---
title: "Speed: Distance and Time"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 35
exercises: 11
source: https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time
---

# Chapter 35 — Speed: Distance and Time

“I’m faster!” — “No, I am!” Every playground settles this the honest way: line up, ready, set, go. But how do you compare a cyclist with a runner who raced yesterday, on another road? [Speed](#def-g5-speed-distance-time-speed) is how physics settles races that can never be run side by side.

## 35.1 Two ways to win a race

**Proposition 35.1 (The fair race rules).**

Two honest ways to compare fast and slow:

1. over the *same distance* , the faster one needs *less time* — first across the line wins the sprint;
2. in the *same time* , the faster one covers *more distance* — after one minute, the faster swimmer is simply farther.

Same distance or same time: fix one, compare the other. Comparing with *both* different — “I ran farther!” “But you ran all afternoon!” — decides nothing.

**Example 35.2 (Unfair and fair).**

Lena cycled $10\,\mathrm{km}$; Marco cycled $15\,\mathrm{km}$. Is Marco faster? No way to tell — perhaps he rode three times as long. But if both rode for exactly one hour, the comparison turns fair: same time, more distance, Marco is faster. That little “in one hour” is about to become our favorite phrase.

## 35.2 Speed: the distance in one hour

**Definition 35.3 (Speed).**

The *speed* of a steady traveler is the distance it covers in one fixed helping of time — usually one hour. “This car travels $60$ kilometres in each hour” names its speed; a brisk walker manages about $5$ kilometres in each hour, a racing cyclist $40$. For crawlers we use friendlier helpings: a snail covers about $5$ [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) in each hour.

![A ladder of speeds: the distance each traveler covers in one hour — from the snail’s five metres to the airliner’s nine hundred kilometres.](https://one-course.com/images/onecourse/chapters/physics-1/g5-speed-distance-time/fig-6b66d84ee258.svg)

*A ladder of [speeds](#def-g5-speed-distance-time-speed): the distance each traveler covers in one hour — from the snail’s five [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) to the airliner’s nine hundred kilometres.*

**Method 35.4 (Measuring your own speed).**

1. Measure or find a stretch of exactly $100\,\mathrm{m}$ (many sports tracks mark one);
2. walk it at your natural pace while a friend times you — suppose it takes one minute;
3. one hour holds $60$ of those minutes, so in an hour you would cover $60 \times 100 = 6000\,\mathrm{m}$ ;
4. said in kilometres: your walking [speed](#def-g5-speed-distance-time-speed) is $6$ kilometres in each hour.

Repeat at a run and compare — most children roughly double.

## 35.3 Predicting distances

**Proposition 35.5 (Distance from speed).**

A steady traveler covers its one-hour distance again in every further hour. So:

$$
\text{distance covered} = \text{distance in one hour} \times
\text{number of hours}.
$$

A car doing $60$ kilometres each hour covers, in $3$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock), $60 \times 3 = 180\,\mathrm{km}$.

**Example 35.6 (Journey sums).**

The walker at $4$ kilometres each hour, out for $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock): $4 \times 2 = 8\,\mathrm{km}$. The cyclist at $15$ kilometres each hour, riding $4$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock): $15 \times 4 = 60\,\mathrm{km}$. The airliner at $900$ kilometres each hour, flying $5$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock): $900 \times 5 =
4500\,\mathrm{km}$ — a continent crossed between two meals.

![Steady journeys drawn as lines: every hour adds the same helping of kilometres. The faster traveler’s line climbs more steeply.](https://one-course.com/images/onecourse/chapters/physics-1/g5-speed-distance-time/fig-38775cf56106.svg)

*Steady journeys drawn as lines: every hour adds the same helping of kilometres. The faster traveler’s line climbs more steeply.*

**Remark 35.7 (Steady is a simplification).**

Real travelers are rarely steady: the car slows in villages, the cyclist flies downhill and toils up, you dawdle past the bakery. Our sums pretend the whole journey runs at one faithful pace — a useful pretense, good enough for planning. How to be honest about a journey of many paces — and what “average” really means — is a story for a later year, once you own the right mathematics.

**Example 35.8 (Reading speeds around you).**

Road signs and car dashboards speak in kilometres-per-one-hour — the round sign saying $50$ tells drivers: cover at most $50$ kilometres in each hour, a pace chosen so that a child chasing a ball can still be spared. The [high-speed](#def-g5-speed-distance-time-speed) train’s screen may boast $300$; walkers’ signposts in the mountains prefer [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) over kilometres — “lake: 2 h” — trusting the walker to know their own [speed](#def-g5-speed-distance-time-speed).

## 35.4 Exercises

**Exercise 35.1 ★.**

State the two fair ways to compare fast and slow. Why does “I ran farther than you” alone prove nothing?

**Solution of Exercise 35.1.**

Same distance — less time wins; same time — more distance wins. “I ran farther” fixes neither: with unequal times, more distance proves nothing about being fast.

**Exercise 35.2 ★.**

Two swimmers cross the same pool; Ana needs $30$ seconds, Bo needs $25$. Who is faster, and by which fair-race rule?

**Solution of Exercise 35.2.**

Bo — same distance (one pool), less time ($25 < 30$ seconds): rule one.

**Exercise 35.3 ★.**

After one hour, a tractor has covered $20\,\mathrm{km}$ and a scooter $45\,\mathrm{km}$. Who is faster, and by which rule?

**Solution of Exercise 35.3.**

The scooter — same time (one hour), more distance ($45 > 20$ kilometres): rule two.

**Exercise 35.4 ★.**

What does “this train travels $200$ kilometres in each hour” mean? How far does it get in $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock)? In half an hour?

**Solution of Exercise 35.4.**

In every hour it covers $200\,\mathrm{km}$. In $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock): $200 \times 2
= 400\,\mathrm{km}$. In half an hour: half of $200$, so $100\,\mathrm{km}$.

**Exercise 35.5 ★.**

Rank by [speed](#def-g5-speed-distance-time-speed): a walker ($5$ kilometres each hour); a pigeon ($80$ kilometres each hour); a snail ($5$ *[metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)* each hour); a city bus ($30$ kilometres each hour).

**Solution of Exercise 35.5.**

Snail ($5$ [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) each hour), walker ($5$ kilometres), city bus ($30$), pigeon ($80$ kilometres each hour).

**Exercise 35.6 ★.**

You stroll a marked $100\,\mathrm{m}$ in $2$ minutes. How many [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) would you cover in an hour at that pace? Say your [speed](#def-g5-speed-distance-time-speed) in kilometres in each hour.

**Solution of Exercise 35.6.**

An hour holds $30$ helpings of $2$ minutes: $30 \times 100 =
3000\,\mathrm{m}$ — a [speed](#def-g5-speed-distance-time-speed) of $3$ kilometres in each hour.

**Exercise 35.7 ★.**

A car keeps a steady $80$ kilometres each hour for $3$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock). How far does it travel? And a cyclist at $15$ kilometres each hour for the same $3$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock)?

**Solution of Exercise 35.7.**

The car: $80 \times 3 = 240\,\mathrm{km}$. The cyclist: $15 \times 3 =
45\,\mathrm{km}$.

**Exercise 35.8 ★.**

On the journey graph of the chapter, how do you spot the faster traveler at a glance? Read from it: how far has the cyclist gone after $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock)?

**Solution of Exercise 35.8.**

The faster traveler’s line climbs more steeply. After $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) the cyclist’s line stands at $30\,\mathrm{km}$.

**Exercise 35.9 ★★.**

A ferry covers $30$ kilometres in each hour. The crossing is $90\,\mathrm{km}$. Using [Proposition 35.5](#prop-g5-speed-distance-time-distance) backwards — try [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) one by one — how long does the crossing take?

**Solution of Exercise 35.9.**

Hour by hour: $30$, $60$, $90$ — after $3$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) the ferry has covered exactly $90\,\mathrm{km}$: the crossing takes $3$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock).

**Exercise 35.10 ★★.**

Two towns are $60\,\mathrm{km}$ apart. Mara cycles from one at $20$ kilometres each hour; at the same moment Jon cycles from the other toward her at $10$ kilometres each hour. How many kilometres does the gap between them shrink each hour? After how many [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) do they meet?

**Solution of Exercise 35.10.**

Each hour Mara closes $20\,\mathrm{km}$ and Jon $10\,\mathrm{km}$: the gap shrinks by $30\,\mathrm{km}$ each hour. From $60\,\mathrm{km}$, it reaches zero after $2$ [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) — they meet, $40\,\mathrm{km}$ from Mara’s town.

**Exercise 35.11 ★★.**

The old fable, with numbers: the race is $120\,\mathrm{m}$ long. The hare runs $60$ [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) in each minute — but after one minute of running it stops for a $70$-minute nap, then runs on. The tortoise plods $2$ [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) in each minute and never stops. Where is the hare when it starts its nap? At what time does each cross the finish line? Who wins?

**Solution of Exercise 35.11.**

After its one running minute the hare stands at $60\,\mathrm{m}$ — halfway — and falls asleep. The tortoise needs $120 \div 2 = 60$ minutes and finishes at time $60$. The hare wakes at time $1 + 70 =
71$, runs its last $60\,\mathrm{m}$ in one minute, and finishes at time $72$. The tortoise wins by $12$ minutes — steady beats speedy, now with arithmetic.