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Meters per second means meters in one second, and that sentence contains the whole calculation.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will work out a speed from a distance and a time, find a distance or a time when you are given the speed, and compare two journeys fairly by turning both into meters per second.
From Grade 3 you can read a scale — which is how you get a time off a stopwatch — and convert between millimeters, centimeters, meters and kilometers, and between seconds, minutes and hours. Both are used here: a measurement has to be in meters and seconds before it can become a speed in meters per second. You can also already divide.
And you know what fast and slow feel like. A car on the highway is faster than a bike, and a bike is faster than a walker. This lesson puts a number on that feeling, so that two speeds can be compared exactly.
| Term | What it means |
|---|---|
| Speed | How far something goes in a given amount of time. |
| Meters per second (m/s) | How many meters it covers in one second. |
| Per | In each. It is the reason the calculation is a division. |
| Kilometers per hour (km/h) | The same idea in bigger units: kilometers in each hour. |
| Miles per hour (mph) | The speed unit on American road signs: miles in each hour. |
| Steady speed | A speed that does not change during the trip. |
| Stopwatch | A clock you start and stop to time how long something takes. |
Speed answers how far in a given time. To compare two journeys fairly you have to put them on the same footing, and the footing everyone uses is one second.
So the speed is the distance shared out over the seconds:
$$\text{speed} = \frac{\text{distance}}{\text{time}}$$
A runner who covers $150$ m in $30$ s is going $150 \div 30 = 5$ m/s — five meters in each second.
The same sentence, rearranged, answers the other two questions. Five meters in each second for $30$ seconds is $5 \times 30 = 150$ m. And $150$ m at five meters a second takes $150 \div 5 = 30$ s.
The unit carries the method. Meters per second has meters first and seconds second, and the division goes the same way the words do: meters divided by seconds.
Another way: action
Walk across a room while somebody counts seconds out loud. Now walk it again twice as fast. The room did not change length; what changed is how much of it you got through in each count.
Another way: steps
There are only three, and which one you have is decided by which number is missing.
| You are given | You want | What to do |
|---|---|---|
| distance and time | speed | divide distance by time |
| speed and time | distance | multiply |
| distance and speed | time | divide distance by speed |
The only one worth memorizing is the first, because the other two are just it read backward. If $150 \div 30 = 5$, then $5 \times 30 = 150$ and $150 \div 5 = 30$ — the same three numbers, arranged the three ways.
And before any of it: check the units match. A speed in meters per second with a time in minutes is a wrong answer waiting to happen, and the fix is to turn the minutes into seconds first.
You can measure a speed with two tools you already know: a tape measure and a stopwatch.
Timing by hand is never perfect, because the person with the stopwatch reacts a little late at each end. That is why scientists time the same run several times, as you will do in the last unit of this course, and why a longer track gives a better answer: the reaction time is a smaller part of a longer time.
A few speeds are worth knowing, so you can check whether an answer makes sense.
| Moving thing | About | In km/h |
|---|---|---|
| a garden snail | 0.01 m/s | 0.036 |
| a person walking | 1.5 m/s | 5 |
| a person running | 5 m/s | 18 |
| a car in town | 11 m/s | 40 |
| a car on the highway | 30 m/s | 108 |
If you work out that a walker is going $50$ m/s, something has gone wrong, because that is faster than a car on the highway. If a car comes out at $0.3$ m/s, it is slower than a walker. Comparing with a speed you can picture catches a division done the wrong way round.
In the United States, road signs and car speedometers show miles per hour, written mph. In most other countries they show kilometers per hour. Scientists use meters per second. All three measure the same thing: how far in a given time.
A speed limit of $30$ mph is about $48$ km/h, and about $13$ m/s. The numbers are different because the distance units and the time units are different, not because the speed changed. Whatever the units, the method is the same: the distance divided by the time, with the distance unit first and the time unit second.
When you compare two speeds, make sure they are in the same unit first. $20$ m/s is faster than $50$ km/h, even though $20$ is the smaller number.
Real journeys are rarely steady. A car stops at lights, speeds up, and slows down for corners. When a question gives the whole distance and the whole time of a journey like that, dividing them gives the average speed: the steady speed that would have covered the same distance in the same time.
Suppose a family drives $90$ km to visit a lake, and the trip takes $2$ hours including a stop for lunch. The average speed is $90 \div 2 = 45$ km/h. On the highway they went much faster than that, and while eating lunch their speed was zero. The average smooths all of that into one number.
The average speed is still useful. It tells you how long a similar trip will take next time. It just does not tell you the speed at any one moment. For that, you would need the distance and time of a short piece of the journey, which is what a car's speedometer works out many times every second.
Every speed answer should pass three checks.
The first check catches arithmetic slips. The second catches dividing the wrong way round, which usually gives an answer that is far too big or far too small. The third catches a speed given in the wrong units, which is one of the most common mistakes in all of physics, even for grown-up scientists.
At field day, a class times a $50$ m dash. One runner crosses the line in $10$ s, so her speed is $50 \div 10 = 5$ m/s. Another runner takes $12.5$ s, so his speed is $50 \div 12.5 = 4$ m/s. The first runner covers one more meter in every second.
The teacher also times a $400$ m run around the track. The first runner finishes in $100$ s, which is $400 \div 100 = 4$ m/s. She ran slower over the long race than over the dash, which is what happens to every runner: nobody can keep up their fastest speed for long.
Comparing the dash and the long run by time alone would be meaningless, because the distances were different. Turning both into meters per second puts them on the same footing, which is exactly what speed is for.
Scientists from the U.S. Geological Survey measure how fast rivers flow, to warn towns about floods. One simple way is to drop a floating orange into the river and time how long it takes to drift between two markers on the bank.
Suppose the markers are $30$ m apart and the orange takes $20$ s. The water's speed at the surface is $30 \div 20 = 1.5$ m/s, about as fast as a person walks. After heavy rain, the same orange might take only $6$ s, a speed of $30 \div 6 = 5$ m/s — faster than most people can run, and dangerous to wade into.
The scientists repeat the drop several times and take a best estimate, just as you will learn to do at the end of this course. Then they use the speed to work out how much water is flowing past each second, which tells them whether the river will rise over its banks.
Asked which of two journeys was faster, most people first look at the distances. A bus that went 60 km surely beat a cyclist who went 20 km.
Not if the bus took four hours and the cyclist took one. The bus did 15 km in each hour and the cyclist did 20. Neither number alone means anything; speed is made of both, and until you have divided you do not know.
The other slip is dividing the wrong way round — time by distance instead of distance by time. The check against it is the unit. Meters per second has meters on top and seconds underneath, in that order, and the division goes the same way the words do. Say meters per second out loud before every speed calculation, and the order of the division comes with it.
Read the distance and the time.
$150\ \text{m in } 30\ \text{s}$
Both are already in meters and seconds.
Decide which one is missing.
$\text{the speed}$
Distance and time are given.
Share the meters over the seconds.
$150 \div 30 = 5$
Distance divided by time.
Write the speed with its unit.
$5\ \text{m/s}$
Five meters in each second.
Check by multiplying back.
$5 \times 30 = 150\ \text{m}$
Five meters, thirty times over, is the whole distance.
Read the distance and the time.
$9\ \text{m in } 90\ \text{s}$
A small distance over a long time.
Decide which one is missing.
$\text{the speed}$
Distance and time are given.
Share the meters over the seconds.
$9 \div 90 = 0.1$
A small number over a big one.
Write the speed with its unit.
$0.1\ \text{m/s}$
A tenth of a meter in each second: a speed can be less than one.
Check by multiplying back.
$0.1 \times 90 = 9\ \text{m}$
It agrees.
Check it against a picture.
$0.1\ \text{m/s} < 1.5\ \text{m/s walking}$
Much slower than a walker, as a tortoise should be.
Read the distance and the time.
$6\ \text{km in } 10\ \text{min}$
The units do not match meters per second.
Convert the distance to meters.
$6 \times 1000 = 6000\ \text{m}$
A kilometer is a thousand meters.
Convert the time to seconds.
$10 \times 60 = 600\ \text{s}$
A minute is sixty seconds.
Share the meters over the seconds.
$6000 \div 600 = 10$
Distance divided by time.
Write the speed with its unit.
$10\ \text{m/s}$
Ten meters in each second.
Check it against a picture.
$10\ \text{m/s} \approx \text{a car in town}$
A sensible speed for a bus on town streets.
Find the time for a longer route.
$9000 \div 10 = 900\ \text{s} = 15\ \text{min}$
Distance divided by speed gives a time.
Decide which one is missing.
$\text{the speed}$
Distance and time are given.
Share the meters over the seconds.
Write the speed with its unit.
a walker covers $100$ m in $80$ s. What is the speed?
Answer: unit: m/s / km/h
Complete the worked solution: a walker covers $36$ m in $18$ s at a steady speed. Find the speed, how far the walker goes in a whole minute, and how long $108$ m would take.
Find the speed.
$36 \div 18 =$ a m/s
Meters shared over seconds.
Find the distance in a minute.
$\text{speed} \times 60 =$ b m
A minute is sixty seconds, each adding the same distance.
Find the time for the longer walk.
$108 \div \text{speed} =$ c s
Three times as far takes three times as long at the same speed.
a walker goes $100$ m in $80$ s. a cyclist goes $150$ m in $30$ s. Which is going faster?
a rowing boat keeps up $1.5$ m/s for $30$ s. How far is that?
Answer: unit: m / cm / km / mm
Three journeys, each with its distance and its time. Work out the speed of each, in meters per second.
| distance (m) | time (s) | speed (m/s) | |
|---|---|---|---|
| a walker | 100 | 80 | |
| a tortoise | 9 | 90 | |
| a skateboarder | 200 | 25 |
a walker travels at $1.25$ m/s. How long does it take to cover $100$ m?
Answer: unit: h / s / min
a ferry covers $45$ km in $3$ hours. What is its speed in kilometers per hour?
Answer: unit: m/s / km/h
A cyclist rides $0.9$ km along a bike trail in $3$ minutes at a steady speed. What is her speed in meters per second?
Answer: unit: m/s / km/h
A cable car at a ski resort crosses a valley at a steady $4$ m/s. How far does it travel in $6$ minutes? Answer in meters.
Answer: unit: m / cm / km / mm
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Three journeys, each with its distance and its time. Work out the speed of each, in meters per second.
| distance (m) | time (s) | speed (m/s) | |
|---|---|---|---|
| a sprinter | 180 | 20 | |
| a rowing boat | 45 | 30 | |
| a cyclist | 150 | 30 |
You can find a speed, a distance or a time when you have the other two. Tell someone why going further does not always mean going faster.
17. Your turn: 240 m in 120 s. What is the speed?, step 2
$240 \div 120 = 2$
Meters on top, seconds underneath.
17. Your turn: 240 m in 120 s. What is the speed?, step 3
$2\ \text{m/s}$
Two meters in each second.