Speed, distance, and time

Of all the proportional reasoning a student does, none is more useful than the speed-distance-time triple. Trains, cars, runners, light, and even sound , every motion in the universe is described by these three quantities.

Concept

The relationship.

$\text{distance} = \text{speed} \times \text{time}, \quad d = v \cdot t.$

Three derived facts:

• $v = \dfrac{d}{t}$ (speed is distance per time).
• $t = \dfrac{d}{v}$ (time is distance per speed).

Fixed distance. With a fixed $d$, speed and time are in inverse proportion. Doubling speed halves time.

Fixed time. With a fixed $t$, speed and distance are in direct proportion.

Fixed speed. With a fixed $v$, distance and time are in direct proportion.

Units. Always make sure units are consistent before computing.

• $5$ km/h means $5$ km in $1$ hour. To convert to m/s: $5 \times \dfrac{1000}{3600} = \dfrac{25}{18} \approx 1.39$ m/s.
• $36$ km/h $= 10$ m/s. ($\times 5/18$ converts km/h to m/s.)

Average speed. When speed is not constant, average speed is

$\bar v = \dfrac{\text{total distance}}{\text{total time}}.$

Trap: if equal distances are covered at speeds $v_1$ and $v_2$, the average speed is the harmonic mean:

$\bar v = \dfrac{2 v_1 v_2}{v_1 + v_2}.$

(Not the simple average $(v_1 + v_2)/2$ , that would only be right for equal times, not equal distances.)

Relative speed. Two objects moving on the same path:

• Same direction: relative speed $= v_1 - v_2$ (if $v_1 > v_2$).
• Opposite directions: relative speed $= v_1 + v_2$.

This is used in train problems: a faster train overtaking a slower one, two trains crossing each other.

Worked examples

Example 1. A bus covers $240$ km in $4$ hours. Find its speed.

• $v = 240/4 = 60$ km/h.

Example 2. A man walks $5$ km/h. How long to walk $7.5$ km?

• $t = 7.5/5 = 1.5$ hours.

Example 3. A car goes $90$ km at $30$ km/h then returns at $60$ km/h. Average speed for the trip?

• Equal distance each way; use harmonic mean.
• $\bar v = \dfrac{2 \cdot 30 \cdot 60}{30 + 60} = \dfrac{3600}{90} = 40$ km/h.

Example 4. Two trains, $80$ km/h and $60$ km/h, run in opposite directions. They cross each other in $0.5$ min. Combined length?

• Relative speed $= 80 + 60 = 140$ km/h $= 140 \cdot 1000/3600 \approx 38.89$ m/s.
• Time $= 30$ s. Combined length $\approx 38.89 \cdot 30 \approx 1167$ m.

Try it yourself

1. A train covers $315$ km in $4.5$ hours. Find its speed.
2. A jogger runs at $9$ km/h. Time to cover $4.5$ km?
3. Convert $90$ km/h to m/s.
4. A car goes $120$ km at $40$ km/h and $120$ km at $60$ km/h. Average speed?
5. Two cars start towards each other, $200$ km apart, at $50$ and $30$ km/h. When do they meet?
6. A train $100$ m long crosses a $200$ m platform in $30$ s. Find its speed.
7. A man can row $5$ km/h in still water. Stream is $2$ km/h. Time to row $14$ km downstream and back?
8. A boy cycles to school at $12$ km/h and returns at $4$ km/h. Average speed?

Activity

Daily speed log. For one week, record the distance and time of any one journey you make regularly (walk to school, ride in a bus). Compute the average speed daily. Plot it on a graph. You'll see how traffic, weather, and your own pace produce a surprising amount of variation.