Introduction to Physics and Measurement of Physical Quantities

Lessons

Senior 1 · Physics

Introduction to Physics and Measurement of Physical Quantities

5 lessons

Lesson 5 of 5

Measuring Time; Scalar and Vector Quantities

22 min

Two puzzles

Puzzle 1. In a school 100 m race, three teachers time the winner with stopwatches. They get 13.2 s, 13.4 s and 13.3 s. An electronic timer, started by the gun and stopped by a beam of light at the finish, gives 13.28 s. Why do the teachers disagree, with each other and with the machine?

Puzzle 2. A moto driver phones you: "I am 2 km from you." Can you tell where he is? You know how far, but not which way. Some quantities need a direction as well as a size.

Definition

Time ( $t$ ) is the duration between two events. Its SI unit is the second (s). Larger units: $1\,\mathrm{min} = 60\,\mathrm{s}$ , $1\,\mathrm{h} = 3600\,\mathrm{s}$ , $1\,\text{day} = 86\,400\,\mathrm{s}$ . Smaller: $1\,\mathrm{ms} = 10^{-3}\,\mathrm{s}$ , $1\,\mu\mathrm{s} = 10^{-6}\,\mathrm{s}$ .

Reaction time is the delay between seeing (or hearing) a signal and responding to it, usually 0.15 s to 0.3 s for a person.

Instruments for measuring time

Instrument	Smallest reading	Used for
Clock or wrist-watch	1 s (or 1 min)	the length of a lesson or a journey
Stopwatch (hand-operated)	0.1 s or 0.01 s shown, but limited by your reaction time	races, pendulums, experiments lasting seconds or minutes
Phone timer	0.01 s shown	the same as a stopwatch
Light gates with an electronic timer	0.001 s	fast events: athletics finishes, trolleys in a laboratory

A hand-operated stopwatch may SHOW hundredths of a second, but the person pressing it cannot react that precisely. That explains Puzzle 1: each teacher's reaction time was slightly different, and the electronic timer has no reaction time at all.

Experiment: measure your reaction time with a ruler

Figure 5.1: The ruler-drop test. The further the ruler falls before you catch it, the longer your reaction time.

Your partner holds a 30 cm ruler by its top end, hanging straight down.
Hold your thumb and first finger open on either side of the 0 cm mark, not touching the ruler.
Without warning, your partner lets go. Catch the ruler as fast as you can.
Read the mark level with the top of your thumb: this is the distance the ruler fell.
Repeat five times, find the mean distance, and use the table to find your reaction time.

Distance fallen (cm)	5	10	15	20	25	30
Reaction time (s)	0.10	0.14	0.17	0.20	0.23	0.25

The times in the table come from the way an object speeds up as it falls, which you will study in a later unit. One pupil's five catches were 17, 19, 16, 18 and 20 cm: a mean of 18 cm, which is a reaction time of about 0.19 s.

Important

Safety. Use a wooden or plastic ruler, not a metal one with sharp edges. Keep your fingers open until the ruler is released, and do not snatch.

Worked example 1: converting time units

A bus journey from Kigali to Huye takes 2 h 15 min. Express this in seconds.

t = 2 \times 3600\,\mathrm{s} + 15 \times 60\,\mathrm{s} = 7200\,\mathrm{s} + 900\,\mathrm{s} = 8100\,\mathrm{s}

Worked example 2: why we time many swings

A timekeeper's reaction time adds an error of about 0.2 s. Compare the percentage error in timing ONE swing of 1.5 s with timing 20 swings (30 s).

\text{one swing: } \frac{0.2\,\mathrm{s}}{1.5\,\mathrm{s}} \times 100\% \approx 13\% \qquad \text{20 swings: } \frac{0.2\,\mathrm{s}}{30\,\mathrm{s}} \times 100\% \approx 0.7\%

The same reaction error is a much smaller fraction of a long time.

Definition

A scalar quantity has size (magnitude) only. Examples: mass, time, distance, speed, volume, density, temperature, energy.

A vector quantity has both size AND direction. Examples: displacement, velocity, acceleration, force, weight.

Scalars and vectors

Scalar (size only)	Matching vector (size and direction)
distance: 2 km	displacement: 2 km north of the school
speed: $15\,\mathrm{m\,s^{-1}}$	velocity: $15\,\mathrm{m\,s^{-1}}$ towards Musanze
mass: 50 kg	weight: 490 N downwards
time, volume, density, temperature	force, acceleration

That solves Puzzle 2: "2 km" is a distance, a scalar. To find the driver you need his displacement, a vector: "2 km east of you".

Drawing a vector

A vector is drawn as an arrow. The arrow's length shows the size, using a stated scale, and the arrowhead shows the direction.

Figure 5.2: Four force vectors drawn to the scale 1 square = 10 N. Arrow D runs 3 squares east and 3 squares north, so it points north-east, and its length (about 4.2 squares) represents about 42 N.

Worked example 3: drawing a vector to scale

Draw a force of 250 N acting towards the north-east, using the scale $1\,\mathrm{cm} : 50\,\mathrm{N}$ .

\text{length} = \frac{250\,\mathrm{N}}{50\,\mathrm{N\,cm^{-1}}} = 5\,\mathrm{cm}

Draw a 5 cm arrow at $45^\circ$ between north and east, with its head at the north-east end, and write the scale beside it.

Worked example 4: adding vectors along one line

Two forces act on a box along the same straight line: 30 N east and 20 N west. Find the combined (resultant) force.

Take east as positive: $+30\,\mathrm{N} + (-20\,\mathrm{N}) = +10\,\mathrm{N}$ , so the resultant is 10 N east. If both forces pointed east, the resultant would be $30 + 20 = 50\,\mathrm{N}$ east. Scalars simply add: 30 kg and 20 kg always make 50 kg.

Warning

Common mistakes: giving a vector without its direction ("the force is 10 N" is incomplete), and adding vectors like ordinary numbers when they point in different directions. A minus sign on a vector means "the opposite direction", not "less than nothing".

Tip

Ask "which way?" If the question makes sense for a quantity, it is a vector. "Which way is your mass?" makes no sense, so mass is a scalar. "Which way is the wind blowing?" makes sense, so wind velocity is a vector.

Vectors in the sky over Kigali

Weather reports from Meteo Rwanda give the wind as a speed AND a direction, because a pilot needs both. Aircraft at Kigali International Airport take off and land facing into the wind, which gives the wings more lift at a lower ground speed. A wind reported as "easterly, $15\,\mathrm{km\,h^{-1}}$ " blows FROM the east TOWARDS the west.

Scalar and vector quantities (Cognito). Watch for the list of pairs, distance and displacement, speed and velocity, and how a vector is drawn as an arrow.

✦

Practice

Convert to seconds: (a) 3 min 20 s; (b) 1.5 h; (c) 250 ms.
In the ruler-drop test a pupil catches the ruler at 12 cm. Use the table to estimate her reaction time.
Sort into scalars and vectors: mass, force, speed, velocity, time, displacement, temperature, weight.
Using the scale $1\,\mathrm{cm} : 20\,\mathrm{N}$ , how long is the arrow for a force of 90 N?
Two forces act on a cart along one line: 45 N east and 30 N west. Find the resultant force.
A timekeeper's reaction error is 0.2 s. Find the percentage error when timing (a) one swing of 2.0 s, (b) 20 swings.
Explain why the winner's time in a national athletics final is measured electronically, not with hand-held stopwatches.

Show answersHide answers

(a) 200 s; (b) $1.5 \times 3600 = 5400\,\mathrm{s}$ ; (c) 0.25 s.
Between 0.14 s (10 cm) and 0.17 s (15 cm): about 0.16 s.
Scalars: mass, speed, time, temperature. Vectors: force, velocity, displacement, weight.
$\frac{90}{20} = 4.5\,\mathrm{cm}$
$45 - 30 = 15\,\mathrm{N}$ east.
(a) $\frac{0.2}{2.0} \times 100\% = 10\%$ ; (b) 20 swings take 40 s, so $\frac{0.2}{40} \times 100\% = 0.5\%$ .
A human's reaction time (about 0.2 s) varies from person to person and from try to try, while an electronic timer started by the gun and stopped at the line has no reaction time and reads to 0.001 s. Races are often won by less than 0.1 s.

Key points

Time is measured in seconds; hand timing is limited by reaction time (about 0.2 s), so time long intervals or use electronic timers.
A scalar has size only (mass, time, distance, speed); a vector has size and direction (displacement, velocity, force, weight).
A vector is drawn as an arrow: its length shows the size to a stated scale, and the arrowhead shows the direction.

1Two puzzles

2Instruments for measuring time

3Experiment: measure your reaction time with a ruler

4Scalars and vectors