Compound measures: speed, density and pressure

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Lesson notes

What are compound measures?

A compound measure combines two different units, such as distance and time, or mass and volume.
Examples include speed (distance per time), density (mass per volume) and pressure (force per area).
The units of a compound measure are written as a division, for example m/s, km/h, g/cm³ or N/m².
Compound measures are rates: they tell you how much of one quantity corresponds to one unit of another.

Speed

Speed is the rate at which distance changes with time.
The formula is v = \frac{d}{t} where v = speed, d = distance, t = time.
The SI unit of speed is the metre per second (m/s).
Other common units are kilometres per hour (km/h) and miles per hour (mph).
Average speed is total distance divided by total time; it does not show changes in speed during the journey.
Instantaneous speed is the speed at a particular moment, like the reading on a car speedometer.

Density

Density is the mass of a substance per unit volume.
The formula is \rho = \frac{m}{V} where \rho = density, m = mass, V = volume.
Common units are g/cm³ for solids and liquids, and kg/m³ for gases.
A denser material has more mass packed into the same volume.
For example, if a block has a mass of 200 g and a volume of 25 cm³, its density is 8 g/cm³.

Pressure

Pressure is the force acting per unit area.
The formula is P = \frac{F}{A} where P = pressure, F = force, A = area.
The SI unit of pressure is the pascal (Pa), which is one newton per square metre (N/m²).
For the same force, a smaller area gives a larger pressure.
For example, if a force of 100 N acts on an area of 2 m², the pressure is 50 Pa.

Rearranging the formulae

To find distance from speed and time: d = v \times t.
To find time from distance and speed: t = \frac{d}{v}.
To find mass from density and volume: m = \rho \times V.
To find volume from mass and density: V = \frac{m}{\rho}.
To find force from pressure and area: F = P \times A.
To find area from force and pressure: A = \frac{F}{P}.

Converting between units

To convert km/h to m/s, divide by 3.6 (or multiply by \frac{1000}{3600}).
To convert m/s to km/h, multiply by 3.6.
Example: 72 km/h = 72 ÷ 3.6 = 20 m/s.
Example: 15 m/s = 15 × 3.6 = 54 km/h.
When converting, make sure the units of distance and time match before you calculate.

Unit pricing and rates of pay

Unit pricing compares the cost of items by finding the price per unit, such as price per 100 g or per litre.
To find the unit price, divide the total cost by the quantity.
Example: 500 g of cheese costs £4.50, so the unit price is £4.50 ÷ 5 = £0.90 per 100 g.
Rates of pay are compound measures, for example pounds per hour (£/h).
To find total pay, multiply the hourly rate by the number of hours worked.
To find the hourly rate, divide total pay by hours worked.

Choosing and checking units

Always write the units with your answer; a number without units is incomplete.
Check that the units in the answer match the compound measure: speed in m/s or km/h, density in g/cm³ or kg/m³, pressure in Pa or N/m².
Make sure all quantities are in compatible units before substituting into a formula.
For example, if distance is in metres and time in seconds, speed will be in m/s.
If distance is in kilometres and time in hours, speed will be in km/h.
Convert units first if they are mixed, such as minutes to hours or grams to kilograms.

Solving problems step by step

Read the problem carefully and identify which compound measure is involved.
Write down the formula you need.
Substitute the given values, making sure units are consistent.
Calculate the result and round sensibly if needed.
Write the final answer with the correct units.
Check that your answer is reasonable in the context of the problem.

Slides

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Practice questions

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1.Which of these is the SI unit of speed?
Easy
- Am/s
- Bkm/h
- Cmph
- Dknot
2.A cyclist covers 30 metres in 2 seconds. What is the cyclist's speed?
Easy
- A15 m/s
- B60 m/s
- C28 m/s
- D32 m/s
3.A car travels 320 kilometres in 4 hours. What is its average speed?
Medium
- A80 km/h
- B316 km/h
- C324 km/h
- D1280 km/h
4.A car travels at a constant speed of 50 km/h. How far does it travel in half an hour?
Medium
- A25 km
- B50 km
- C100 km
- D12.5 km
5.A car travels at 50 km/h. Approximately how far does it travel in one minute?
Medium
- A833 m
- B50 m
- C3000 m
- D500 m
6.Which formula gives density?
Medium
- A\rho = \frac{m}{V}
- B\rho = \frac{V}{m}
- C\rho = m \times V
- D\rho = \frac{F}{A}
7.Which formula gives pressure?
Medium
- AP = \frac{F}{A}
- BP = \frac{A}{F}
- CP = F \times A
- DP = \frac{m}{V}
8.A block has a mass of 240 g and a volume of 30 cm³. What is its density?
Medium
- A8 g/cm³
- B7200 g/cm³
- C0.125 g/cm³
- D210 g/cm³

Revision notes

## Compound Measures: Speed, Density and Pressure A **compound measure** combines two or more quantities, so its unit is a compound unit (e.g. m/s, g/cm^3, N/m^2). ### Speed Speed is the rate at which distance changes with time. - Formula: \(v = \frac{d}{t}\) (v = speed, d = distance, t = time) - Rearranged: \(d = vt\) and \(t = \frac{d}{v}\) - **Average speed** = total distance ÷ total time. - **Instantaneous speed** is the speed at one instant (what a speedometer shows). - Example: 30 m in 2 s gives v = 30 ÷ 2 = 15 m/s. - Units: m/s (SI), km/h, mph, knots. ### Density Density is mass per unit volume. - Formula: \(\rho = \frac{m}{V}\) (ρ = density, m = mass, V = volume) - Rearranged: \(m = \rho V\) and \(V = \frac{m}{\rho}\) - Units: kg/m^3 or g/cm^3. ### Pressure Pressure is force per unit area. - Formula: \(P = \frac{F}{A}\) (P = pressure, F = force, A = area) - Rearranged: \(F = PA\) and \(A = \frac{F}{P}\) - Units: N/m^2 (pascals, Pa). ### Converting Units - km/h to m/s: divide by 3.6. m/s to km/h: multiply by 3.6. - Example: 72 km/h = 72 ÷ 3.6 = 20 m/s. ### Unit Pricing and Rates of Pay - Unit price = total cost ÷ quantity (e.g. £/kg) — use it to compare deals. - Rate of pay = total pay ÷ hours worked (e.g. £/hour). ### Choosing and Checking Units - Divide the units along with the numbers: km ÷ h gives km/h. - Always check the answer's unit matches what the question asks for. ### Exam Tips - Write the formula, substitute, then solve. - Rearrange before substituting to reduce errors. - Watch for mixed units (e.g. minutes with km/h) — convert first.

Compound measure: A measure made from two or more other measures, giving a compound unit such as m/s, g/cm^3 or N/m^2.
Speed: The rate at which distance travelled changes with time; v = d / t (v = speed, d = distance, t = time).
Average speed: Total distance covered divided by the total time interval.
Instantaneous speed: The speed at a particular instant, or assumed constant over a very short time; shown by a speedometer.
Density: Mass per unit volume; ρ = m / V (ρ = density, m = mass, V = volume).
Pressure: Force per unit area; P = F / A (P = pressure, F = force, A = area).
Unit price: The cost of one unit of an item, found by dividing total cost by quantity; used to compare value.
Rate of pay: Earnings per unit of time, found by dividing total pay by hours worked.

Flashcards

What is the formula for speed?

v = d / t, where v is speed, d is distance and t is time.

What is the formula for density?

ρ = m / V, where ρ is density, m is mass and V is volume.

What is the formula for pressure?

P = F / A, where P is pressure, F is force and A is area.

What is the SI unit of speed?

The metre per second (m/s).

What is the formula for average speed?

Average speed = total distance ÷ total time.

How do you find distance if you know average speed and time?

d = v × t, where d is distance, v is average speed and t is time.

How do you find time if you know distance and speed?

t = d / v, where t is time, d is distance and v is speed.

How do you find mass if you know density and volume?

m = ρ × V, where m is mass, ρ is density and V is volume.

How do you find volume if you know mass and density?

V = m / ρ, where V is volume, m is mass and ρ is density.

How do you find force if you know pressure and area?

F = P × A, where F is force, P is pressure and A is area.

How do you find area if you know force and pressure?

A = F / P, where A is area, F is force and P is pressure.

How do you convert a speed from km/h to m/s?

Multiply by 1000/3600, which simplifies to 5/18.

How do you convert a speed from m/s to km/h?

Multiply by 3600/1000, which simplifies to 18/5.

What is the difference between average speed and instantaneous speed?

Average speed is total distance divided by total time; instantaneous speed is the speed at a particular instant.

What is a compound unit?

A unit made by combining two or more different units, such as m/s, km/h, kg/m³ or N/m².

What is unit pricing?

Finding the cost of one unit of an item, such as the price per kilogram or per litre, to compare value.

How do you calculate a rate of pay?

Divide total pay by the number of hours worked to find the pay per hour.

What does the unit of density tell you?

It tells you the mass per unit volume, for example kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³).

What does the unit of pressure tell you?

It tells you the force per unit area, for example newtons per square metre (N/m²) or pascals (Pa).

Why is it important to check units in an answer?

To make sure the units match the quantity being measured and that calculations are consistent.

What is the speed of an object that travels 30 metres in 2 seconds?

15 metres per second (15 m/s).

If a car travels at 80 km/h for 4 hours, how far does it go?

320 kilometres.

What is the speed of light in vacuum?

Approximately 299792458 metres per second (about 1079000000 km/h).

What is the difference between speed and velocity?

Speed is the magnitude of velocity; velocity also includes direction.

What is the formula for speed in terms of distance and time?

v = d / t, where v is speed, d is distance and t is time.

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