Speed, Density and Pressure
Speed, density and pressure are compound measures: each combines two different quantities into a single rate. Units matter just as much as the numbers.
Foundation & Higher. Rearranging formulae is useful here: Rearranging Formulae. For scientific notation, see Standard Form.
Speed
speed = distance ÷ time. Rearranged: distance = speed × time, and time = distance ÷ speed.
If a car travels 150 km in 2.5 h, average speed = 150 ÷ 2.5 = 60 km/h.
Worked example — find distance or time
At 72 km/h for 25 minutes, use 25/60 hours: distance = 72 × 25/60 = 30 km. To travel 18 km at 12 km/h, time = 18/12 = 1.5 hours = 1 h 30 min. A decimal fraction of an hour must be multiplied by 60 to find minutes.
To convert speed, 72 km/h = 72,000 m / 3600 s = 20 m/s. Thus divide km/h by 3.6 to obtain m/s; reverse the conversion by multiplying by 3.6. These factors come from converting both distance and time.
Worked example — mixed journey
A van travels 90 km in 1.5 hours, stops for 30 minutes, then travels another 60 km in 1 hour.
Total distance = 150 km. Total elapsed time = 1.5 + 0.5 + 1 = 3 hours.
Average speed over the whole journey = 150 ÷ 3 = 50 km/h.
Whether a stop is included depends on the wording. ‘Average speed for the whole journey’ usually uses total elapsed time.
Average speed
Average speed is total distance ÷ total time. It is not usually the ordinary mean of two speeds.
If 60 km is travelled at 30 km/h and then 60 km at 60 km/h, times are 2 h and 1 h. Average speed = 120 ÷ 3 = 40 km/h, not 45 km/h.
Density
density = mass ÷ volume. Rearranged: mass = density × volume; volume = mass ÷ density.
A 540 g object with volume 200 cm³ has density 540 ÷ 200 = 2.7 g/cm³.
Worked example — use density in reverse
A material has density 2.7 g/cm³. For volume 40 cm³, mass = 2.7 × 40 = 108 g. If another piece has mass 81 g, its volume is 81/2.7 = 30 cm³. The units match the density in both calculations.
For compound-unit conversion, 1 g/cm³ = 0.001 kg / 0.000001 m³ = 1000 kg/m³. Hence 2.7 g/cm³ = 2700 kg/m³; convert the numerator and denominator, not just the mass.
Pressure
pressure = force ÷ area. In SI units, pressure is measured in pascals, where 1 Pa = 1 N/m².
A force of 600 N over 0.3 m² gives pressure 600 ÷ 0.3 = 2000 Pa.
Rearrange P = F/A to F = PA or A = F/P. For P = 3000 Pa on 0.2 m², F = 3000 × 0.2 = 600 N. For F = 600 N and P = 3000 Pa, A = 600/3000 = 0.2 m².
Worked example — pressure with an area conversion
A 200 N force acts on 50 cm². Since 1 m² = 10,000 cm², the area is 50/10,000 = 0.005 m². Pressure = 200/0.005 = 40,000 Pa. Equivalently it is 4 N/cm²: 1 N/cm² = 10,000 Pa.
Convert before calculating
If speed is required in m/s, use metres and seconds. If density is in g/cm³, use grams and cubic centimetres.
A correct formula with mismatched units still gives a meaningless result.
Worked example — convert units before using the formula
A block has mass 1.8 kg and volume 600 cm³. Find density in g/cm³.
Convert 1.8 kg to 1800 g. Then density = 1800 ÷ 600 = 3 g/cm³.
If the required answer were in kg/m³ instead, convert the volume as well. Unit choice is part of the mathematics, not an afterthought.
Higher application — bounds
When measurements are rounded, the greatest or least possible compound measure can require bounds. Link to Bounds and Error Intervals rather than reteaching the full method here.
Think Like a Mathematician
Units reveal the structure of a formula. km/h literally means kilometres divided by hours; kg/m³ means kilograms divided by cubic metres.
Common Mistakes
Averaging speeds directly: use total distance ÷ total time.
Mixing cm² and m²: convert areas carefully.
Using the wrong rearrangement: identify which quantity is unknown before substituting.
Practice — compound measures
1. A runner covers 12 km in 1.5 h. Find average speed.
12÷1.5=8 km/h.
2. A train travels at 90 km/h for 2.4 h. Find distance.
90×2.4=216 km.
3. A cyclist travels 45 km at 15 km/h. Find time.
45÷15=3 h.
4. A block has mass 780 g and volume 300 cm³. Find density.
780÷300=2.6 g/cm³.
5. A material has density 7.8 g/cm³ and volume 50 cm³. Find mass.
7.8×50=390 g.
6. A force of 1200 N acts over 0.4 m². Find pressure.
1200÷0.4=3000 Pa.
7. Pressure is 2500 Pa over 0.6 m². Find force.
2500×0.6=1500 N.
8. A car travels 80 km at 40 km/h, then 80 km at 80 km/h. Find average speed.
Times: 2 h and 1 h. Total 160 km in 3 h, so 53.3 km/h to 1 d.p.
9. Convert 72 km/h to m/s.
72 km/h = 72,000 m per 3600 s = 20 m/s.
10. A material has density 1250 kg/m³. What volume does a 2.5 kg sample occupy? Also find the area supporting a 200 N force at 40,000 Pa.
Volume = mass/density = 2.5/1250 = 0.002 m³. Area = force/pressure = 200/40,000 = 0.005 m² = 50 cm².
Frequently Asked Questions
When is the ordinary mean of two speeds valid?
For two stages lasting equal times. In general, calculate total distance divided by total time; equal distances do not justify the ordinary mean.
Must pressure always be in pascals?
No. Match the stated force and area units. N/m² gives Pa, while N/cm² is a different unit and must be converted when required.