Non-uniform motion

Definition

Non-uniform motion

Motion in which the speed of an object changes during its journey.

Definition

Instantaneous speed

The speed of an object at one particular moment.

An object moving at a constant speed covers equal distances in equal times, so one value of speed describes its whole journey.
Most real journeys are non-uniform, because the speed changes as the object speeds up, slows down and stops.
A bus on a town route is a typical example, because it pulls away from stops, slows at junctions, waits at traffic lights and picks up speed again on clear stretches of road.
The instantaneous speed is the speed at one particular moment, which is the value a car's speedometer shows.
For non-uniform motion the instantaneous speed changes from moment to moment, so no single reading of it describes the whole journey.
The average speed gives one value for the whole journey, and it equals the steady speed that would cover the same total distance in the same total time.
The average speed is lower than the highest instantaneous speed reached and higher than the lowest, because the fast and slow parts of the journey are combined into one value.

Calculating average speed

Definition

Average speed

The total distance travelled by an object divided by the total time taken.

Average speed is calculated using $taken\text{average speed}=\dfrac{\text{total distance travelled}}{\text{total time taken}}$ .
For a journey made in stages, the distances of all the stages are added to give the total distance.
The times of all the stages are added to give the total time, including every period when the object was stationary.
A stop is included because the journey is still taking time while the object is not moving, and leaving it out makes the average speed too large.
All the distances must be in the same unit and all the times must be in the same unit before they are added.
The average speed is not the mean of the stage speeds, unless every stage lasts for exactly the same time.
Over equal distances, a slow stage lasts longer than a fast stage, so the slow stage pulls the average speed down more than a simple mean of the speeds suggests.
On a distance-time graph, the total distance and the total time are read from the start and end points of the line, and the average speed is the gradient of a straight line joining those two points.

Example

Average speed for a journey with a stop

A cyclist rides $m2400\ \text{m}$ in $s300\ \text{s}$ , waits at traffic lights for $s60\ \text{s}$ and then rides another $m1200\ \text{m}$ in $s240\ \text{s}$ .
Add the distances to find the total distance:

2400\ \text{m}+1200\ \text{m}=3600\ \text{m}

Add the times, including the wait, to find the total time:

300\ \text{s}+60\ \text{s}+240\ \text{s}=600\ \text{s}

Divide the total distance by the total time:

\text{average speed}=\frac{3600\ \text{m}}{600\ \text{s}}=6.0\ \text{m/s}

Leaving out the wait would give $m/s\dfrac{3600\ \text{m}}{540\ \text{s}}=6.7\ \text{m/s}$ , which is too high.

Example

Why stage speeds cannot be averaged

A walker covers $m1200\ \text{m}$ uphill at $m/s1.0\ \text{m/s}$ and then the same $m1200\ \text{m}$ back downhill at $m/s2.0\ \text{m/s}$ .
The time for each stage is its distance divided by its speed:

t_1=\frac{1200\ \text{m}}{1.0\ \text{m/s}}=1200\ \text{s}\qquad t_2=\frac{1200\ \text{m}}{2.0\ \text{m/s}}=600\ \text{s}

The average speed uses the totals:

\text{average speed}=\frac{2400\ \text{m}}{1800\ \text{s}}=1.3\ \text{m/s}

The mean of the two stage speeds would be $m/s1.5\ \text{m/s}$ , which is wrong because the walker spends twice as long on the slow uphill stage.

Example

Average speed with mixed units

A train travels $km18\ \text{km}$ in $min12\ \text{min}$ , stops at a station for $min3\ \text{min}$ and then travels $km9\ \text{km}$ in $min5\ \text{min}$ .
Convert the total distance into metres:

18\ \text{km}+9\ \text{km}=27\ \text{km}=27\,000\ \text{m}

Convert the total time, including the stop, into seconds:

12\ \text{min}+3\ \text{min}+5\ \text{min}=20\ \text{min}=1200\ \text{s}

Divide the total distance by the total time:

\text{average speed}=\frac{27\,000\ \text{m}}{1200\ \text{s}}=22.5\ \text{m/s}

Common Mistake

Do not leave stationary time out of the total time, because the clock keeps running while the object waits.
Do not find the mean of the stage speeds, because stages that last for different times do not count equally.
Do not add a time in minutes to a time in seconds, because the totals only make sense when every value is in the same unit.

Exam technique

Setting out an average speed calculation

Write the total distance and the total time as separate lines of working before you divide, so each total can earn credit on its own.
Read every stage from the table or graph, including any flat section of a distance-time graph, because the flat sections are the stops.
Check that your answer lies between the slowest and the fastest stage speeds, since an answer outside that range shows a mistake.

Self review

State the equation used to calculate average speed.
Explain why a speedometer reading is usually different from the average speed for a journey.
Explain why time spent stationary is included when calculating average speed.
A bus travels $m1800\ \text{m}$ in $s240\ \text{s}$ , stops for $s60\ \text{s}$ and then travels $m600\ \text{m}$ in $s100\ \text{s}$ , so calculate its average speed.
Explain why the average speed of a journey is not usually the mean of the stage speeds.

Non-uniform motion

Definition

Non-uniform motion

Motion in which the speed of an object changes during its journey.

Definition

Instantaneous speed

The speed of an object at one particular moment.

An object moving at a constant speed covers equal distances in equal times, so one value of speed describes its whole journey.
Most real journeys are non-uniform, because the speed changes as the object speeds up, slows down and stops.
A bus on a town route is a typical example, because it pulls away from stops, slows at junctions, waits at traffic lights and picks up speed again on clear stretches of road.
The instantaneous speed is the speed at one particular moment, which is the value a car's speedometer shows.
For non-uniform motion the instantaneous speed changes from moment to moment, so no single reading of it describes the whole journey.
The average speed gives one value for the whole journey, and it equals the steady speed that would cover the same total distance in the same total time.
The average speed is lower than the highest instantaneous speed reached and higher than the lowest, because the fast and slow parts of the journey are combined into one value.

Calculating average speed

Definition

Average speed

The total distance travelled by an object divided by the total time taken.

Average speed is calculated using $taken\text{average speed}=\dfrac{\text{total distance travelled}}{\text{total time taken}}$ .
For a journey made in stages, the distances of all the stages are added to give the total distance.
The times of all the stages are added to give the total time, including every period when the object was stationary.
A stop is included because the journey is still taking time while the object is not moving, and leaving it out makes the average speed too large.
All the distances must be in the same unit and all the times must be in the same unit before they are added.
The average speed is not the mean of the stage speeds, unless every stage lasts for exactly the same time.
Over equal distances, a slow stage lasts longer than a fast stage, so the slow stage pulls the average speed down more than a simple mean of the speeds suggests.
On a distance-time graph, the total distance and the total time are read from the start and end points of the line, and the average speed is the gradient of a straight line joining those two points.

Example

Average speed for a journey with a stop

A cyclist rides $m2400\ \text{m}$ in $s300\ \text{s}$ , waits at traffic lights for $s60\ \text{s}$ and then rides another $m1200\ \text{m}$ in $s240\ \text{s}$ .
Add the distances to find the total distance:

2400\ \text{m}+1200\ \text{m}=3600\ \text{m}

Add the times, including the wait, to find the total time:

300\ \text{s}+60\ \text{s}+240\ \text{s}=600\ \text{s}

Divide the total distance by the total time:

\text{average speed}=\frac{3600\ \text{m}}{600\ \text{s}}=6.0\ \text{m/s}

Leaving out the wait would give $m/s\dfrac{3600\ \text{m}}{540\ \text{s}}=6.7\ \text{m/s}$ , which is too high.

Example

Why stage speeds cannot be averaged

A walker covers $m1200\ \text{m}$ uphill at $m/s1.0\ \text{m/s}$ and then the same $m1200\ \text{m}$ back downhill at $m/s2.0\ \text{m/s}$ .
The time for each stage is its distance divided by its speed:

t_1=\frac{1200\ \text{m}}{1.0\ \text{m/s}}=1200\ \text{s}\qquad t_2=\frac{1200\ \text{m}}{2.0\ \text{m/s}}=600\ \text{s}

The average speed uses the totals:

\text{average speed}=\frac{2400\ \text{m}}{1800\ \text{s}}=1.3\ \text{m/s}

The mean of the two stage speeds would be $m/s1.5\ \text{m/s}$ , which is wrong because the walker spends twice as long on the slow uphill stage.

Example

Average speed with mixed units

A train travels $km18\ \text{km}$ in $min12\ \text{min}$ , stops at a station for $min3\ \text{min}$ and then travels $km9\ \text{km}$ in $min5\ \text{min}$ .
Convert the total distance into metres:

18\ \text{km}+9\ \text{km}=27\ \text{km}=27\,000\ \text{m}

Convert the total time, including the stop, into seconds:

12\ \text{min}+3\ \text{min}+5\ \text{min}=20\ \text{min}=1200\ \text{s}

Divide the total distance by the total time:

\text{average speed}=\frac{27\,000\ \text{m}}{1200\ \text{s}}=22.5\ \text{m/s}

Common Mistake

Do not leave stationary time out of the total time, because the clock keeps running while the object waits.
Do not find the mean of the stage speeds, because stages that last for different times do not count equally.
Do not add a time in minutes to a time in seconds, because the totals only make sense when every value is in the same unit.

Exam technique

Setting out an average speed calculation

Write the total distance and the total time as separate lines of working before you divide, so each total can earn credit on its own.
Read every stage from the table or graph, including any flat section of a distance-time graph, because the flat sections are the stops.
Check that your answer lies between the slowest and the fastest stage speeds, since an answer outside that range shows a mistake.

Self review

State the equation used to calculate average speed.
Explain why a speedometer reading is usually different from the average speed for a journey.
Explain why time spent stationary is included when calculating average speed.
A bus travels $m1800\ \text{m}$ in $s240\ \text{s}$ , stops for $s60\ \text{s}$ and then travels $m600\ \text{m}$ in $s100\ \text{s}$ , so calculate its average speed.
Explain why the average speed of a journey is not usually the mean of the stage speeds.

2.1.7 Average speed for non-uniform motion

Non-uniform motion

Calculating average speed

Average speed for a journey with a stop

Why stage speeds cannot be averaged

Average speed with mixed units

Setting out an average speed calculation

Review 2.1.7 Average speed for non-uniform motion by teaching Genie

2.1.7 Average speed for non-uniform motion

Non-uniform motion

Calculating average speed

Average speed for a journey with a stop

Why stage speeds cannot be averaged

Average speed with mixed units

Setting out an average speed calculation

Review 2.1.7 Average speed for non-uniform motion by teaching Genie

2.1.7 Average speed for non-uniform motion

Non-uniform motion

Calculating average speed

Average speed for a journey with a stop

Why stage speeds cannot be averaged

Average speed with mixed units

Setting out an average speed calculation

Review 2.1.7 Average speed for non-uniform motion by teaching Genie

2.1.7 Average speed for non-uniform motion Revision Guide

2.1.7 Average speed for non-uniform motion

Non-uniform motion

Calculating average speed

Average speed for a journey with a stop

Why stage speeds cannot be averaged

Average speed with mixed units

Setting out an average speed calculation

Review 2.1.7 Average speed for non-uniform motion by teaching Genie

2.1.7 Average speed for non-uniform motion Revision Guide