Calculating speed

Definition

Speed

The distance travelled by an object per unit time.

Speed tells you how far an object travels in each unit of time, so a speed of $m/s3\ \text{m/s}$ means the object covers $m3\ \text{m}$ in every second.
Speed is calculated from a measured distance and a measured time using $taken\text{speed}=\dfrac{\text{distance travelled}}{\text{time taken}}$ .
In symbols this is $v=stv=\dfrac{s}{t}$ , where $v$ is the speed, $s$ is the distance travelled and $t$ is the time taken.
A distance in metres divided by a time in seconds gives a speed in metres per second ( $m/s\text{m/s}$ ), which is the SI unit of speed.
The distance and the time must belong to the same part of the motion, so the time used is the time taken to cover exactly the distance that was measured.
Over the same distance, a shorter time means a greater speed, so a pupil who runs $m100\ \text{m}$ in $s14\ \text{s}$ is faster than a pupil who takes $s18\ \text{s}$ .
In the same time, a longer distance means a greater speed, so a cyclist who rides $m150\ \text{m}$ in $s30\ \text{s}$ is faster than one who rides $m90\ \text{m}$ in $s30\ \text{s}$ .
Speed is found by dividing the distance by the time, and never the other way round, because time divided by distance gives the number of seconds needed for each metre.
A speed calculated from one distance and one time describes the whole stretch that was timed, so any speeding up or slowing down inside that stretch does not show in the answer.

Example

Speed of a runner

A pupil runs $m60\ \text{m}$ along a marked track in a mean time of $s9.6\ \text{s}$ .
Write the equation that links speed, distance and time:

v=\frac{s}{t}

Substitute the measured distance and time:

v=\frac{60\ \text{m}}{9.6\ \text{s}}=6.25\ \text{m/s}

The data are given to two significant figures, so the speed is $m/s6.3\ \text{m/s}$ .

Example

Speed of a trolley between two light gates

Two light gates are placed $m0.50\ \text{m}$ apart on a bench, and a trolley takes $s0.40\ \text{s}$ to travel from the first beam to the second.
Substitute the gap between the gates and the time between them into the equation:

v=\frac{0.50\ \text{m}}{0.40\ \text{s}}=1.25\ \text{m/s}

The trolley's speed between the two gates is $m/s1.3\ \text{m/s}$ to two significant figures.

Planning a speed investigation

Definition

Independent variable

The variable that is deliberately changed in an investigation.

Definition

Dependent variable

The variable that is measured in an investigation to find out how it responds to changes in the independent variable.

Definition

Control variable

A variable that is kept the same throughout an investigation so that it cannot affect the result.

Every speed measurement needs two readings, a distance measured with a suitable instrument and a time measured over exactly that distance.
The distance is marked out first, and the timing then starts and stops as the object passes the marks.
In an investigation, the independent variable is the one you change on purpose, such as the angle of a ramp.
The dependent variable is the one you measure to see the effect, such as the speed of a trolley.
The control variables are kept the same, such as the trolley, the timed distance and the release point, so that only the independent variable can change the result.
A longer timed distance gives a longer time, which makes the reaction-time error a smaller fraction of each reading.
Each speed is calculated from the mean time of repeated runs, which reduces the effect of random errors.
Results are recorded in a table with the independent variable in the first column and units in the column headings, not beside each number.
When the results are plotted, the independent variable goes on the horizontal axis and the dependent variable goes on the vertical axis.

Practical

Walking and running speeds

Aim: to measure the walking speed and the running speed of a person over a measured distance and compare them.
Apparatus: a $m30\ \text{m}$ tape measure, four traffic cones or playground chalk, a stopwatch, a results table and a pencil.
Method:
- Choose a flat, straight stretch of playground and lay the tape measure along the ground.
- Place a cone at the $m0\ \text{m}$ mark of the tape as the start line and another at the $m20\ \text{m}$ mark as the finish line.
- Ask the walker to begin a few metres behind the start line, so that they are already moving at a steady pace when they cross it.
- Stand to one side where you can see both lines, start the stopwatch as the walker's chest crosses the start line and stop it as their chest crosses the finish line.
- Record the time to $s0.1\ \text{s}$ , because reaction time makes the hundredths digit meaningless.
- Repeat the walk twice more so that there are three times for walking.
- Repeat the whole procedure three times with the same person running instead of walking.
- Calculate the mean time for each type of movement, leaving out any anomalous time, and use $v=stv=\dfrac{s}{t}$ with $ms=20\ \text{m}$ to find each speed.
Variables: the type of movement is the independent variable, the speed is the dependent variable, and the person, the distance and the surface are control variables.
Results: the running times are much shorter than the walking times over the same $m20\ \text{m}$ .
Conclusion: the distance is the same for both types of movement, so the one with the shorter mean time, running, has the greater speed.
Watch out: a walker who starts from a standstill on the start line is still speeding up during the first few metres, and reaction time affects both the start and the stop of every reading.
Safety: use a dry, level surface clear of obstacles, wear suitable footwear and leave plenty of space beyond the finish line for the runner to slow down.

Practical

Ramp angle and trolley speed

Aim: to investigate whether the angle of a ramp affects the speed of a trolley rolling down it.
Apparatus: a smooth wooden ramp about $m1\ \text{m}$ long, a stack of wooden blocks, a protractor, a dynamics trolley, a metre rule, masking tape, two light gates on clamp stands connected to an electronic timer or data logger, and a box of foam to catch the trolley.
Method:
- Rest one end of the ramp on the blocks so that it makes an angle of $10∘10^\circ$ with the bench, checking the angle with the protractor.
- Stick a strip of masking tape across the ramp near the top as the release line, so the trolley starts from the same point every time.
- Clamp the first light gate $cm20\ \text{cm}$ below the release line and the second light gate $cm50\ \text{cm}$ further down the ramp, measuring the gap between the beams with the metre rule.
- Place the foam box at the bottom of the ramp to stop the trolley.
- Hold the trolley with its front edge on the release line and let it go without pushing it.
- Record the time the trolley takes to travel from the first beam to the second.
- Repeat the run twice more at the same angle and calculate the mean time.
- Increase the angle to $15∘15^\circ$ , $20∘20^\circ$ , $25∘25^\circ$ and $30∘30^\circ$ in turn, keeping both light gates in the same place on the ramp.
- Calculate the speed at each angle using $v=stv=\dfrac{s}{t}$ with $ms=0.50\ \text{m}$ , then plot speed on the vertical axis against angle on the horizontal axis.
Variables: the angle of the ramp is the independent variable, the speed of the trolley is the dependent variable, and the trolley, the release line, the gap between the gates and the ramp surface are control variables.
Results: the time between the gates becomes shorter as the angle increases, so the calculated speed becomes larger.
Conclusion: the steeper the ramp, the greater the speed of the trolley over the timed section.
Watch out: a push at release gives the trolley extra speed, a trolley that drifts sideways can miss a beam, and a stopwatch is a poor choice here because each time is under one second.
Safety: stop the trolley with the foam box so it cannot run off the bench, keep fingers clear of the wheels and build the stack of blocks on a firm, level surface.

Common Mistake

Do not divide the time by the distance, because speed is the distance divided by the time.
Do not say that a longer time over the same distance means a greater speed, because it means a smaller speed.
Do not leave the unit off a speed, because a bare number such as $6.3$ does not state a speed.

Exam technique

Calculating speed from data

Write the equation, then the substitution, then the answer with its unit, because each step can earn a mark even if a later step goes wrong.
Take the distance and the time from the same part of the motion, such as the gap between two light gates and the time between them.
Use the mean time when a table gives repeated readings, leaving out any reading that clearly does not fit the others.
Round the final answer to the same number of significant figures as the least precise value in the data.

Self review

State the equation that links speed, distance and time.
Give the SI unit of speed.
Calculate the speed of a trolley that travels $m0.60\ \text{m}$ between two light gates in $s0.50\ \text{s}$ .
Name the independent, dependent and control variables in the ramp investigation.
Explain why a walker should start a few metres behind the start line.

Calculating speed

Definition

Speed

The distance travelled by an object per unit time.

Speed tells you how far an object travels in each unit of time, so a speed of $m/s3\ \text{m/s}$ means the object covers $m3\ \text{m}$ in every second.
Speed is calculated from a measured distance and a measured time using $taken\text{speed}=\dfrac{\text{distance travelled}}{\text{time taken}}$ .
In symbols this is $v=stv=\dfrac{s}{t}$ , where $v$ is the speed, $s$ is the distance travelled and $t$ is the time taken.
A distance in metres divided by a time in seconds gives a speed in metres per second ( $m/s\text{m/s}$ ), which is the SI unit of speed.
The distance and the time must belong to the same part of the motion, so the time used is the time taken to cover exactly the distance that was measured.
Over the same distance, a shorter time means a greater speed, so a pupil who runs $m100\ \text{m}$ in $s14\ \text{s}$ is faster than a pupil who takes $s18\ \text{s}$ .
In the same time, a longer distance means a greater speed, so a cyclist who rides $m150\ \text{m}$ in $s30\ \text{s}$ is faster than one who rides $m90\ \text{m}$ in $s30\ \text{s}$ .
Speed is found by dividing the distance by the time, and never the other way round, because time divided by distance gives the number of seconds needed for each metre.
A speed calculated from one distance and one time describes the whole stretch that was timed, so any speeding up or slowing down inside that stretch does not show in the answer.

Example

Speed of a runner

A pupil runs $m60\ \text{m}$ along a marked track in a mean time of $s9.6\ \text{s}$ .
Write the equation that links speed, distance and time:

v=\frac{s}{t}

Substitute the measured distance and time:

v=\frac{60\ \text{m}}{9.6\ \text{s}}=6.25\ \text{m/s}

The data are given to two significant figures, so the speed is $m/s6.3\ \text{m/s}$ .

Example

Speed of a trolley between two light gates

Two light gates are placed $m0.50\ \text{m}$ apart on a bench, and a trolley takes $s0.40\ \text{s}$ to travel from the first beam to the second.
Substitute the gap between the gates and the time between them into the equation:

v=\frac{0.50\ \text{m}}{0.40\ \text{s}}=1.25\ \text{m/s}

The trolley's speed between the two gates is $m/s1.3\ \text{m/s}$ to two significant figures.

Planning a speed investigation

Definition

Independent variable

The variable that is deliberately changed in an investigation.

Definition

Dependent variable

The variable that is measured in an investigation to find out how it responds to changes in the independent variable.

Definition

Control variable

A variable that is kept the same throughout an investigation so that it cannot affect the result.

Every speed measurement needs two readings, a distance measured with a suitable instrument and a time measured over exactly that distance.
The distance is marked out first, and the timing then starts and stops as the object passes the marks.
In an investigation, the independent variable is the one you change on purpose, such as the angle of a ramp.
The dependent variable is the one you measure to see the effect, such as the speed of a trolley.
The control variables are kept the same, such as the trolley, the timed distance and the release point, so that only the independent variable can change the result.
A longer timed distance gives a longer time, which makes the reaction-time error a smaller fraction of each reading.
Each speed is calculated from the mean time of repeated runs, which reduces the effect of random errors.
Results are recorded in a table with the independent variable in the first column and units in the column headings, not beside each number.
When the results are plotted, the independent variable goes on the horizontal axis and the dependent variable goes on the vertical axis.

Practical

Walking and running speeds

Aim: to measure the walking speed and the running speed of a person over a measured distance and compare them.
Apparatus: a $m30\ \text{m}$ tape measure, four traffic cones or playground chalk, a stopwatch, a results table and a pencil.
Method:
- Choose a flat, straight stretch of playground and lay the tape measure along the ground.
- Place a cone at the $m0\ \text{m}$ mark of the tape as the start line and another at the $m20\ \text{m}$ mark as the finish line.
- Ask the walker to begin a few metres behind the start line, so that they are already moving at a steady pace when they cross it.
- Stand to one side where you can see both lines, start the stopwatch as the walker's chest crosses the start line and stop it as their chest crosses the finish line.
- Record the time to $s0.1\ \text{s}$ , because reaction time makes the hundredths digit meaningless.
- Repeat the walk twice more so that there are three times for walking.
- Repeat the whole procedure three times with the same person running instead of walking.
- Calculate the mean time for each type of movement, leaving out any anomalous time, and use $v=stv=\dfrac{s}{t}$ with $ms=20\ \text{m}$ to find each speed.
Variables: the type of movement is the independent variable, the speed is the dependent variable, and the person, the distance and the surface are control variables.
Results: the running times are much shorter than the walking times over the same $m20\ \text{m}$ .
Conclusion: the distance is the same for both types of movement, so the one with the shorter mean time, running, has the greater speed.
Watch out: a walker who starts from a standstill on the start line is still speeding up during the first few metres, and reaction time affects both the start and the stop of every reading.
Safety: use a dry, level surface clear of obstacles, wear suitable footwear and leave plenty of space beyond the finish line for the runner to slow down.

Practical

Ramp angle and trolley speed

Aim: to investigate whether the angle of a ramp affects the speed of a trolley rolling down it.
Apparatus: a smooth wooden ramp about $m1\ \text{m}$ long, a stack of wooden blocks, a protractor, a dynamics trolley, a metre rule, masking tape, two light gates on clamp stands connected to an electronic timer or data logger, and a box of foam to catch the trolley.
Method:
- Rest one end of the ramp on the blocks so that it makes an angle of $10∘10^\circ$ with the bench, checking the angle with the protractor.
- Stick a strip of masking tape across the ramp near the top as the release line, so the trolley starts from the same point every time.
- Clamp the first light gate $cm20\ \text{cm}$ below the release line and the second light gate $cm50\ \text{cm}$ further down the ramp, measuring the gap between the beams with the metre rule.
- Place the foam box at the bottom of the ramp to stop the trolley.
- Hold the trolley with its front edge on the release line and let it go without pushing it.
- Record the time the trolley takes to travel from the first beam to the second.
- Repeat the run twice more at the same angle and calculate the mean time.
- Increase the angle to $15∘15^\circ$ , $20∘20^\circ$ , $25∘25^\circ$ and $30∘30^\circ$ in turn, keeping both light gates in the same place on the ramp.
- Calculate the speed at each angle using $v=stv=\dfrac{s}{t}$ with $ms=0.50\ \text{m}$ , then plot speed on the vertical axis against angle on the horizontal axis.
Variables: the angle of the ramp is the independent variable, the speed of the trolley is the dependent variable, and the trolley, the release line, the gap between the gates and the ramp surface are control variables.
Results: the time between the gates becomes shorter as the angle increases, so the calculated speed becomes larger.
Conclusion: the steeper the ramp, the greater the speed of the trolley over the timed section.
Watch out: a push at release gives the trolley extra speed, a trolley that drifts sideways can miss a beam, and a stopwatch is a poor choice here because each time is under one second.
Safety: stop the trolley with the foam box so it cannot run off the bench, keep fingers clear of the wheels and build the stack of blocks on a firm, level surface.

Common Mistake

Do not divide the time by the distance, because speed is the distance divided by the time.
Do not say that a longer time over the same distance means a greater speed, because it means a smaller speed.
Do not leave the unit off a speed, because a bare number such as $6.3$ does not state a speed.

Exam technique

Calculating speed from data

Write the equation, then the substitution, then the answer with its unit, because each step can earn a mark even if a later step goes wrong.
Take the distance and the time from the same part of the motion, such as the gap between two light gates and the time between them.
Use the mean time when a table gives repeated readings, leaving out any reading that clearly does not fit the others.
Round the final answer to the same number of significant figures as the least precise value in the data.

Self review

State the equation that links speed, distance and time.
Give the SI unit of speed.
Calculate the speed of a trolley that travels $m0.60\ \text{m}$ between two light gates in $s0.50\ \text{s}$ .
Name the independent, dependent and control variables in the ramp investigation.
Explain why a walker should start a few metres behind the start line.

2.1.2 Measuring speed

Calculating speed

Speed of a runner

Speed of a trolley between two light gates

Planning a speed investigation

Walking and running speeds

Ramp angle and trolley speed

Calculating speed from data

Review 2.1.2 Measuring speed by teaching Genie

2.1.2 Measuring speed

Calculating speed

Speed of a runner

Speed of a trolley between two light gates

Planning a speed investigation

Walking and running speeds

Ramp angle and trolley speed

Calculating speed from data

Review 2.1.2 Measuring speed by teaching Genie

2.1.2 Measuring speed

Calculating speed

Speed of a runner

Speed of a trolley between two light gates

Planning a speed investigation

Walking and running speeds

Ramp angle and trolley speed

Calculating speed from data

Review 2.1.2 Measuring speed by teaching Genie

2.1.2 Measuring speed Revision Guide

2.1.2 Measuring speed

Calculating speed

Speed of a runner

Speed of a trolley between two light gates

Planning a speed investigation

Walking and running speeds

Ramp angle and trolley speed

Calculating speed from data

Review 2.1.2 Measuring speed by teaching Genie

2.1.2 Measuring speed Revision Guide