6.3 Changing Rates
Understanding Speed
Speed measures how fast something is moving. It's the rate of change of distance with time.
Speed measures how fast something is moving. It's the rate of change of distance with time.
Distance
D = S × T
Distance = Speed × Time
Speed
S = D ÷ T
Speed = Distance ÷ Time
Time
T = D ÷ S
Time = Distance ÷ Speed
Example 1: Finding Distance
A car travels at 60 km/h for 3 hours. How far does it travel?
Given: Speed = 60 km/h, Time = 3 hours
Find: Distance
Formula: $D = S \times T$
$D = 60 \times 3 = 180$ km
Given: Speed = 60 km/h, Time = 3 hours
Find: Distance
Formula: $D = S \times T$
$D = 60 \times 3 = 180$ km
🚗
60 km/h
Speed
⏱️
3 hours
Time
📏
180 km
Distance
Example 2: Finding Speed
A cyclist travels 45 km in 2.5 hours. What is their average speed?
Given: Distance = 45 km, Time = 2.5 hours
Find: Speed
Formula: $S = D \div T$
$S = 45 \div 2.5 = 18$ km/h
Answer: The cyclist's average speed is 18 km/h
Given: Distance = 45 km, Time = 2.5 hours
Find: Speed
Formula: $S = D \div T$
$S = 45 \div 2.5 = 18$ km/h
Answer: The cyclist's average speed is 18 km/h
Example 3: Finding Time
How long does it take to travel 240 miles at 80 mph?
Given: Distance = 240 miles, Speed = 80 mph
Find: Time
Formula: $T = D \div S$
$T = 240 \div 80 = 3$ hours
Answer: The journey takes 3 hours
Given: Distance = 240 miles, Speed = 80 mph
Find: Time
Formula: $T = D \div S$
$T = 240 \div 80 = 3$ hours
Answer: The journey takes 3 hours
Example 4: Average Speed with Different Sections
A car travels 60 km at 40 km/h, then 90 km at 60 km/h. Find the average speed for the whole journey.
Step 1: Find time for each section
Section 1: $T_1 = 60 \div 40 = 1.5$ hours
Section 2: $T_2 = 90 \div 60 = 1.5$ hours
Step 2: Find totals
Total distance = $60 + 90 = 150$ km
Total time = $1.5 + 1.5 = 3$ hours
Step 3: Calculate average speed
Average speed = $150 \div 3 = 50$ km/h
⚠️ Note: Average speed is NOT the average of the two speeds!
Step 1: Find time for each section
Section 1: $T_1 = 60 \div 40 = 1.5$ hours
Section 2: $T_2 = 90 \div 60 = 1.5$ hours
Step 2: Find totals
Total distance = $60 + 90 = 150$ km
Total time = $1.5 + 1.5 = 3$ hours
Step 3: Calculate average speed
Average speed = $150 \div 3 = 50$ km/h
⚠️ Note: Average speed is NOT the average of the two speeds!
💡 Common Speed Units:
(road speeds in most of the world)
(walking/running)
(UK/US roads)
km/h
Kilometers per hour(road speeds in most of the world)
m/s
Meters per second(walking/running)
mph
Miles per hour(UK/US roads)
🎯 Speed, Distance, Time
Understanding Density
Density is the mass per unit volume. It measures how compact a substance is.
Pressure is the force per unit area. It measures how concentrated a force is.
Density is the mass per unit volume. It measures how compact a substance is.
Density
D = M ÷ V
Density = Mass ÷ Volume
Mass
M = D × V
Mass = Density × Volume
Volume
V = M ÷ D
Volume = Mass ÷ Density
Example 1: Finding Density
A block of wood has a mass of 600g and a volume of 800 cm³. Find its density.
Given: Mass = 600 g, Volume = 800 cm³
Find: Density
Formula: $D = M \div V$
$D = 600 \div 800 = 0.75$ g/cm³
Answer: The density is 0.75 g/cm³
Given: Mass = 600 g, Volume = 800 cm³
Find: Density
Formula: $D = M \div V$
$D = 600 \div 800 = 0.75$ g/cm³
Answer: The density is 0.75 g/cm³
Example 2: Finding Mass
A metal cube has a volume of 250 cm³ and density of 8 g/cm³. Find its mass.
Given: Volume = 250 cm³, Density = 8 g/cm³
Find: Mass
Formula: $M = D \times V$
$M = 8 \times 250 = 2000$ g = 2 kg
Answer: The mass is 2000 g or 2 kg
Given: Volume = 250 cm³, Density = 8 g/cm³
Find: Mass
Formula: $M = D \times V$
$M = 8 \times 250 = 2000$ g = 2 kg
Answer: The mass is 2000 g or 2 kg
💡 Common Densities:
Understanding PressureWater
1 g/cm³ or 1000 kg/m³Ice
0.92 g/cm³ (floats on water!)Iron
7.87 g/cm³Gold
19.3 g/cm³Air
0.0013 g/cm³Lead
11.3 g/cm³Pressure is the force per unit area. It measures how concentrated a force is.
Pressure
P = F ÷ A
Pressure = Force ÷ Area
Force
F = P × A
Force = Pressure × Area
Area
A = F ÷ P
Area = Force ÷ Pressure
Example 3: Calculating Pressure
A force of 500 N acts on an area of 2.5 m². Calculate the pressure.
Given: Force = 500 N, Area = 2.5 m²
Find: Pressure
Formula: $P = F \div A$
$P = 500 \div 2.5 = 200$ N/m² or 200 Pa (Pascals)
Answer: The pressure is 200 Pa
Given: Force = 500 N, Area = 2.5 m²
Find: Pressure
Formula: $P = F \div A$
$P = 500 \div 2.5 = 200$ N/m² or 200 Pa (Pascals)
Answer: The pressure is 200 Pa
Example 4: Why Stilettos Damage Floors
A 60 kg person (weight ≈ 600 N) stands:
• On flat shoes with area 200 cm² (0.02 m²)
• On stiletto heels with area 2 cm² (0.0002 m²)
Flat shoes: $P = 600 \div 0.02 = 30{,}000$ Pa
Stilettos: $P = 600 \div 0.0002 = 3{,}000{,}000$ Pa
Conclusion: Stilettos create 100 times more pressure! 👠
• On flat shoes with area 200 cm² (0.02 m²)
• On stiletto heels with area 2 cm² (0.0002 m²)
Flat shoes: $P = 600 \div 0.02 = 30{,}000$ Pa
Stilettos: $P = 600 \div 0.0002 = 3{,}000{,}000$ Pa
Conclusion: Stilettos create 100 times more pressure! 👠
🎯 Density & Pressure Practice
Converting Between Units
Understanding how to convert between different units is essential for working with rates and compound measures.
Understanding how to convert between different units is essential for working with rates and compound measures.
⚡ Common Conversions:
1 m = 100 cm
1 cm = 10 mm
1 mile ≈ 1.6 km
1 minute = 60 seconds
1 hour = 3600 seconds
1 g = 1000 mg
1 tonne = 1000 kg
1 litre = 1000 ml
1 cm³ = 1 ml
Distance
1 km = 1000 m1 m = 100 cm
1 cm = 10 mm
1 mile ≈ 1.6 km
Time
1 hour = 60 minutes1 minute = 60 seconds
1 hour = 3600 seconds
Mass
1 kg = 1000 g1 g = 1000 mg
1 tonne = 1000 kg
Volume
1 m³ = 1000 litres1 litre = 1000 ml
1 cm³ = 1 ml
Example 1: Converting Speed Units
Convert 20 m/s to km/h
Step 1: Convert meters to kilometers
20 m = 0.02 km (divide by 1000)
Step 2: Convert seconds to hours
1 second = $\frac{1}{3600}$ hours
Step 3: Calculate
$20 \text{ m/s} = \frac{0.02 \text{ km}}{\frac{1}{3600} \text{ h}} = 0.02 \times 3600 = 72$ km/h
Quick rule: To convert m/s to km/h, multiply by 3.6
Step 1: Convert meters to kilometers
20 m = 0.02 km (divide by 1000)
Step 2: Convert seconds to hours
1 second = $\frac{1}{3600}$ hours
Step 3: Calculate
$20 \text{ m/s} = \frac{0.02 \text{ km}}{\frac{1}{3600} \text{ h}} = 0.02 \times 3600 = 72$ km/h
20 m/s
→
× 3.6
→
72 km/h
Quick rule: To convert m/s to km/h, multiply by 3.6
Example 2: Converting Density Units
Convert 2.5 g/cm³ to kg/m³
Step 1: Convert grams to kilograms
2.5 g = 0.0025 kg (divide by 1000)
Step 2: Convert cm³ to m³
1 cm³ = 0.000001 m³ (or $10^{-6}$ m³)
Step 3: Calculate
$2.5 \text{ g/cm}^3 = \frac{0.0025 \text{ kg}}{0.000001 \text{ m}^3} = 2500$ kg/m³
Quick rule: To convert g/cm³ to kg/m³, multiply by 1000
Step 1: Convert grams to kilograms
2.5 g = 0.0025 kg (divide by 1000)
Step 2: Convert cm³ to m³
1 cm³ = 0.000001 m³ (or $10^{-6}$ m³)
Step 3: Calculate
$2.5 \text{ g/cm}^3 = \frac{0.0025 \text{ kg}}{0.000001 \text{ m}^3} = 2500$ kg/m³
2.5 g/cm³
→
× 1000
→
2500 kg/m³
Quick rule: To convert g/cm³ to kg/m³, multiply by 1000
Example 3: Time Conversions
A journey takes 2.75 hours. Express this in hours and minutes.
Step 1: Separate whole hours
2 hours (whole part)
Step 2: Convert decimal to minutes
0.75 hours = $0.75 \times 60 = 45$ minutes
Answer: 2 hours 45 minutes
Step 1: Separate whole hours
2 hours (whole part)
Step 2: Convert decimal to minutes
0.75 hours = $0.75 \times 60 = 45$ minutes
Answer: 2 hours 45 minutes
2.75 hours
=
2 hours 45 minutes
Example 4: Area and Volume Conversions
Area:
1 m² = 100 cm × 100 cm = 10,000 cm²
1 km² = 1000 m × 1000 m = 1,000,000 m²
Volume:
1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³
1 km³ = 1000 m × 1000 m × 1000 m = 1,000,000,000 m³
⚠️ Remember: Square and cube the conversion factor!
1 m² = 100 cm × 100 cm = 10,000 cm²
1 km² = 1000 m × 1000 m = 1,000,000 m²
Volume:
1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³
1 km³ = 1000 m × 1000 m × 1000 m = 1,000,000,000 m³
⚠️ Remember: Square and cube the conversion factor!
💡 Conversion Tips:
• Drawing a conversion ladder can help visualize
• Check your answer makes sense - is it bigger or smaller?
• For compound units (like m/s), convert each part separately
• Check your answer makes sense - is it bigger or smaller?
• For compound units (like m/s), convert each part separately
🧮 Unit Converter:
Speed Converter
Real Life Uses:
• Travel: Converting between mph and km/h when driving abroad
• Science: Converting measurements for experiments
• Cooking: Converting recipe measurements
• Engineering: Working with different measurement systems
• Sports: Comparing speeds and times in different units
• Construction: Converting between metric and imperial units
• Travel: Converting between mph and km/h when driving abroad
• Science: Converting measurements for experiments
• Cooking: Converting recipe measurements
• Engineering: Working with different measurement systems
• Sports: Comparing speeds and times in different units
• Construction: Converting between metric and imperial units
🎯 Unit Conversions Practice