Converting Units of Speed

Changing a speed from km/h to m/s and back, and into m/min, cm/s and km/min, by reasoning from what the speed means; comparing speeds given in different units; and converting other rates in the same way.

Before you start

Speed is a rate: the distance travelled in one unit of time. Speed = distance ÷ time, so a cyclist who rides 150 m in 30 s rides at $150 \div 30 = 5$ m/s: 5 metres in every second.

Units of length: 1 km = 1000 m and 1 m = 100 cm. Units of time: 1 h = 60 min and 1 min = 60 s, so 1 h = $60 \times 60 = 3600$ s.

A fraction is simplified by dividing its numerator and denominator by a common factor: $\frac{1000}{3600} = \frac{5}{18}$ , dividing both by 200.

Units of Speed

The unit of a speed is a unit of length per a unit of time. Which unit is used depends on what is moving. A car on the expressway is timed over hours and its distances are long, so its speed is given in kilometres per hour (km/h). A sprinter is timed in seconds over a short track, so her speed is given in metres per second (m/s). A walk may be given in metres per minute (m/min) and something slow, such as a robot vacuum cleaner, in centimetres per second (cm/s); the speed of something fast, such as an aeroplane, can also be written in kilometres per minute (km/min).

One speed can be written in any of these units, and to compare two speeds, or to use a speed with a distance and a time given in other units, it must often be changed from one unit to another. The table shows some speeds you meet every day in Singapore, each in two units. Remember a few of them: they are the sense check for every conversion in this lesson.

What is moving	About	The same speed
A person walking	5 km/h	about 1.4 m/s
A cyclist on a park connector	18 km/h	5 m/s
An MRT train	72 km/h	20 m/s
A car on the expressway	90 km/h	25 m/s
An aeroplane at cruising height	840 km/h	14 km/min

From km/h to m/s

A speed of 90 km/h means 90 km in every hour. To write it in m/s, ask: how many metres does the car go in one second? Change the distance to metres and the time to seconds, and divide.

Example 1: A car on the expressway

Mr Tan drives along the expressway at a constant speed of 90 km/h. Express his speed in m/s.

Solution

In 1 hour the car goes 90 km. Change both units: 90 km = $90 \times 1000 = 90\,000$ m, and 1 h = 3600 s.

So the car goes 90 000 m in 3600 s.

In 1 second it goes $90\,000 \div 3600 = 25$ m.

Check: at 90 km/h the car goes 90 km in 1 hour. At 25 m/s it goes 90 000 m in $90\,000 \div 25 = 3600$ s, which is also 1 hour. The two speeds take the same time over the same distance, so they are the same speed.

Mr Tan's speed is 25 m/s. So 90 km/h and 25 m/s are the same speed.

Why the number gets smaller

The speed is the same, but 90 has become 25. Two things happened to the number.

Changing kilometres into metres made it 1000 times larger: 90 km is 90 000 m, so the car goes 90 000 m per hour.

Changing "per hour" into "per second" made it 3600 times smaller: a second is $\frac{1}{3600}$ of an hour, so in one second the car goes only $\frac{1}{3600}$ of the distance it goes in an hour.

The number is multiplied by 1000 and divided by 3600. Since 3600 is larger than 1000, the result is smaller: it is the number divided by $3600 \div 1000 = 3.6$ . A speed in m/s is always a smaller number than the same speed in km/h.

Converting km/h to m/s

To convert a speed from km/h to m/s, multiply by 1000 (kilometres to metres) and divide by 3600 (per hour to per second).

Since $\frac{1000}{3600} = \frac{5}{18}$ , this is the same as multiplying by $\frac{5}{18}$ , or dividing by 3.6.

$1 \text{ km/h} = \frac{1000}{3600} \text{ m/s} = \frac{5}{18} \text{ m/s}$

Use the short cut once you can say why it works. Multiplying by $\frac{5}{18}$ is often easiest when the speed is a multiple of 18; dividing by 3.6 is easiest with a calculator.

Example 2: Using the short cut

A bus travels at 54 km/h. Express its speed in m/s.

Solution

To change km/h to m/s, multiply by $\frac{5}{18}$ .

$54 \times \frac{5}{18} = \frac{54}{18} \times 5 = 3 \times 5 = 15$

Or divide by 3.6: $54 \div 3.6 = 15$ .

Sense check: 15 is smaller than 54, as a speed in m/s must be.

The bus travels at 15 m/s.

Example 3: An answer that is not exact

A cheetah can run at 110 km/h for a short time. Express this speed in m/s, correct to 3 significant figures.

Solution

Multiply by $\frac{5}{18}$ : $110 \times \frac{5}{18} = \frac{550}{18} = \frac{275}{9} = 30\frac{5}{9}$

Or divide by 3.6: $110 \div 3.6 = 30.555...$

Round only at the end: $30.555... \approx 30.6$ (correct to 3 significant figures).

The cheetah runs at $30\frac{5}{9}$ m/s, which is about 30.6 m/s (correct to 3 significant figures).

From m/s to km/h

Going the other way, ask: how many kilometres does it go in one hour? A speed of 1 m/s means 1 m in every second. In one hour, which is 3600 seconds, that is 3600 m, which is 3.6 km. So 1 m/s is 3.6 km/h, and the number gets larger.

Converting m/s to km/h

To convert a speed from m/s to km/h, multiply by 3600 (per second to per hour) and divide by 1000 (metres to kilometres).

Since $\frac{3600}{1000} = \frac{18}{5} = 3.6$ , this is the same as multiplying by $\frac{18}{5}$ , or multiplying by 3.6.

$1 \text{ m/s} = \frac{3600}{1000} \text{ km/h} = \frac{18}{5} \text{ km/h} = 3.6 \text{ km/h}$

Example 4: From a distance and a time

Ravi runs 200 m in 20 s. Find his average speed (a) in m/s, (b) in km/h.

Solution

(a) Speed = distance ÷ time = $200 \div 20 = 10$ m/s.

(b) At 10 m/s, Ravi would go 10 m in every second. In 1 hour, which is 3600 s, he would go $10 \times 3600 = 36\,000$ m.

36 000 m = $36\,000 \div 1000 = 36$ km, so the speed is 36 km in 1 hour.

By the short cut: $10 \times 3.6 = 36$ , or $10 \times \frac{18}{5} = 36$ .

Sense check: 36 is larger than 10, as a speed in km/h must be. (Ravi cannot keep this up for an hour; the speed in km/h says how far he would go in an hour at the same rate.)

(a) Ravi's average speed is 10 m/s.
(b) This is 36 km/h.

Example 5: Two speeds in m/s

(a) A racing car travels at 50 m/s. Express its speed in km/h.
(b) A sprinter's average speed over 100 m is 10.4 m/s. Express this speed in km/h.

Solution

(a) Multiply by $\frac{18}{5}$ : $50 \times \frac{18}{5} = 10 \times 18 = 180$

(b) Multiply by 3.6: $10.4 \times 3.6 = 37.44$

From what the speed means: in 1 hour the sprinter would go $10.4 \times 3600 = 37\,440$ m, which is $37\,440 \div 1000 = 37.44$ km.

Both answers are exact, so they are not rounded.

(a) The racing car travels at 180 km/h.
(b) The sprinter's average speed is 37.44 km/h.

Other Units: m/min, cm/s and km/min

Every conversion of speed is made in the same way: change the unit of length and the unit of time one at a time, asking each time whether the number gets larger or smaller.

To change	Do this to the number	Because
km into m (per the same time)	$\times\ 1000$	each km is 1000 m, so there are more of them
m into cm (per the same time)	$\times\ 100$	each m is 100 cm
per hour into per minute	$\div\ 60$	in one minute you go $\frac{1}{60}$ of the distance you go in an hour
per minute into per second	$\div\ 60$	in one second you go $\frac{1}{60}$ of the distance you go in a minute
per hour into per second	$\div\ 3600$	in one second you go $\frac{1}{3600}$ of the distance you go in an hour

To go the other way (m into km, per second into per minute, and so on), do the opposite: divide where the table multiplies, and multiply where it divides.

Example 6: A walking speed in m/min and m/s

Mrs Goh walks to the hawker centre at 4.8 km/h. Express her speed (a) in m/min, (b) in m/s, correct to 3 significant figures.

Solution

(a) Change the length: 4.8 km = $4.8 \times 1000 = 4800$ m, so she walks 4800 m per hour.

Change the time: in 1 minute she walks $\frac{1}{60}$ of that, $4800 \div 60 = 80$ m. So 4.8 km/h = 80 m/min.

(b) In 1 second she walks $\frac{1}{60}$ of 80 m: $80 \div 60 = \frac{4}{3} = 1\frac{1}{3}$ m.

$1\frac{1}{3} = 1.333... \approx 1.33$ (correct to 3 significant figures).

Check with the short cut: $4.8 \div 3.6 = 1.333...$ , the same.

(a) Mrs Goh walks at 80 m/min.
(b) This is $1\frac{1}{3}$ m/s, about 1.33 m/s (correct to 3 significant figures).

Example 7: A slow speed in cm/s

A robot vacuum cleaner moves across the floor at 25 cm/s. Express its speed (a) in m/min, (b) in km/h.

Solution

(a) Change the length: 25 cm = $25 \div 100 = 0.25$ m, so it moves 0.25 m per second.

Change the time: in 1 minute (60 s) it moves 60 times as far, $0.25 \times 60 = 15$ m. So 25 cm/s = 15 m/min.

(b) In 1 hour (60 min) it moves $15 \times 60 = 900$ m, which is $900 \div 1000 = 0.9$ km.

Check with the short cut: 25 cm/s = 0.25 m/s, and $0.25 \times 3.6 = 0.9$ .

(a) The robot moves at 15 m/min.
(b) This is 0.9 km/h.

Example 8: A fast speed in km/min

An aeroplane flies at 840 km/h. Express its speed (a) in km/min, (b) in m/s, correct to 3 significant figures.

Solution

(a) The length stays in km. Change the time: in 1 minute the aeroplane flies $\frac{1}{60}$ of 840 km, $840 \div 60 = 14$ km. So 840 km/h = 14 km/min.

(b) From 14 km/min: 14 km = 14 000 m, flown in 60 s. In 1 second it flies $14\,000 \div 60 = 233.33...$ m.

Check with the short cut: $840 \times \frac{5}{18} = \frac{4200}{18} = 233\frac{1}{3}$ , the same.

$233.33... \approx 233$ (correct to 3 significant figures).

(a) The aeroplane flies at 14 km/min.
(b) This is $233\frac{1}{3}$ m/s, about 233 m/s (correct to 3 significant figures).

Comparing Speeds Given in Different Units

Speeds can be compared only when they are in the same unit: 18 km/h is not faster than 5 m/s, because they are the same speed. Choose one unit, convert every speed to it, then compare. Any unit gives the same order; choose the one that gives the easiest numbers.

Example 9: Three speeds in three units

On a park connector, Siti cycles at 18 km/h, Ravi runs at 4.5 m/s and Mei Ling skates at 280 m/min. Arrange the three speeds in order, fastest first.

Solution

Choose m/min: one speed is already in it, and the others become whole numbers.

Siti: 18 km/h = 18 000 m per hour. In 1 minute: $18\,000 \div 60 = 300$ m. So 300 m/min.

Ravi: 4.5 m/s means 4.5 m every second. In 1 minute (60 s): $4.5 \times 60 = 270$ m. So 270 m/min.

Mei Ling: 280 m/min.

Compare: 300 > 280 > 270.

Check in m/s: Siti $18 \div 3.6 = 5$ m/s; Mei Ling $280 \div 60 = 4\frac{2}{3}$ m/s; Ravi 4.5 m/s. The same order.

Fastest first: Siti (18 km/h), Mei Ling (280 m/min), Ravi (4.5 m/s).

Example 10: A race

Ali and Hui Min run a 200 m race, starting together. Ali runs at a constant speed of 18 km/h and Hui Min at a constant speed of 4.8 m/s.
(a) Who wins, and by how many seconds? Give the time difference correct to 3 significant figures.
(b) How far behind is the other runner when the winner finishes?

Solution

(a) The distance is in metres and the time is wanted in seconds, so use m/s.

Ali: $18 \times \frac{5}{18} = 5$ m/s. Hui Min: 4.8 m/s. Ali is faster, so Ali wins.

Ali's time = distance ÷ speed = $200 \div 5 = 40$ s.

Hui Min's time = $200 \div 4.8 = \frac{2000}{48} = \frac{125}{3} = 41\frac{2}{3}$ s.

Difference = $41\frac{2}{3} - 40 = 1\frac{2}{3}$ s $= 1.666... \approx 1.67$ s (correct to 3 significant figures).

(b) When Ali finishes, 40 s have passed. In 40 s Hui Min runs $4.8 \times 40 = 192$ m.

She is $200 - 192 = 8$ m from the finish.

(a) Ali wins, by $1\frac{2}{3}$ s, about 1.67 s (correct to 3 significant figures).
(b) Hui Min is 8 m behind when Ali finishes.

Converting Other Rates

Every rate is one quantity per unit of another, so every rate is converted in the same way as a speed: change one unit at a time and ask whether the number gets larger or smaller. A tap that runs 12 L in one minute runs 60 times as much in an hour; a price of 2.4 cents for one gram is 1000 times as much for a kilogram.

Example 11: Filling a tank

Water flows from a tap at 12 L/min. Express this rate (a) in L/h, (b) in mL/s.

Solution

(a) The volume stays in litres. In 1 hour (60 min) the tap runs 60 times as much as in 1 minute: $12 \times 60 = 720$ L. So 12 L/min = 720 L/h.

(b) Change the volume: 12 L = $12 \times 1000 = 12\,000$ mL, so the tap runs 12 000 mL per minute.

Change the time: in 1 second it runs $\frac{1}{60}$ of that, $12\,000 \div 60 = 200$ mL.

(a) The rate is 720 L/h.
(b) The rate is 200 mL/s.

Example 12: The better buy

At a market, Stall A sells cashew nuts at 2.4 cents per gram. Stall B sells the same nuts at $22 per kilogram. Which stall is cheaper, and by how much per kilogram?

Solution

Write Stall A's price per kilogram. 1 kg = 1000 g, so 1 kg costs 1000 times as much as 1 g: $2.4 \times 1000 = 2400$ cents.

2400 cents = $2400 \div 100 = 24$ dollars, so Stall A charges $24 per kg.

Compare: $22 per kg is less than $24 per kg. Difference: $24 − $22 = $2.

Check the other way: $22 per kg is 2200 cents for 1000 g, which is $2200 \div 1000 = 2.2$ cents per gram, less than 2.4 cents per gram.

Stall B is cheaper, by $2 per kilogram.

Common Mistakes

Multiplying by 3.6 when changing km/h to m/s. "18 km/h = $18 \times 3.6 = 64.8$ m/s" is wrong: 64.8 m/s is over 230 km/h, faster than any car on the expressway. A speed in m/s is a smaller number than the same speed in km/h, so km/h to m/s divides by 3.6 (18 km/h = 5 m/s), and m/s to km/h multiplies by 3.6.
Changing only one unit. 90 km/h is 90 000 m per hour, not 90 000 m/s; and 72 km/h is 1.2 km per minute, not 1.2 m/s. Change the length and the time, and write the unit after each step.
Dividing by 60 only once for hours into seconds. An hour is $60 \times 60 = 3600$ seconds, not 60 seconds.
Rounding too early. $110 \div 3.6 = 30.555...$ ; if 30.6 is used in later working, $30.6 \times 3.6 = 110.16$ , not 110. Keep the exact value ( $30\frac{5}{9}$ ) and round only the final answer.
Not checking the answer for sense. A person walks at about 1.4 m/s (5 km/h), a cyclist at about 5 m/s (18 km/h), a car on the expressway at about 25 m/s (90 km/h). An answer far from these for such a thing has gone the wrong way.

Practice

Try each question before you open its solution. The questions get harder as you go. Give answers that are not exact correct to 3 significant figures.

Question 1

Convert (a) 72 km/h to m/s, (b) 25 m/s to km/h.

ShowHide solution

(a) $72 \times \frac{5}{18} = \frac{72}{18} \times 5 = 4 \times 5 = 20$ , or $72 \div 3.6 = 20$ .

(b) $25 \times 3.6 = 90$ , or $25 \times \frac{18}{5} = 5 \times 18 = 90$ .

Sense check: 20 is smaller than 72, and 90 is larger than 25.

(a) 72 km/h = 20 m/s
(b) 25 m/s = 90 km/h

Question 2

Convert 100 km/h to m/s from what the speed means, showing each unit. Give the answer as a mixed number and correct to 3 significant figures.

ShowHide solution

100 km in 1 hour is $100 \times 1000 = 100\,000$ m in 3600 s.

In 1 second: $\frac{100\,000}{3600} = \frac{1000}{36} = \frac{250}{9} = 27\frac{7}{9}$ m.

$27\frac{7}{9} = 27.777... \approx 27.8$ (correct to 3 significant figures).

100 km/h = $27\frac{7}{9}$ m/s, about 27.8 m/s.

Question 3

Siti cycles at 18 km/h. Her brother writes: "18 km/h = $18 \times 3.6 = 64.8$ m/s." Explain how you can tell that his answer is wrong without working it out, and give the correct speed in m/s.

ShowHide solution

The same speed in m/s is a smaller number than in km/h, because a second is 3600 times shorter than an hour while a metre is only 1000 times shorter than a kilometre. His answer, 64.8, is larger than 18.

Also, 64.8 m/s is far faster than a car on the expressway (about 25 m/s), which no cyclist can ride.

Correct working: $18 \div 3.6 = 5$ , or $18 \times \frac{5}{18} = 5$ .

The number in m/s must be smaller than 18, so he should have divided by 3.6. Siti's speed is 5 m/s.

Question 4

Convert (a) 72 m/min to m/s, (b) 5 cm/s to m/min.

ShowHide solution

(a) 72 m in 1 minute is 72 m in 60 s. In 1 second: $72 \div 60 = 1.2$ m.

(b) In 1 minute (60 s): $5 \times 60 = 300$ cm. Change the length: 300 cm = $300 \div 100 = 3$ m.

(a) 72 m/min = 1.2 m/s
(b) 5 cm/s = 3 m/min

Question 5

An MRT train travels at 66 km/h. Express its speed (a) in km/min, (b) in m/s.

ShowHide solution

(a) In 1 minute the train goes $\frac{1}{60}$ of 66 km: $66 \div 60 = 1.1$ km.

(b) 1.1 km = 1100 m, in 60 s. In 1 second: $1100 \div 60 = \frac{55}{3} = 18\frac{1}{3}$ m.

Check: $66 \div 3.6 = 18.333...$ , the same.

$18.333... \approx 18.3$ (correct to 3 significant figures).

(a) 1.1 km/min
(b) $18\frac{1}{3}$ m/s, about 18.3 m/s

Question 6

A bus travels at a constant speed of 54 km/h.
(a) How many metres does it travel in 1 minute?
(b) How many metres does it travel in 1 second?
(c) How many seconds does it take to travel 450 m between two bus stops?

ShowHide solution

(a) 54 km/h is 54 000 m per hour. In 1 minute: $54\,000 \div 60 = 900$ m.

(b) In 1 second: $900 \div 60 = 15$ m.

(a) 900 m
(b) 15 m
(c) 30 seconds

Question 7

Hui Min runs 1000 m in 3 minutes 20 seconds. Mei Ling runs at half of Hui Min's average speed. Find Mei Ling's average speed in km/h.

ShowHide solution

Hui Min's time: 3 min 20 s = $3 \times 60 + 20 = 200$ s.

Hui Min's average speed = $1000 \div 200 = 5$ m/s.

Mei Ling's average speed = $5 \div 2 = 2.5$ m/s.

In km/h: $2.5 \times 3.6 = 9$ .

Mei Ling's average speed is 9 km/h.

Question 8

Mr Tan and his son leave home together for the same shop. Mr Tan walks at 1.75 m/s and his son cycles at 17.5 km/h. Who reaches the shop first?

ShowHide solution

Write both speeds in km/h. Mr Tan: $1.75 \times 3.6 = 6.3$ km/h.

His son: 17.5 km/h.

17.5 km/h is faster than 6.3 km/h. They cover the same distance, so the faster one takes less time.

His son reaches the shop first.

Question 9

A tap that has not been turned off properly lets out water at 4 mL/s. (a) Express this rate in L/h. (b) How many litres of water are lost in one day?

ShowHide solution

(a) In 1 hour (3600 s): $4 \times 3600 = 14\,400$ mL.

Change the volume: 14 400 mL = $14\,400 \div 1000 = 14.4$ L. So 4 mL/s = 14.4 L/h.

(b) One day is 24 hours: $14.4 \times 24 = 345.6$ L.

(a) 14.4 L/h
(b) 345.6 L of water is lost in one day.

Question 10

At one fruit stall, grapes cost $0.90 per 100 g. At another, the same grapes cost $8.50 per kg. Which stall is cheaper, and by how much per kilogram?

ShowHide solution

1 kg = 1000 g, which is 10 lots of 100 g.

At the first stall, 1 kg costs $10 \times 0.90 = 9$ dollars, so $9.00 per kg.

Compare: $8.50 per kg is less than $9.00 per kg. Difference: $9.00 − $8.50 = $0.50.

The second stall is cheaper, by 50 cents per kilogram.

Question 11

On the expressway, Mrs Goh drives at a constant 81 km/h. A taxi in the next lane travels at a constant 23 m/s, and at one moment the two are side by side.
(a) Which is faster, and by how many m/s?
(b) How far ahead is the faster one after 1 minute?

ShowHide solution

(a) Mrs Goh: $81 \div 3.6 = 22.5$ m/s. The taxi: 23 m/s.

The taxi is faster, by $23 - 22.5 = 0.5$ m/s.

(b) Each second the taxi goes 0.5 m further than Mrs Goh's car. In 1 minute (60 s): $0.5 \times 60 = 30$ m.

Check: in 60 s the taxi goes $23 \times 60 = 1380$ m and Mrs Goh's car $22.5 \times 60 = 1350$ m; $1380 - 1350 = 30$ m.

(a) The taxi is faster, by 0.5 m/s.
(b) The taxi is 30 m ahead after 1 minute.

Question 12

Ali walks the 1.5 km from home to school at a constant 0.075 km/min.
(a) Express his speed in m/s.
(b) How long does he usually take?
(c) One day he leaves home 5 minutes later than usual. At what constant speed, in km/h, must he walk to reach school at his usual time?

ShowHide solution

(a) 0.075 km = $0.075 \times 1000 = 75$ m, so he walks 75 m per minute. In 1 second: $75 \div 60 = 1.25$ m.

(b) Time = distance ÷ speed. In metres and minutes: 1.5 km = 1500 m, and $1500 \div 75 = 20$ min.

Speed = distance ÷ time = $1.5 \div \frac{1}{4} = 1.5 \times 4 = 6$ km/h.

Check: at 6 km/h he walks 6000 m in 60 min, so 100 m per minute, and $1500 \div 100 = 15$ min.

(a) 1.25 m/s
(b) 20 minutes
(c) He must walk at 6 km/h.

Key Points

A unit of speed is a unit of length per a unit of time: km/h, m/s, m/min, cm/s, km/min.
To convert a speed, change the unit of length and the unit of time one at a time, and ask each time whether the number gets larger or smaller.
km/h to m/s: multiply by 1000 and divide by 3600, which is $\times\ \frac{5}{18}$ or $\div\ 3.6$ .
m/s to km/h: multiply by 3600 and divide by 1000, which is $\times\ \frac{18}{5}$ or $\times\ 3.6$ .
The same speed is a smaller number in m/s than in km/h, because a second is 3600 times shorter than an hour while a metre is only 1000 times shorter than a kilometre.
To compare speeds, convert them all to one unit first.
Other rates (L/min, mL/s, cents per gram, dollars per kilogram) are converted in the same way.
Sense check: walking about 1.4 m/s (5 km/h), cycling about 5 m/s (18 km/h), a car on the expressway about 25 m/s (90 km/h). Keep exact values and round only at the end.