IGCSE Physics · Unit 1: Motion, Forces and Energy · Core + Extended

Measurement of Physical Quantities CoreExtended adds Scalars and Vectors

Every physics measurement you will ever take — a length, a time, a force — is only useful if it’s taken correctly and expressed with the right precision. This note covers how to measure length, volume and time properly, how to get a reliable value from something too small or too fast to measure directly, and (Extended only) the crucial distinction between scalar and vector quantities, including how to combine two vectors at right angles.

Why this matters Almost every practical exam question — and every Paper 5/6 experiment — starts with “measure this accurately.” Examiners are very particular about reading instruments correctly (to half a division), avoiding parallax error, and knowing when to measure multiples rather than a single value. Get this foundation solid and it pays off across the whole syllabus.

1. Measuring Length and Volume

Length is measured using a ruler (metre rule or 30 cm ruler, graduated in mm). To avoid parallax error, your eye must look straight down (perpendicular) at the scale — viewing at an angle makes the reading appear shifted. Always check for a zero error too: if the ruler’s zero mark is worn or the end is damaged, start your measurement from a clean, clearly marked point instead of the very end.

Volume of a liquid is measured using a measuring cylinder. Liquids form a curved surface called a meniscus; the reading should be taken at eye level, looking at the bottom of the meniscus for most liquids (water curves downward in glass).

Worked example — reading a ruler correctly A pencil is placed against a ruler. The left end lines up with the 2.0 cm mark and the right end lines up with the 14.6 cm mark.
Length = 14.6 − 2.0 = 12.6 cm
Notice we take a difference between two readings rather than assuming the object starts exactly at zero — this removes any small zero error.
Worked example — reading a measuring cylinder A student pours water into a measuring cylinder. The bottom of the meniscus sits exactly on the 37 cm³ mark.
Volume = 37 cm³
If the scale is marked every 1 cm³, and the meniscus sits between two lines, estimate to the nearest half division — e.g. 37.5 cm³.
Try it — Length and Volume
  1. A student measures a metal rod. The left end is at the 3.2 cm mark, the right end is at the 19.5 cm mark. What is the length of the rod?
  2. Why should you always view a measuring cylinder’s scale at eye level rather than from above?
  3. A ruler’s zero end is chipped and worn. Explain how a student can still get an accurate length measurement.
Show answers
  1. 19.5 − 3.2 = 16.3 cm
  2. Viewing from above or below introduces parallax error — the position of the meniscus against the scale appears to shift depending on the angle of viewing, giving a false reading.
  3. Start the measurement from a clean, undamaged point further along the scale (not the worn zero), and subtract that starting value from the final reading, rather than assuming the object begins exactly at 0.

2. Measuring Time

Time intervals are measured using clocks and digital timers (stopwatches). For a stopwatch, human reaction time (typically around 0.2–0.3 s) introduces a small random error each time you start and stop it — this is one reason why timing is often repeated and averaged, or why multiple events are timed together rather than just one.

Worked example — timing with a stopwatch A student times a ball rolling down a ramp three times: 2.1 s, 2.3 s, 2.2 s.
Average time = (2.1 + 2.3 + 2.2) ÷ 3 = 2.2 s
Repeating and averaging reduces the effect of random reaction-time errors on the final result.

3. Determining an Average Value for a Small Distance or Short Time (Measuring Multiples)

Some quantities are too small or too fast to measure accurately in a single reading — the thickness of one sheet of paper, or the time for one swing of a pendulum. The solution is to measure a convenient multiple of the quantity and divide.

Worked example — thickness of a single sheet of paper A student measures the thickness of a stack of 250 sheets of paper using a ruler: the stack is 22.5 mm thick.
Thickness of one sheet = 22.5 mm ÷ 250 = 0.09 mm
Measuring a single sheet directly would be far too imprecise on a mm-scale ruler — the stack method turns a tiny, hard-to-measure quantity into an easily measurable one.
Worked example — period of a pendulum A student sets up a simple pendulum and uses a stopwatch to time 20 complete oscillations, recording 36.4 s.
Period, T = 36.4 s ÷ 20 = 1.82 s
Timing just one swing would make reaction-time error a huge fraction of the result; timing 20 swings spreads that error out and makes it negligible in comparison.
Try it — Measuring Multiples
  1. A pile of 40 identical washers has a total mass of 68 g. What is the mass of one washer?
  2. A student times 25 oscillations of a pendulum and records 42.5 s. Calculate the period of one oscillation.
  3. Explain why timing 20 oscillations gives a more reliable value for the period than timing 1 oscillation.
Show answers
  1. 68 g ÷ 40 = 1.7 g
  2. 42.5 s ÷ 25 = 1.7 s
  3. Reaction time when starting and stopping the stopwatch is roughly constant (a fixed error of a fraction of a second) regardless of how many oscillations are timed. Spread across 20 oscillations, that fixed error becomes a much smaller fraction of the total, and dividing by 20 shrinks its effect on the final period — giving a more precise and reliable value.

4. Scalars and Vectors Extended only

Every physical quantity falls into one of two categories. A scalar quantity has magnitude (size) only — a number and a unit is all you need to describe it fully. A vector quantity has both magnitude and direction — you cannot fully describe it without stating which way it points.

ScalarsVectors
distanceforce
speedweight
timevelocity
massacceleration
energymomentum
temperatureelectric field strength
—gravitational field strength
Why this distinction matters “Speed” and “velocity” sound interchangeable in everyday language, but in physics they are not. A car travelling at 60 km/h is a speed (scalar) — but “60 km/h due north” is a velocity (vector). This matters enormously once you start adding quantities together: two speeds of 5 m/s don’t simply add to 10 m/s unless they’re in the same direction, but two velocities in opposite directions can cancel out completely.
Try it — Classifying Quantities
  1. State whether each of the following is a scalar or a vector: mass, acceleration, temperature, momentum, distance, weight.
  2. A student says “the object has a velocity of 10 m/s.” Explain what extra piece of information is missing for this to be a complete description of a velocity.
Show answers
  1. mass — scalar; acceleration — vector; temperature — scalar; momentum — vector; distance — scalar; weight — vector.
  2. A direction is missing — e.g. “10 m/s to the east.” Without a direction, only the speed (the scalar part) has been given.

5. The Resultant of Two Vectors at Right Angles Extended only

When two vectors act at right angles to each other (limited to forces or velocities in this syllabus), they can be combined into a single equivalent vector called the resultant. This can be found either graphically (scale drawing) or by calculation, using Pythagoras’ theorem for the magnitude and trigonometry for the direction.

resultant magnitude: R = √(A² + B²)
direction: tan θ = opposite / adjacent
Worked example — two forces at right angles A box is pulled by a force of 3 N to the east and a force of 4 N to the north, acting at the same point.
Magnitude: R = √(3² + 4²) = √(9 + 16) = √25 = 5 N
Direction: tan θ = 4 / 3 → θ = tan⁻¹(1.33) = 53.1° north of east
Answer: 5 N at 53.1° north of east
Worked example — velocities at right angles A boat can travel at 4 m/s in still water and heads directly across a river. The river current flows at 3 m/s at right angles to the boat’s heading.
Resultant speed: R = √(4² + 3²) = √(16 + 9) = √25 = 5 m/s
Direction: tan θ = 3 / 4 → θ = tan⁻¹(0.75) = 36.9° from the boat’s original heading, in the direction the current pushes it
Answer: 5 m/s, at 36.9° off the boat’s heading — this is the boat’s actual velocity relative to the riverbank.
Worked example — the graphical method The same 3 N (east) and 4 N (north) forces can also be combined by scale drawing:
1. Choose a scale, e.g. 1 cm = 1 N.
2. Draw the 3 N force as a 3 cm arrow pointing east.
3. From the tip of that arrow, draw the 4 N force as a 4 cm arrow pointing north.
4. Draw a straight line from the start point to the tip of the second arrow — this is the resultant.
5. Measure its length (5 cm → 5 N) and its angle with a protractor (53° from east).
Calculation and scale drawing should give the same answer — use whichever method the question asks for.
Try it — Resultant of Two Vectors
  1. Two forces act on a point object: 6 N to the east and 8 N to the north. Calculate the magnitude and direction of the resultant force.
  2. A swimmer swims at 1.2 m/s directly across a river. The river flows at 0.5 m/s at right angles to the swimmer’s direction. Calculate the swimmer’s resultant velocity.
  3. An aircraft flies due north at 200 km/h through air that is itself moving (a crosswind) at 40 km/h due east. Calculate the aircraft’s resultant speed and direction relative to the ground.
Show answers
  1. R = √(6² + 8²) = √(36+64) = √100 = 10 N. tan θ = 8/6 → θ = 53.1° north of east. Resultant = 10 N at 53.1° north of east.
  2. R = √(1.2² + 0.5²) = √(1.44+0.25) = √1.69 = 1.3 m/s. tan θ = 0.5/1.2 → θ = 22.6° from the swimmer’s original direction.
  3. R = √(200² + 40²) = √(40000+1600) = √41600 ≈ 203.96 km/h. tan θ = 40/200 → θ = tan⁻¹(0.2) = 11.3° east of north. Resultant ≈ 204 km/h at 11.3° east of north.
Common mistakes to watch for
  • Reading a ruler or measuring cylinder from an angle instead of straight-on — always check for parallax error.
  • Assuming an object starts exactly at the “0” mark on a ruler instead of subtracting a start reading from an end reading.
  • Trying to measure a single tiny or fast quantity directly instead of measuring a convenient multiple and dividing.
  • Forgetting that dividing by the number of oscillations/items only works if all the items or oscillations are identical (or assumed identical).
  • Calling speed and velocity the same thing — a scalar answer needs no direction, but a vector answer is incomplete without one.
  • Adding two vectors at right angles as if they were simply along the same line (e.g. wrongly writing 3 N + 4 N = 7 N instead of using Pythagoras to get 5 N).
  • Mixing up which side is “opposite” and which is “adjacent” when finding the angle of a resultant with tan θ.

6. Quick-Fire Challenge Round

True or False?
  1. A vector quantity is fully described by a number and a unit alone.
  2. Measuring 20 oscillations of a pendulum and dividing by 20 gives a more reliable period than timing a single oscillation.
  3. Mass is a vector quantity.
  4. The resultant of two forces at right angles can be found using Pythagoras’ theorem.
Show answers
  1. False — a vector needs both a magnitude and a direction; a number and unit alone only describes a scalar.
  2. True — it spreads the fixed reaction-time error over many oscillations, shrinking its effect on the final result.
  3. False — mass is a scalar; weight (the force due to gravity on that mass) is the vector.
  4. True — Pythagoras’ theorem gives the magnitude, and trigonometry (tan θ) gives the direction.
Spot the Error
  1. “A stack of 100 sheets of card is 15 mm thick, so one sheet is 15 mm thick.”
  2. “Velocity is a scalar quantity because it just tells you how fast something is going.”
  3. “Two forces of 5 N and 12 N act at right angles, so the resultant is 5 + 12 = 17 N.”
Show answers
  1. The thickness of one sheet should be found by dividing: 15 mm ÷ 100 = 0.15 mm, not by taking the stack’s thickness as the thickness of a single sheet.
  2. Velocity is a vector, not a scalar — it includes direction as well as speed. Speed alone (without direction) is the scalar quantity.
  3. Forces at right angles cannot be added directly like this. Using Pythagoras: R = √(5² + 12²) = √(25+144) = √169 = 13 N.

7. Mixed Practice — Bringing It All Together

  1. A student measures a wire with a ruler: one end is at 1.5 cm, the other end is at 24.5 cm. What is the length of the wire?
  2. A pile of 50 coins has a total mass of 275 g. Find the mass of one coin.
  3. A student times 30 oscillations of a pendulum, recording 45.9 s. Find the period of one oscillation.
  4. State whether each is a scalar or vector: acceleration, energy, gravitational field strength, time.
  5. Two forces of 9 N and 12 N act at right angles on an object. Find the magnitude and direction of the resultant force.
Show answers
  1. 24.5 − 1.5 = 23.0 cm
  2. 275 g ÷ 50 = 5.5 g
  3. 45.9 s ÷ 30 = 1.53 s
  4. acceleration — vector; energy — scalar; gravitational field strength — vector; time — scalar
  5. R = √(9² + 12²) = √(81+144) = √225 = 15 N. tan θ = 12/9 → θ = 53.1° from the 9 N force. Resultant = 15 N at 53.1° from the 9 N direction.

Practice at Home

Work through these steadily — Part D (vectors) is Extended-only content.

Part A — Length, volume and time
  1. A ruler measurement gives a start reading of 2.4 cm and an end reading of 17.9 cm. Find the length.
  2. A measuring cylinder reads 84 cm³ when a stone is added to 50 cm³ of water, and the water level rises to 84 cm³. If the water alone was at 50 cm³, what is the volume of the stone?
  3. A student times an event three times: 5.4 s, 5.6 s, 5.5 s. Find the average time.
Part B — Measuring multiples
  1. A stack of 60 identical plastic sheets is 90 mm thick. Find the thickness of one sheet.
  2. A pendulum completes 15 oscillations in 27.0 s. Find its period.
  3. Explain, in your own words, why measuring one oscillation of a pendulum directly with a stopwatch is a poor method for finding an accurate period.
Part C — Scalars and vectors
  1. List three examples of scalar quantities and three examples of vector quantities, choosing different ones from those given in this note.
  2. Explain the key difference between a scalar and a vector quantity, using an example of each.
Part D — Resultant of two vectors (Extended)
  1. Two forces act on an object: 7 N to the east and 24 N to the north. Find the magnitude and direction of the resultant.
  2. A cyclist travels at 5 m/s due north while a crosswind blows at 2 m/s due east. Find the cyclist’s resultant velocity relative to the ground.
  3. A resultant force of 10 N is made up of two forces at right angles. One of the forces is 6 N. Find the size of the other force. (Hint: rearrange Pythagoras’ theorem.)

Key Vocabulary Recap

Parallax error — an error in reading a scale caused by viewing it from an angle rather than straight on
Meniscus — the curved surface of a liquid in a container, read at eye level for an accurate volume
Scalar — a quantity with magnitude (size) only, e.g. distance, mass, time
Vector — a quantity with both magnitude and direction, e.g. force, velocity, acceleration
Resultant — a single vector that has the same overall effect as two or more combined vectors
Measuring multiples — measuring a convenient number of identical items or repeated events together, then dividing, to get an accurate value for one

Syllabus Reference

This note covers Cambridge IGCSE Physics (0625) section 1.1, Physical quantities and measurement techniques — Core points 1-3 (using rulers and measuring cylinders, measuring time intervals, determining an average value for a small distance or short time by measuring multiples) and Supplement points 4-7 (scalars and vectors, examples of each, and the resultant of two vectors at right angles, limited to forces or velocities).

Leave a Comment

Your email address will not be published. Required fields are marked *