel matching units top and bottom, leaving m/s.
5
Calculate the number (72 × 1000 ÷ 3600 = 20) and write 20 m/s.
Part 5 of 7Worked examples
Worked examples
A car travels at 90 km/h. Express this speed in m/s.
- 90 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 90 000 m / 3600 s = 25 m/s.
- (Check: 90 ÷ 3.6 = 25.) Units km and h cancel, leaving m/s.
A metal block has mass 540 g and volume 200 cm3. Find its density in kg/m3.
- Convertm = 540 g = 0.540 kg; V = 200 cm3 = 200 × 10−6 m3 = 2.00 × 10−4 m3. ρ = m/V = 0.540 / 2.00 × 10−4 = 2.70 × 103 kg/m3 (3 s.f.; this is aluminium).What the symbols mean: ρ = density (kg m−3 or g cm−3) · m = mass (kg) · V = volume (m3)
- Check540/200 = 2.70 g/cm3 × 1000 = 2700 kg/m3.
Calculate (4.2 × 103 m) ÷ (1.5 × 10−2 s) and give the answer to correct significant figures.
- Divide the numbers4.2 / 1.5 = 2.8.
- Divide powers103 / 10−2 = 103−(−2) = 105.
- Answer = 2.8 × 105 m/s .
- (2 s.f., because both data values have 2 s.f.)
Part 6 of 7Summary
1
Physics uses mathematics as its language to describe and predict nature.
2
Every measurement needs a number and a unit; use SI base units in calculations.
3
Dimensional analysis checks equations and converts units by cancelling units like algebra.
4
Give answers to the correct number of significant figures — no more precise than the data.
5
Laws describe patterns; theories explain them.
Part 7 of 7Compare: Compare
| Prefix | Symbol | Multiplier | Example |
|---|---|---|---|
| giga | G | 109 | 1 GHz = 1 × 109 Hz |
| mega | M | 106 | 1 MJ = 1 × 106 J |
| kilo | k | 103 | 1 km = 1000 m |
| centi | c | 10−2 | 1 cm = 0.01 m |
| milli | m | 10−3 | 1 mm = 0.001 m |
| micro | μ | 10−6 | 1 μs = 1 × 10−6 s |
| nano | n | 10−9 | 1 nm = 1 × 10−9 m |
Capital M (mega) and lower-case m (milli) differ by a factor of 109 — always check the case.
| Rule | Example | Sig. figs |
|---|---|---|
| All non-zero digits count | 4.72 m | 3 |
| Zeros between non-zero digits count | 2.05 kg | 3 |
| Leading zeros never count | 0.0036 s | 2 |
| Trailing zeros after a decimal point count | 5.40 cm | 3 |
| Multiply/divide: answer has fewest sig. figs of the data | 2.5 × 3.14 = 7.9 | 2 |
| Add/subtract: answer has fewest decimal places of the data | 12.1 + 0.25 = 12.4 | 1 d.p. |
Key terms
Command terms
Command terms
Identify / Describe (FRQ)
A short, specific statement; describing requires relevant characteristics, not a single word.
Explain / Justify (FRQ)
Give the cause-and-effect reasoning or evidence; a restated fact does not score.
Calculate
Show setup, substitution and the answer with units.
Predict
State the expected outcome; add a reason only if asked to justify.
MCQ technique
Read every option; eliminate distractors that are true statements but do not answer the question; watch for NOT/EXCEPT.
Exam technique
Keywords the mark scheme looks for in this lesson
Keywords the mark scheme looks for in this lesson
Physics · Physical quantity · SI (International System of Units) · Base unit · Derived unit · Metric prefix · Scientific notation · Dimensional analysis · Conversion factor · Significant figures · Model · Scientific law · Scientific theory
Learn each definition in the Key words tab exactly; a missing keyword in a definition usually loses the mark.
How to get full marks (examiner guidance)
1
Answer exactly what is asked; extra correct material does not earn credit, and contradictory statements can cancel credit (AP FRQ).
2
Use precise vocabulary, never “it/stuff/things”.
3
Show units and working in calculations.
4
In MCQs, predict the answer before reading the options, then eliminate.
Words to avoid
1
the answer is 25the speed is 25 m/s (always include the unit).
2
2.6666667 m/s when data were given to 2 s.f.2.7 m/s.
3
a theory is just a guessa theory is a well-tested explanation supported by much evidence.
4
5 Km or 5 KG5 km, 5 kg (prefix and unit symbols have fixed cases).
5
a law explains why something happensa law describes a pattern; a theory explains it.
Instant feedbackCheck yourself
Q1. Which is an SI base unit?
- A. newton
- B. joule
- C. kilogram
- D. watt
Show answer
C. The kilogram is one of the seven base units; the others listed are derived units.
Q2. How many significant figures are in 0.004050 m?
- A. 3
- B. 4
- C. 6
- D. 7
Show answer
B. Leading zeros do not count; 4, 0, 5 and the trailing 0 after the decimal point count: 4 s.f.
Q3. What is 36 km/h in m/s?
- A. 3.6 m/s
- B. 10 m/s
- C. 36 m/s
- D. 130 m/s
Show answer
B. 36 ÷ 3.6 = 10 m/s.
Q4. An equation gives a result with units m·s. The quantity is supposed to be a speed. What does this show?
1
A. The equation is correct
2
B. The answer needs rounding
3
C. The equation has been rearranged incorrectly
4
D. The unit should be km
Show answer
C. Speed must come out in m/s; mismatched units reveal an algebra error.
Q5. 5.0 × 104 × 3.0 × 10−6 equals
1
A. 1.5 × 10−1
2
B. 1.5 × 10−2
3
C. 15 × 10−24
4
D. 8.0 × 10−2
Show answer
A. 5.0 × 3.0 = 15; 104 × 10−6 = 10−2; 15 × 10−2 = 1.5 × 10−1.
Q6. Which statement best describes a scientific theory?
1
A. A guess that has not been tested
2
B. An explanation supported by much evidence that can be revised
3
C. A rule that can never change
4
D. A single experimental result
Show answer
B. Theories explain observations, are well supported, but remain open to revision.
Marked practiceQuizzes for this lesson
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