Distance travelled, average speed maintained and time elapsed are related.
Science classes often use the vectors form.
Key formulas: Maths
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
\[ \text{Distance (d) } = \text{Speed (s)} , \text{Time (t)} \]
\[
d = st,\qquad s = \frac{d}{t},\qquad t = \frac{d}{s}
\]
Key formulas: Physics
\[
\text{Displacement} = \text{Velocity} \times \text{Time}
\]
\[
\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}
\]
\[
\text{Time} = \frac{\text{Displacement}}{\text{Velocity}}
\]
\[ \text{Displacement (s) } = \text{Velocity (v)} , \text{Time (t)} \]
\[
s = vt,\qquad v = \frac{s}{t},\qquad t = \frac{s}{v}
\]
DST graphs
Maths vs Physics (Distance–Speed–Time)
Maths (Scalars)
- Uses distance, speed, time
- All are scalars → size only
- Direction is ignored
\( \text{distance} = \text{speed} \times \text{time} \)
Physics (Vectors)
- Uses displacement and velocity
- These are vectors → size + direction
- Time stays scalar
\( \vec{s} = \vec{v} \, t \)
Scalars and Vectors
Remember:
Converting time units
Decimal hours × 60 = minutes
Example: \(1.15\) hours = \(1\) hour \(9\) minutes
(because \(0.15 \times 60 = 9\))
Minutes ÷ 60 = decimal hours
Example: \(3\) hours \(45\) minutes = \(3.75\) hours
(because \(45/60 = 0.75\))
Example
A man walks for 3 hours at 3 mph. How far has he travelled?
\[
\text{Distance} = 3 \times 3 = 9\text{ miles}
\]
Example
A man walks for 45 minutes at 4 mph. How far has he travelled?
Convert minutes to hours:
Convert
\[
45\text{ minutes} = \frac{45}{60} = 0.75\text{ hours}
\]
\[
\text{Distance} = 4 \times 0.75 = 3\text{ miles}
\]
Example
A man covers 21 miles in 7 hours. Find his average speed.
\[
\text{Speed} = \frac{21}{7} = 3\text{ mph}
\]
Example
A man covers 232 miles in 7 hours 15 minutes.
Convert minutes to hours:
\[
7\text{ h }15\text{ min}
= 7 + \frac{15}{60}
= 7.25\text{ hours}
\]
\[
\text{Speed} = \frac{232}{7.25} = 32\text{ mph (approx)}
\]
Example
How long does it take an aircraft travelling at 580 km/h to travel
232 km?
\[
\text{Time} = \frac{232}{580}
= 0.4\text{ hours}
\]
Convert to minutes:
\[
0.4 \times 60 = 24\text{ minutes}
\]
Example
A man drives 250 miles at 70 mph.
Give the time in hours, minutes and seconds.
\[
\text{Time} = \frac{250}{70}
\approx 3.5714\text{ hours}
\]
Convert decimal hours:
\[
0.5714 \times 60 = 34.2857\text{ minutes}
\]
Convert decimal minutes:
\[
0.2857 \times 60 \approx 17\text{ seconds}
\]
Final answer:
\[
3\text{ hours }34\text{ minutes }17\text{ seconds}
\]
Unit Conversions
Unitary Conversions
km/h → m/s
Converting km/h to m/s
Key facts:
- 1 kilometre = 1000 metres
- 1 hour = 3600 seconds
So:
\[
1 \text{ km/h} = \frac{1000}{3600} \text{ m/s} = \frac{5}{18} \text{ m/s}
\]
How to Convert
- Multiply the speed in km/h by \(\tfrac{5}{18}\).
- This gives the speed in m/s.
Example
\[
72 \text{ km/h} = 72 \times \frac{5}{18}
\]
\[
= 4 \times 5 = 20 \text{ m/s}
\]
Quick Reference
- \(18 \text{ km/h} = 5 \text{ m/s}\)
- \(36 \text{ km/h} = 10 \text{ m/s}\)
- \(54 \text{ km/h} = 15 \text{ m/s}\)
- \(90 \text{ km/h} = 25 \text{ m/s}\)
m/s → km/h
Converting m/s to km/h
Key facts:
- 1 metre = 0.001 kilometres
- 1 second = 1/3600 hours
So:
\[
1 \text{ m/s} = \frac{1}{1000} \text{ km per second}
\]
\[
= \frac{1}{1000} \times 3600 \text{ km/h}
\]
\[
= 3.6 \text{ km/h}
\]
How to Convert
- Multiply the speed in m/s by \(3.6\).
- This gives the speed in km/h.
Example
\[
20 \text{ m/s} = 20 \times 3.6
\]
\[
= 72 \text{ km/h}
\]
Quick Reference
- \(5 \text{ m/s} = 18 \text{ km/h}\)
- \(10 \text{ m/s} = 36 \text{ km/h}\)
- \(15 \text{ m/s} = 54 \text{ km/h}\)
- \(25 \text{ m/s} = 90 \text{ km/h}\)
Miles ↔ Kilometres
Converting Miles to Kilometres
Key fact:
\[
1 \text{ mile} \approx 1.6 \text{ km}
\]
How to Convert
- Multiply the number of miles by \(1.6\).
- This gives the distance in kilometres.
Example
\[
12 \text{ miles} = 12 \times 1.6
\]
\[
= 19.2 \text{ km}
\]
Quick Reference
- \(5 \text{ miles} = 8 \text{ km}\)
- \(10 \text{ miles} = 16 \text{ km}\)
- \(20 \text{ miles} = 32 \text{ km}\)
Kilometres↔ Miles
Converting Kilometres to Miles
Key fact:
\[
1 \text{ km} \approx 0.625 \text{ miles}
\]
(Because \(1 \text{ mile} \approx 1.6 \text{ km}\), so \(1 \div 1.6 = 0.625\))
How to Convert
- Multiply the number of kilometres by \(0.625\).
- This gives the distance in miles.
Example
\[
20 \text{ km} = 20 \times 0.625
\]
\[
= 12.5 \text{ miles}
\]
Quick Reference
- \(5 \text{ km} = 3.125 \text{ miles}\)
- \(10 \text{ km} = 6.25 \text{ miles}\)
- \(40 \text{ km} = 25 \text{ miles}\)
km/h → mph
Converting km/h to mph
Key fact:
\[
1 \text{ km/h} \approx 0.625 \text{ mph}
\]
(Because \(1 \text{ mile} \approx 1.6 \text{ km}\), so \(1 \div 1.6 = 0.625\))
How to Convert
- Multiply the speed in km/h by \(0.625\).
- This gives the speed in mph.
Example
\[
80 \text{ km/h} = 80 \times 0.625
\]
\[
= 50 \text{ mph}
\]
Quick Reference
- \(40 \text{ km/h} = 25 \text{ mph}\)
- \(60 \text{ km/h} = 37.5 \text{ mph}\)
- \(100 \text{ km/h} = 62.5 \text{ mph}\)
- \(120 \text{ km/h} = 75 \text{ mph}\)
mph → km/h
Converting mph to km/h
Key fact:
\[
1 \text{ mph} \approx 1.6 \text{ km/h}
\]
(Because \(1 \text{ mile} \approx 1.6 \text{ km}\))
How to Convert
- Multiply the speed in mph by \(1.6\).
- This gives the speed in km/h.
Example
\[
50 \text{ mph} = 50 \times 1.6
\]
\[
= 80 \text{ km/h}
\]
Quick Reference
- \(30 \text{ mph} = 48 \text{ km/h}\)
- \(40 \text{ mph} = 64 \text{ km/h}\)
- \(60 \text{ mph} = 96 \text{ km/h}\)
- \(70 \text{ mph} = 112 \text{ km/h}\)
