Compound measures: speed, density and pressure

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Apuntes de la lección

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.

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Preguntas de práctica

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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

Más preguntas

9.A force of 60 N acts on an area of 4 m². What is the pressure?
Medium
- A15 N/m²
- B240 N/m²
- C0.067 N/m²
- D64 N/m²
10.Which of the following are valid units for speed? (select all that apply)
Medium
- Am/s
- Bkm/h
- Cmph
- Dknot
- Ekg/m³
11.Which of these statements about speed are correct? (select all that apply)
Medium
- AAverage speed = total distance ÷ total time
- BSpeed is the magnitude of velocity
- CSpeed is a vector quantity
- DInstantaneous speed can be read from a speedometer
12.Average speed is calculated by dividing the total distance travelled by the total time taken.
Easy
True or false?
13.Speed is a vector quantity because it has both magnitude and direction.
Easy
True or false?
14.If a car travels at a constant 80 km/h for 4 hours, it covers 320 km.
Medium
True or false?
15.Match each compound measure to its formula.
Medium
- Speed
- Density
- Pressure
- \frac{d}{t}
- \frac{m}{V}
- \frac{F}{A}
16.Arrange these steps in the correct order to find the average speed of a journey.
Medium
- Measure the total distance travelled
- Measure the total time taken
- Divide the total distance by the total time
- State the answer with the correct compound unit
17.What is the average speed for the whole journey, including the stop?
Medium
- A60 km/h
- B90 km/h
- C72 km/h
- D80 km/h
18.The average speed for the whole journey is the same as the average speed for the first part.
Medium
True or false?
19.Match each compound measure to the quantity it is measured in.
Medium
- Speed
- Density
- Pressure
- metres per second
- grams per cubic centimetre
- newtons per square metre
20.Which of the following are compound units? (select all that apply)
Medium
- Am/s
- Bkg
- Cg/cm³
- DN/m²
- Es
21.Arrange these steps in the correct order to find the density of a metal block given its mass and volume.
Medium
- Measure the mass of the block in grams.
- Measure the volume of the block in cubic centimetres.
- Divide the mass by the volume.
- Write the density with the unit g/cm³.
22.Which of the following are valid units for density? (select all that apply)
Medium
- Ag/cm³
- Bkg/m³
- Cm/s
- DN/m²
- Ekg/m²
23.Arrange these speeds in increasing order (slowest first).
Medium
- 10 m/s
- 36 km/h
- 1 km/min
- 500 m/min
24.Which of the following statements about pressure are correct? (select all that apply)
Medium
- APressure increases if the force increases while the area stays the same.
- BPressure increases if the area increases while the force stays the same.
- CPressure is measured in newtons per square metre.
- DPressure is calculated by multiplying force by area.
- EPressure is inversely proportional to area when force is constant.
25.Match each formula to the compound measure it calculates.
Medium
- Speed
- Density
- Pressure
- v = \frac{d}{t}
- \rho = \frac{m}{V}
- P = \frac{F}{A}
26.A car travels 120 km in 1.5 hours. Which of the following are true? (select all that apply)
Medium
- AThe average speed is 80 km/h.
- BThe average speed is 180 km/h.
- CThe car could have travelled at a constant speed of 80 km/h.
- DThe car must have travelled at a constant speed of 80 km/h.
- EIf the car continued at 80 km/h for another 0.5 hours, it would travel a further 40 km.
27.Arrange these steps in the correct order to convert a speed from km/h to m/s.
Medium
- Write down the speed in km/h.
- Multiply by 1000 to convert kilometres to metres.
- Divide by 3600 to convert hours to seconds.
- Simplify to give the speed in m/s.
28.Which of the following could be units of speed? (select all that apply)
Medium
- Am/s
- Bkm/h
- Cmph
- Dknot
- Ekg/m³
29.Match each quantity to its correct unit.
Medium
- Force
- Area
- Pressure
- Density
- Speed
- N
- m²
- N/m²
- g/cm³
- m/s
30.Arrange these pressures in increasing order (smallest first).
Medium
- 100 N/m²
- 0.5 kPa
- 2000 Pa
- 0.1 N/cm²
31.A rectangular block has a mass of 600 g and a volume of 200 cm³. Which of the following are true? (select all that apply)
Hard
- AThe density is 3 g/cm³.
- BThe density is 0.33 g/cm³.
- CThe density in kg/m³ is 3000 kg/m³.
- DThe block will float in water (density 1 g/cm³).
- EThe block will sink in water (density 1 g/cm³).
32.Match each rearranged formula to the quantity it gives.
Hard
- t = \frac{d}{v}
- m = \rho V
- F = PA
- A = \frac{F}{P}
- time
- mass
- force
- area
33.Arrange these steps in the correct order to find the pressure exerted by a force of 500 N on an area of 0.2 m².
Hard
- Write down the formula P = \frac{F}{A}.
- Substitute F = 500 and A = 0.2.
- Calculate 500 ÷ 0.2.
- State the pressure as 2500 N/m².
34.Which of the following statements about speed, density and pressure are correct? (select all that apply)
Hard
- ASpeed is the magnitude of velocity.
- BDensity is directly proportional to volume for a fixed mass.
- CPressure is inversely proportional to area for a fixed force.
- DThe SI unit of speed is the metre per second.
- EDensity is measured in newtons per cubic metre.
35.A car travels 60 km at an average speed of 40 km/h and then 60 km at an average speed of 60 km/h. Which of the following are true? (select all that apply)
Hard
- AThe total distance is 120 km.
- BThe total time is 2.5 hours.
- CThe average speed for the whole journey is 50 km/h.
- DThe average speed for the whole journey is 48 km/h.
- EThe average speed for the whole journey is 45 km/h.
36.Arrange these steps in the correct order to find the average speed for a journey of two parts.
Hard
- Calculate the distance for each part of the journey.
- Calculate the time for each part using time = distance ÷ speed.
- Add the distances to get the total distance.
- Add the times to get the total time.
- Divide the total distance by the total time.
37.Match each scenario to the compound measure it involves.
Hard
- A car travelling 150 km in 2 hours
- A metal block with mass 500 g and volume 100 cm³
- A force of 200 N acting on 0.5 m²
- A tap dripping 3 litres per minute
- speed
- density
- pressure
- rate of flow
38.Which of the following are correct conversions? (select all that apply)
Hard
- A10 m/s = 36 km/h
- B20 m/s = 72 km/h
- C1 g/cm³ = 1000 kg/m³
- D1 N/m² = 1 Pa
- E1 km/h = 1 m/s
39.Arrange these densities in increasing order (smallest first).
Challenge
- 0.8 g/cm³
- 900 kg/m³
- 1.2 g/cm³
- 1500 kg/m³
40.A rectangular tank is 2 m long, 1 m wide and 0.5 m deep. It is filled with a liquid of density 800 kg/m³. Which of the following are true? (select all that apply)
Challenge
- AThe volume of liquid is 1 m³.
- BThe mass of liquid is 800 kg.
- CThe mass of liquid is 1600 kg.
- DThe weight of liquid is 8000 N (using g = 10 N/kg).
- EThe pressure at the bottom depends only on the depth and density of the liquid.
41.A car travels 150 km in 2 hours. Calculate its average speed in km/h.
Medium
42.A metal block has a mass of 500 g and a volume of 100 cm³. Calculate its density in g/cm³.
Medium
43.A force of 200 N is applied over an area of 0.5 m². Calculate the pressure in N/m².
Medium
44.A train travels at a constant speed of 90 km/h for 20 minutes. Calculate the distance travelled in kilometres.
Medium
45.Complete the sentence with the correct compound unit.
Easy
The SI unit of speed is ____.
46.Complete the formula.
Medium
Density = ____ ÷ volume.
47.Complete the sentence about average speed.
Medium
Average speed = total ____ ÷ total ____.
48.What name is given to the speed of an object at a particular instant, such as the reading on a speedometer?
Easy
49.What is the name of the physical quantity defined as the magnitude of velocity?
Medium
50.Calculate the average speed for the first part of the journey in km/h.
Medium
51.Explain the difference between average speed and instantaneous speed, and give one example of when each might be used.
Hard
52.Describe how to calculate the density of a rectangular block if you know its mass and its volume, and state the units you would use if mass is in grams and volume in cubic centimetres.
Medium
53.A car travels 150 km in 2 hours. Calculate its average speed in km/h.
Medium
- A75 km/h
- B67 km/h
- C91 km/h
- D83 km/h
54.A metal block has a mass of 500 g and a volume of 100 cm³. Calculate its density in g/cm³.
Medium
- A5 g/cm³
- B4 g/cm³
- C7 g/cm³
- D6 g/cm³
55.A force of 200 N is applied over an area of 0.5 m². Calculate the pressure in N/m².
Medium
- A400 N/m²
- B440 N/m²
- C360 N/m²
- D480 N/m²
56.A train travels at a constant speed of 90 km/h for 20 minutes. Calculate the distance travelled in kilometres.
Medium
- A36 km
- B27 km
- C30 km
- D33 km
57.Calculate the average speed for the first part of the journey in km/h.
Medium
- A96 km/h
- B72 km/h
- C80 km/h
- D88 km/h

Apuntes de repaso

## 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.

Tarjetas

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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