### Collection Time, distance and speed

## Time, distance and speed

for 14-16

To analyze motion, you must know about positions and times. These are introductory experiments, covering some basic techniques and measurement issues.

**Class practical**

This is a useful introduction to the use of ticker-timers.

Apparatus and Materials

*For each student or student group*

- Ticker-timer
- Ticker-tape
- Stopwatch or stopclock
- Mechanics trolley or wind-up/pull-back toy car OPTIONAL

Health & Safety and Technical Notes

In crowded laboratories, estimate the space needed by the tape puller and arrange the groups to avoid collisions.

Read our standard health & safety guidance

The ticker-timer should come with a recommended power supply unit (low voltage AC as specified, with on/off switch).

Some ticker-timers use light-developed tape rather than carbon discs. After the tape has been struck with the vibrating arm, it takes a few minutes for the dots to become visible.

Procedure

- Thread a short length of ticker-tape through the ticker-timer. If there is a carbon paper disc, make sure the tape goes underneath the disc.
- Turn the ticker-timer on for a few seconds. It vibrates rapidly and hits the top of the carbon paper. It makes a lot of dots on the tape, at regular intervals.
- Remove the tape from the ticker-timer. If the tape didn't move when the ticker-timer was switched on, then all the dots will be in the same place.
- Thread a longer piece of ticker-tape, about 1 metre long, through the ticker-timer. Switch the ticker-timer on. Pull the tape slowly through the ticker-timer.
- Check the tape to see if you can see each individual dot, with a space between. We can say that each dot-to-dot space stands for a
tick

of time. - Thread another 1 metre piece of tape through the ticker-timer.
- You need a
start

signal and astop

signal. These could be handclaps by one of your group or by your teacher. They should be just a few seconds apart. Pull the tape slowly and switch the ticker-timer on at the start signal. Switch it off at the stop signal. - Count the number of dot-to-dot spaces between the start and the stop. That is the time between the signals, measured in
ticks

. - Use a fresh piece of tape, and a stopwatch or stopclock. Pull the tape through the ticker-timer for 3 seconds. Find out how many
ticks

there are in 3 seconds. Find out how many there are in 1 second. Work out the time in seconds that is the same as 1 tick.

Teaching Notes

- All students could use the same start and stop signals, such as two handclaps by the teacher. This provides the opportunity for comparison of the times obtained by different students. Times can be tabulated, and used for discussion of range, mean and estimated error.
- You could make a bar chart of students' answers for the time, in seconds, that is equivalent to one tick. Most ticker-timers vibrate at 50 Hz, and thus make 50 dots per second. For these, the expected mean value of one tick is 1/50 second, or 0.02 s . (Some ticker-timers will give dot-making frequency of 100 Hz, and for these the value of one tick is equivalent to 1/100 seconds or 0.01 s .)
- You can make a model of the action of the ticker-timer and the ticker-tape. Move a roll of wallpaper along the bench or floor and ask a student to put blobs of paint or ink onto the wallpaper at regular times. These could be indicated by a steady handclap or a metronome. Fast and slow motion by the
wallpaper puller

will produce blobs at different separations for equal time intervals. Each blob-to-blob space represents atick

of time. **How Science Works extension:**This activity can be used as a prompt to discuss the relative merits of the ticker-timer as a timing device compared to a stopwatch. Make a simple speed measurement for a moving object such as a mechanics trolley or a pull-back car using the ticker-timer and conventional stopwatch. Encourage students to discuss or write about the strengths and weaknesses of each method. The key teaching point is how to select appropriate equipment, by using the concepts of accuracy and uncertainty in measurements.- Many students find the ticker-timer an awkward piece of equipment. Some struggle to get a time reading from the paper tape and get confused about what the dot spacings represent. If you plan to use ticker-timers regularly in motion experiments, give students extra time with early investigations so they become familiar and confident with the equipment.

*This experiment was safety-checked in January 2007*

### Up next

### Race time measurement

**Class practical**

This is an introduction to the language of measurement, including concepts of range, reproducibility, mean value, true value, accuracy, instrument resolution and, most important, measurement uncertainty.

Apparatus and Materials

*For each student or student group*

- Stopwatch or stopclock
- String
- Statistics board (see technical notes)
- Masses, 50 g, 5 or 6
- Cones/track markers, 10 OPTIONAL
- Video camera OPTIONAL
- Tape measure, long (at least 10 m) OPTIONAL

Health & Safety and Technical Notes

If working outside, students must be appropriately supervised.

If a trolley is used in the lab, ensure that the trolley cannot land on anyone's feet or legs.

Read our standard health & safety guidance

A statistics board is made from a piece of wooden board about 0.5 m square. Ten slotted channels are glued to it and metal (or other suitable material) discs are cut so that they fit into the channels. The board is supported vertically.

Assign values to each channel. Students drop in a disc for the value they achieve. The distribution of results grows as results are added.

Procedure

- One student runs a distance of 100 metres. You, and other students, all independently time the run.
- Compare all of the measurements. What is their
range

(the difference between the highest and the lowest measured values)? What does this tell you about thereproducibility

of the measured values of time? - What is the
mean

of all the measurements? A mean is a kind of average. Work this out by adding them all together and then dividing by the number of measurements. How closely do you think the mean value agrees with the true value of the run time? In other words, estimate the accuracy of the mean value. - Did everybody use stopwatches with the same
resolution

? For example, were everyone’s stopwatch timeindications

in tenths of seconds or hundredths of seconds? (0.1 s econd is a tenth of a second; 0.01 s econds is a hundredth of a second). - Try to estimate the reaction times involved in pressing a stopwatch to both ‘start’ and ‘end’ the run. The sum of these reaction times is very likely larger than the resolution of the stopwatch.
- How certain can you be about the actual time taken for the run? You can’t be perfectly certain! There must be some uncertainty in the measurements. The mean measurement could be 14.8 s econds. Perhaps you think that the ‘true’ time for the run is in between 14.6 s econds and 15.0 s econds. Then you can say that the uncertainty is ± 0.2 s econds.

Teaching Notes

- The most important term here is measurement uncertainty, a concept that can be introduced early in science education. Additional terminology about measurements should enhance its meaning and not distract from it. For example, you may decide to omit steps 4 and 5 from the Procedure (above).
- In more advanced work, measurement uncertainty is sometimes called measurement ‘error’. Here, the word uncertainty more clearly describes a reasonable doubt about the result obtained.
- Precision is a quality denoting the closeness of agreement between measured values obtained by repeated measurements. If values cluster closely, measurements are called ‘precise’. Reproducibility is the precision obtained when measurements are made by different operators using different instruments.
- Statistical treatment plays very important parts in modern science. In advanced experiments students are expected to treat errors with some statistical care. In kinetic theory the steady pressure of a gas is recognized as an average of innumerable individual bombardments. Statistical methods are used to delve into details of molecular speed or sizes. In modern atomic physics statistical views are of prime importance. So you might well make a gentle start to later science studies by showing how scientists look at a number of measurements of the same thing.
- The times could be collated as lists of numbers or, using a computer, as bar charts, or using a statistics board. Bar charts enable students to understand range, mean and uncertainty visually.
- It is worth pointing out that there is such a thing as too many digits in a quoted value. A student who uses a stopwatch and gives a time of 14.77 s econds is crediting the timing process with less uncertainty than it actually has. Answers of 15 seconds or 14.8 s econds may be acceptable (depending on the timing procedure and the stopwatch).
- ‘Mean’ is here used to indicate a particular kind of average – that found by dividing the sum of values by the sample size.
- You could repeat the activity for a different motion, such as for a trolley pulled across a metre distance on a table, or the fall of a mass. Again, all students should measure the time for the same motion. Range, mean value, and measurement uncertainty can be compared with those for the student’s 100 metre run.
- You may want to compare timings for real sports races. Information on sporting records can be found on the Internet. For example see Usain Bolt's record breaking 100 m run in the 2008 Olympics. Instrument resolution for different sports could be compared, and students could discuss the idea of uncertainty in the measured values...
**How Science Works extension:**This experiment covers concepts of accuracy and precision of data, as well as measurement uncertainty. The scope could be increased further, as follows:- Arrange pairs of students every 5 m or 10 m apart along the 100 m running path. Use some kind of signal (e.g. dropping a raised arm) to start both the runner and everyone’s timers. As the runner passes each student, they stop their timer and record the time taken to reach them.
- Students then plot this data graphically (distance against time). This will make it easier for students to understand average speed and get a feel for the variation in measurements. A true value of velocity can be calculated from the gradient of the best fit line.
- If you placed cones/markers along the track, you might be able to video each student running, with a stopclock also in the camera view. This would generate a second set of results that could be compared numerically or graphically to the class set. Students could comment on whether this method improves on the previous one.

*This experiment was safety-checked in January 2007*

### Up next

### Timing a trolley on a slope

**Class practical**

Graphical presentation of data helps to build concepts. Here, time, distance and velocity can all be measured and displayed, and the idea of acceleration can be introduced.

Apparatus and Materials

*For each student or student group*

- Runway
- Dynamics trolley
- Stopwatch or stopclock

Health & Safety and Technical Notes

Long runways or heavy shorter ones should be handled by two persons. Ensure that a string is tied across the bottom of the runway, to prevent the trolley falling onto anyone.

Read our standard health & safety guidance

Relatively light runways can be made using planks with lengths of angle-iron along each side to produce rigidity. Alternatively, MDF window board, available from builders merchants, is relatively rigid and makes suitable runways. Long lengths are quite heavy, and slopes do require midway support to prevent excess sagging.

For this activity, lengths of 1.5 m are adequate. For other activities, longer boards of about 2.5 m are required.

The boards also need to be about 30 cm wide and 2.5 cm thick so that they do not flex.

Procedure

- Use books or wooden blocks to support one end of the runway and make the board slope. The slope should be big enough for the trolley to gently increase in speed as it travels down it. (About 1 in 10)
- Let the trolley start from rest (standstill). Time the trolley, as well as you can, over the first 25 cm it travels. Time it over the second 25 cm, over the third 25 cm, and so on.
- Repeat the measurements twice again. See how much variation there is in your measurements.
- Write all your measurements in a table. Work out the average (mean) time for the first 25 cm, for the second 25 cm, and so on.
- Make bar charts to show the information about distance and time.
- Divide the distance by the time. That will tell you the mean velocity of the trolley during each part of its journey. Make bar charts of this velocity information.
- Note the patterns you see in the changes that take place to the time, the distance, and the velocity.

Teaching Notes

- Here, we use the word
distance

rather than the more correctdisplacement

. At more advanced levels, the distinction between scalar quantities like distance, and vector quantities like displacement, becomes important. - The experiment shows that the times to cover successive 25 cm distances decrease. It also demonstrates difficulties in measuring short time intervals of less than a second, as well as measuring continuous time.
- Students could decide on their own format for graphic data presentation.
- More than three students working at a board will lead to "passengers" in the group.

*This experiment was safety-tested in November 2004*

### Up next

### Instantaneous and average velocities

**Demonstration**

You can use a sensor datalogging system to experience the difference between instantaneous and average velocity.

Apparatus and Materials

- Light beam sensor assembly (source and sensor)
- Computer with datalogging software
- Dynamics trolley
- Runway, with means to produce a uniform slope
- Card

Health & Safety and Technical Notes

A string tied across the runway will ensure that the trolley does not damage the pulley or a user. Long runways should be handled by two persons.

Read our standard health & safety guidance

Procedure

- Set up the runway so that the trolley can accelerate down it. Set up a source and detector across the runway.
- The detector should be connected to a computer. The computer should be set to record the time during which the beam of light to the detector is interrupted.
- Cut pieces of card of different lengths to attach to the trolley, one at a time. Measure their lengths. The longest should be 25 to 30 centimetres long.
- Tape the longest card to the trolley, so that it can break the light beam. Tape the centre of the card at the centre of the trolley's length.
- Use the computer system to measure the time for the card to move past the light beam. Get the computer to divide the length of the card by this time. The answer is the average velocity of the trolley during that period of time.
- Replace the card by a much shorter one. Again, fix the card centre to the trolley centre. Release the trolley from the same place on the runway. Find out the new time and new velocity. The time is shorter than before. The new velocity is still an average velocity.
- Find out the shortest length of card that will give consistent values of time and velocity.

Teaching Notes

- Students should understand the difference between speed and velocity; a scaler and a vector quantity.
- For the accelerating trolley, its velocity is changing, instant by instant. Students could be invited to think about these problems:
*"What information would the computer tell you if you used a card that was as long as the trolley's journey?"*(Answer: The average velocity for the whole journey, provided that the light beam is set up halfway along the distance of travel.)*"What length of card would you need to measure an instantaneous value of the trolley velocity?"*(Answer: A card so thin that it is disappearing would, if the system could sense it, provide the value of velocity at the instant that it passes through the beam. Note that aninstant

is a time that is so small that it is vanishing.)*"What would be the best way to gather data to produce a smooth and accurate graph of velocity against time: should you use short cards or long ones?"*(Answer: A card of disappearing length used with many sensor systems along the track would produce a set of points, one for each sensor, for a velocity-time graph. Other methods provide only average velocities.)- These ideas relate, of course, to the mathematical concept of infinitesimal values, and hence to the foundations of calculus, without which our understanding of the world would be much narrower than it is.
- A good estimate of instantaneous speed is measured by a car speedometer, while the average speed can be calculated knowing the distance travelled and the time taken.
- As an extension activity, you could use a pair of source-sensor assemblies linked to the same computer to obtain two velocity values and the time between their measurements. Divide the difference in the velocities by this time to work out a value for acceleration.

*This experiment was safety-tested in November 2004*

### Up next

### Ticker-timers for investigating speed

**Class practical**

Making ticker-timer charts can develop an understanding of speed-time graphs.

Apparatus and Materials

*For each student or student group*

- Ticker-tape
- Ticker-timer with power supply unit
- Sellotape

Health & Safety and Technical Notes

In crowded laboratories, estimate the space needed by the tape puller and arrange the groups to avoid collisions. It may be impracticable for all groups to do this.

Read our standard health & safety guidance

Self-adhesive and plain ticker-tape are both available. The self-adhesive version is more convenient for this experiment.

Procedure

- One person should operate the ticker-timer switch while another walks away, pulling tape through the ticker-timer. The walker should speed up and then slow down and stop, while the tape is running through the timer.
- Look at the tape. The dots happen at regular time intervals, but sometimes they are close together and sometimes they are further apart.
- Cut across the tape through the dot that was made when the person started to move. Count ten dot-to-dot spaces. Cut it again. That gives you a short strip, which is a ten-tick-tape. (It might be hard to count spaces when the dots are very close together, and you will have to estimate the number of spaces.)
- Make a horizontal line along the bottom of a sheet of paper. Stick your ten-tick-tape near the start of the line. The first dot of the ten-tick-tape should be on the horizontal line. The line of dots should be vertical.
- Count ten more dot-to-dot spaces along the tape. Cut it again. Stick this 10-tick-tape alongside the first one.
- Cut the whole of the tape into ten-tick-tapes, to make a complete chart.
- Label the chart where the person pulling the tape was:
- Going slowly
- Moving fast
- Speeding up
- Slowing down

- Your chart is a bit like a graph of speed against time.
- The time to make each tape was exactly the same. Your horizontal line represents time for the journey of the tape.
- The length of each ten-tick-tape depends on the speed. The higher the speed then the longer the tape. Add a vertical axis to your chart. This axis represents speed.

- Predict what the chart would look like if the tape travelled at a steady and slow speed. Also predict what it would look like if it travelled at steady high speed. Check your predictions. Make a tape for slow steady motion and use it to make a chart. Do the same for fast steady motion.
- Predict what the chart would look like if the tape travelled at speed that increased steadily. To test this prediction, start by pulling the tape very slowly but then let it get gradually faster.
- The motion of other objects such as toys, bicycle, car, falling weights could also be tried.

Teaching Notes

- Provide students with squared (or graph) paper, so the ticker-timer strips are more easily aligned.
- You will need to explain carefully the idea that the length of tape in each 'ten-tick' of time indicates speed.
- If the speed of the tape increases steadily, the upper edge of the ticker-tape chart should form a straight line slanting upwards. Thinking about this should encourage students towards a difficult idea: a speed - time graph with a positive gradient indicates acceleration.
- With more able students, you may want to develop the idea that the total area of the chart tape represents the total distance travelled.
- The total distance travelled by the object is the full length of the tape and cutting it up does not alter this. If the tape was 1 cm-wide then the area of each strip would just be numerically equal to the distance travelled in the time represented by the strip. Adding together all the areas of the strips would give the area of the chart and so the area of the chart represents the distance travelled (for 1 cm wide strips).
- The graph of velocity against time has time in seconds along the x-axis and speed in cm/s along the vertical axis. The area under the graph measures the distance travelled in centimetres. Areas under graphs measure the product of the values represented on the x and y-axis. On a velocity-time graph each vertical strip represents
*v*Δ*t*and*v*Δ*t*= Δ*s*, the distance travelled in time Δ*t*. - Ticker-tape charts make a good introduction to integration as the area under a curve shows one application of calculus.
- Finally, a different type of chart can be made. The tape of a moving object is placed on the bench and a second blank tape is laid alongside it. From this tape a piece is cut off equal to the distance travelled in the first 10-tick or 1/5 of a second and stuck down to start making a chart. Then a second piece of tape is cut from measuring the distance travelled on the first tape from the beginning of the motion to the end of 2/5 of a second and so on until all the motion tape is used. This tape chart shows a distance travelled against time.
- If the speed is increased steadily, the top edge of this chart is not a straight line but rather it follows a curve (
*s*=*1/2at*^{ 2 }). - This experiment could be followed by Simple motion experiments with a datalogger which uses software to plot velocity - time graphs directly.

Simple motion experiments with a datalogger

*This experiment was safety-tested in April 2006*

### Up next

### Simple motion experiments with a datalogger

**Demonstration**

An introduction to motion using ultrasound position sensors.

Apparatus and Materials

- Ultrasound position sensor
- Trolley, ramp
- Lab jack
- Buffer to rebound trolley

Health & Safety and Technical Notes

Read our standard health & safety guidance

Most position sensors will return the distance from the sensor. Often they can also produce automatically a value for velocity and acceleration. They should be able to produce live

graphs and a data collection rate of 10 Hz is adequate.

Procedure

There is a series of participative experiments you can perform but the details will depend upon the datalogging system you have.

- Get a few students to produce live position-time graphs of themselves as they walk towards and away from the sensor. They should be able to identify which parts of each graph correspond to what they were doing.
- They can then predict the speed-time graph, which should include positive and negative parts. Then this can be compared to what the datalogging software produces.
- Record the position-time graph of a trolley rolling on a friction-compensated ramp. Place the sensor at the top of the ramp.
- Again predict and check the velocity-time graph.
- Repeat 3 but with the ramp steeper so the trolley accelerates. Record the velocity-time graph and predict the position-time graph and acceleration-time. Again, check with the computer output.
- Repeat 3 with a buffer at the bottom of the ramp so the trolley rebounds.
- This last step may confuse even the best students - have them roll the trolley up the ramp towards the motion sensor so it comes to a halt and then starts to roll back. Again compare velocity-time and acceleration-time graphs.

Teaching Notes

- The aim is to get the students understanding the relationships between distance-time and velocity-time graphs.
- This is best done through getting them to predict what graphs will look like before they get the computer to produce them.
- With a high-level group it would be sensible to discuss how the computer produces a velocity-time graph from position data. It simply takes differences in position and divides them by the time interval (not the total time). This helps to emphasize the difference between instantaneous and average speed.
- The direction of motion helps to emphasize the difference between velocity and speed.
- The last experiment will help to emphasize the idea that velocity and acceleration can be in opposite directions.

### Up next

### Using speed-time graphs to find an equation

Imagine a graph plotted with SPEED on the vertical axis against TIME on the horizontal axis.

#### Constant speed

For an object moving along with constant speed *v*, the graph is just a horizontal line at height *v* above the axis. You already know that *s*, the distance travelled, is speed multiplied by time, *vt*; but on your graph *v* x *t* is the AREA of the shaded block of height *v* and length *t*.

#### Constant acceleration

Sketch a graph for an object starting from rest and moving faster and faster with constant acceleration. The line must slant upwards as *v* increases. And if the acceleration is constant the line must be a straight

slanting line.

Take a tiny period of time from *T* to *T'* on the time axis when the speed was, say, *v*_{1} . Look at the pillar that sits on that and runs up to the slanting graph line (Graph III). The area of that pillar is its height *v*_{1} multiplied by the short time *TT'*. That area is the distance travelled in that short time.

How big is the distance travelled in the whole

time, *t*, from rest to final *v*? It is the area of all the pillars from start to finish. That is the area of the triangle (in Graph IV) of height final *v* and base *t*, the total time.

The area of any triangle is ½ (height) x (base).

So distance *s* is ½(height, *v*) x (base, *t*) *s* = ½*v**t*.

Suppose the object *does not start from rest* when the clock starts at 0 but is already moving with speed *u*. It accelerates to speed *v* in time *t*. Then the graph is like graph V below; and the distance travelled is given by the shaded area. That is made up of two patches, a rectangle and a triangle (Graph VI).

The rectangle's area is *u**t*, the triangle's is ½(*v*-*u*)*t*.

Then *s* = *u**t* + ½(*v*-*u*)*t*

- =
*u**t*+ ½*v**t*-½*u**t* - = ½
*v**t*+ ½*u**t* - = (
*v*+*u*2)*t*

Alternatively, since *v*-*u* = *a**t*

*s*=*u**t*+ ½(*v*-*u*)*t*can be expressed as- =
*u**t*+ ½(*a**t*)*t* - =
*u**t*+ ½*a**t*^{ 2}

These formulae are only true for constant acceleration. Look at Graph VII. Is the acceleration constant? Which part of the area for *s* is different now?

What part of *u**t* = ½*a**t*^{ 2} is no longer safe for calculating *s*?

### Up next

### A language for measurements

#### What is a measurement?

A measurement tells you about a property of something you are investigating, giving it a number and a unit. Measurements are always made using an instrument of some kind. Rulers, stopclocks, chemical balances and thermometers are all measuring instruments.

Some processes seem to be measuring, but are not, e.g. comparing two lengths of string to see which one is longer. Tests that lead to a simple yes/no or pass/fail result do not always involve measuring.

#### The quality of measurements

Evaluating the quality of measurements is an essential step on the way to sensible conclusions. Scientists use a special vocabulary that helps them think clearly about their data. Key terms that describe the quality of measurements are:

- Validity
- Accuracy
- Precision (repeatability or reproducibility)
- Measurement uncertainty

Validity: A measurement is ‘valid’ if it measures what it is supposed to be measuring. What is measured must also be relevant to the question being investigated.

If a factor is uncontrolled, the measurements may not be valid. For example, if you were investigating the heating effect of a current ( *P* = *I*^{ 2} *R*) by increasing the current, the resistance of the wire may change as it is heated by the current to different temperatures. This would skew the results.

Correct conclusions can only be drawn from valid data.

Accuracy: This describes how closely a measurement comes to the true value of a physical quantity. The ‘true’ value of a measurement is the value that would be obtained by a perfect measurement, i.e. in an ideal world. As the true value is not known, accuracy is a qualitative term only.

Many measured quantities have a range of values rather than one ‘true’ value. For example, a collection of resistors all marked 1 kΩ. will have a range of values, but the mean value should be 1 kΩ.. You can have more confidence in a number of measurements of a sample rather than an individual measurement. The variation enables you to identify a mean, a range and the distribution of values across the range.

Precision: The closeness of agreement between replicate measurements on the same or similar objects under specified conditions.

Repeatability or reproducibility

(precision): The extent to which a measurement replicated under the same conditions gives a consistent result. Repeatability refers to data collected by the same operator, in the same lab, over a short timescale. Reproducibility refers to data collected by different operators, in different laboratories. You can have more confidence in conclusions and explanations if they are based on consistent data.

Measurement uncertainty: The uncertainty of a measurement is the doubt that exists about its value. For any measurement – even the most careful – there is always a margin of doubt. In everyday speech, this might be expressed as ‘give or take…’, e.g. a stick might be two metres long ‘give or take a centimetre’.

The doubt about a measurement has two aspects:

- the width of the margin, or ‘interval’. This is the range of values one expects the true value to lie within. (Note this is not necessarily the range of values one might obtain when taking measurements of the value, which may include outliers.)
- confidence level’, i.e. how sure the experimenter is that the true value lies within that margin. Discussion of confidence levels is generally appropriate only in advanced level science courses.

Uncertainty in measurements can be reduced by using an instrument that has a scale with smaller scale divisions. For example, if you use a ruler with a centimetre scale then the uncertainty in a measured length is likely to be ‘give or take a centimetre’. A ruler with a millimetre scale would reduce the uncertainty in length to ‘give or take a millimetre’.

#### Measurement errors

It is important not to confuse the terms ‘error’ and ‘uncertainty’. Error refers to the difference between a measured value and the true value of a physical quantity being measured. Whenever possible we try to correct for any known errors: for example, by applying corrections from calibration certificates. But any error whose value we do **not** know is a source of uncertainty.

Measurement errors can arise from two sources:

- a random component, where repeating the measurement gives an unpredictably different result;
- a systematic component, where the same influence affects the result for each of the repeated measurements.

Every time a measurement is taken under what seem to be the same conditions, random effects can influence the measured value. A series of measurements therefore produces a scatter of values about a mean value. The influence of variable factors may change with each measurement, changing the mean value. Increasing the number of observations generally reduces the uncertainty in the mean value.

Systematic errors (measurements that are either consistently too large, or too small) can result from:

- poor technique (e.g. carelessness with parallax when sighting onto a scale);
- zero error of an instrument (e.g. a ruler that has been shortened by wear at the zero end, or a newtonmeter that reads a value when nothing is hung from it);
- poor calibration of an instrument (e.g. every volt is measured too large).

Whenever possible, a good experimenter will try and correct for systematic errors, thus improving accuracy. For example, if it is known that a balance always reads 2 g greater than the true reading it is perfectly possible to compensate for that error by simply subtracting 2 g from all readings taken.

Sometimes you can only find a systematic error by measuring the same value by a different method.

Errors that are not recognized contribute to measurement uncertainty.

#### ASE/Nuffield booklet: The Language of Measurement

In 2010, following a series of meetings with Awarding Organisations, the ASE and Nuffield Foundation jointly published a booklet to enable teachers, publishers, awarding bodies and others in England and Wales to achieve a common understanding of key terms that arise from practical work in secondary science. Order a copy or see extracts from the booklet

#### Acknowledgement

This webpage is based on the National Physical Laboratory's *Good Practice Guide: A Beginner's Guide to Uncertainty of Measurements* written by Stephanie Bell.

A Beginner's Guide to Uncertainty of Measurements