---
title: "Describing Motion: Trajectory and Speed"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 40
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/40-describing-motion-trajectory-and-speed
---

# Chapter 40 — Describing Motion: Trajectory and Speed

A falling leaf, a [high-speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) train, a carousel horse, a penalty kick: the world never sits still. To study motion, physics asks two tidy questions of every moving thing: *along what path?* and *how fast along it?* Path and pace — [trajectory](#def-g6-describing-motion-trajectory) and [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) — are this chapter’s pair of spectacles.

## 40.1 Trajectory: the path taken

**Definition 40.1 (Trajectory).**

The *trajectory* of a moving object is the line traced by its journey — the path it draws through space. Three shapes cover most of everyday life: *straight* (the elevator, the falling apple), *circular* (the carousel horse, the tip of a clock hand), and *curved* in freer ways (the thrown ball’s arch, the swallow’s swoop).

![Three everyday trajectories: the straight climb, the endless circle, the thrown arch.](https://one-course.com/images/onecourse/chapters/physics-1/g6-describing-motion/fig-341b90cceef1.svg)

*Three everyday trajectories: the straight climb, the endless circle, the thrown arch.*

**Example 40.2 (Whose trajectory?).**

Be careful to name *which point* you follow. The carousel horse’s nose draws a circle; the whole carousel’s center draws nothing at all — it stays put. A rolling wheel’s hub glides in a straight line while a point on its rim traces graceful hops. One machine, many trajectories: physics follows one point at a time.

**Remark 40.3 (Moving compared to what?).**

A passenger dozing on the train draws no [trajectory](#def-g6-describing-motion-trajectory) at all for the passenger opposite — and a hundred-kilometre line across the map for a cow watching from the field. Both descriptions are honest: motion is always *motion compared to something*, and the describer must say what. For now we quietly compare everything to the ground; the full power of choosing other viewpoints is a celebrated story kept for the High School volume.

![A camera left open at night draws trajectories: every headlamp traces the exact path its car followed.](https://one-course.com/images/onecourse/chapters/physics-1/g6-describing-motion/img-48f5ff648899.jpg)

*A camera left open at night draws trajectories: every headlamp traces the exact path its car followed.*

## 40.2 Uniform or varied?

**Definition 40.4 (Uniform and varied motion).**

A motion is *uniform* when it covers equal distances in equal times — the steady walk of a metronome listener, the cruise of a train between stations. It is *varied* when the pace changes: speeding up (the sprinter off the blocks), slowing down (the bus braking), or both by turns (city traffic’s endless stop and go).

**Proposition 40.5 (Reading the dotted film).**

Photograph a moving object at equal ticks of time — many quick snapshots on one [image](https://one-course.com/books/physics/1/en/chapter/32-mirrors-and-reflection#prop-g5-mirrors-reflection-image) — and its motion signs its own name in dots:

1. dots evenly spaced: [uniform motion](#def-g6-describing-motion-uniform) ;
2. dots spreading apart along the way: speeding up;
3. dots crowding together: slowing down.

Equal times sit between every pair of dots, so the gaps between dots compare the distances — the whole diagnosis at a glance.

![Three dotted films, one snapshot per tick: even spacing, spreading, crowding — uniform, accelerating, braking.](https://one-course.com/images/onecourse/chapters/physics-1/g6-describing-motion/fig-a7b9bb28c2ee.svg)

*Three dotted films, one snapshot per tick: even spacing, spreading, crowding — uniform, accelerating, braking.*

**Method 40.6 (Making a dotted film).**

Any phone camera can produce these portraits:

1. film the moving object — a friend cycling past, a ball rolling off a ramp — holding the camera still;
2. step through the video, pausing at equal steps (every half-second of playback, say);
3. at each pause, mark the object’s position on a tracing over the screen, or note it against fence posts in the background;
4. lay out your marks in a row and read them with [Proposition 40.5](#prop-g6-describing-motion-dots) .

One warning from the professionals: hold the camera still — a drifting camera adds its own motion to the film and muddles whose [trajectory](#def-g6-describing-motion-trajectory) is whose (exactly the warning of [Remark 40.3](#rem-g6-describing-motion-relative)).

## 40.3 Putting numbers on the pace

**Example 40.7 (Speed by proportionality).**

A tram in [uniform motion](#def-g6-describing-motion-uniform) covers $9\,\mathrm{km}$ in $20$ minutes. What [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) is that, in the familiar kilometres-per-hour? Proportionality answers without any new machinery: an hour is three helpings of $20$ minutes, so the tram covers $3 \times 9 = 27\,\mathrm{km}$ in one hour — its [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) is $27$ kilometres per hour. In a table:

| minutes | $20$ | $60$ |
| --- | --- | --- |
| kilometres | $9$ | $27$ |

[Uniform motion](#def-g6-describing-motion-uniform) *is* proportionality between distance and time — your mathematics course and your physics course have just shaken hands.

**Example 40.8 (The pace board).**

Everyday paces, for judging any answer: strolling, about $4$ kilometres per hour; brisk walking, $6$; easy cycling, $15$; a city bus with stops, $20$; a car on the open road, $90$; a [high-speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) train, up to $300$. When a calculation hands you a walker doing $40$ kilometres per hour, the pace board sends it back.

**Remark 40.9 (Speed at an instant).**

A [varied motion](#def-g6-describing-motion-uniform) has no single [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) — the braking bus is fast, then slow, then still. A car’s speedometer needle answers a subtler question: *how fast right now?* Defining that [now-speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) precisely is genuinely deep — it needs mathematics you will meet in the last year of high school, and it opened one of the greatest chapters in all of science. For this year: [uniform motions](#def-g6-describing-motion-uniform) get one honest number; [varied motions](#def-g6-describing-motion-uniform) get a story.

## 40.4 Exercises

**Exercise 40.1 ★.**

Give the [trajectory](#def-g6-describing-motion-trajectory)’s shape: a raindrop down a still windowpane; the tip of the minute hand; a basketball’s shot toward the hoop; a skier’s slalom.

**Solution of Exercise 40.1.**

Straight (down the pane); circular; curved (an arch); curved (a weaving S-line).

**Exercise 40.2 ★.**

On a rolling bicycle, what [trajectory](#def-g6-describing-motion-trajectory) does the hub of the wheel draw? And the valve on the rim? (One word and one sketch each.)

**Solution of Exercise 40.2.**

The hub: a straight line, gliding parallel to the road. The valve: a curve of graceful hops — rising, arching, touching down with each turn of the wheel.

**Exercise 40.3 ★.**

Uniform or varied: a train cruising between stations; the same train pulling away from the platform; a parachutist drifting down at a steady rate; a puck sliding and slowing on rough ice?

**Solution of Exercise 40.3.**

Uniform; varied (speeding up); uniform; varied (slowing down).

**Exercise 40.4 ★.**

A dotted film shows gaps of $2\,\mathrm{cm}$, $2\,\mathrm{cm}$, $2\,\mathrm{cm}$, $2\,\mathrm{cm}$ between snapshots. Diagnose the motion. Another shows $1\,\mathrm{cm}$, $2\,\mathrm{cm}$, $4\,\mathrm{cm}$, $7\,\mathrm{cm}$: diagnose that one.

**Solution of Exercise 40.4.**

Equal gaps: [uniform motion](#def-g6-describing-motion-uniform). Growing gaps ($1, 2, 4, 7$): speeding up — more distance in each equal tick.

**Exercise 40.5 ★.**

Why must the camera hold still when making a dotted film? Which remark of the chapter is at stake?

**Solution of Exercise 40.5.**

A drifting camera adds its own motion to the record — the dots then mix the object’s journey with the camera’s. Motion is always motion compared to something, and the film must fix its “something”: the remark on viewpoints.

**Exercise 40.6 ★.**

A ferry in [uniform motion](#def-g6-describing-motion-uniform) covers $12\,\mathrm{km}$ in $30$ minutes. Build the proportionality table and give its [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) in kilometres per hour.

**Solution of Exercise 40.6.**

Table: $30$ min $\to$ $12\,\mathrm{km}$, so $60$ min $\to$ $24\,\mathrm{km}$: the ferry’s [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) is $24$ kilometres per hour.

**Exercise 40.7 ★.**

A cyclist claims a steady $14$ kilometres per hour. How far do they ride in $30$ minutes? In $15$ minutes? (Tables, not formulas.)

**Solution of Exercise 40.7.**

$60$ min $\to$ $14\,\mathrm{km}$; $30$ min $\to$ $7\,\mathrm{km}$; $15$ min $\to$ $3.5\,\mathrm{km}$.

**Exercise 40.8 ★.**

Against the pace board of [Example 40.8](#ex-g6-describing-motion-paces), judge these claims: a stroller covering $12\,\mathrm{km}$ in one hour; a city bus covering $10\,\mathrm{km}$ in $30$ minutes; a [high-speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) train covering $75\,\mathrm{km}$ in $15$ minutes.

**Solution of Exercise 40.8.**

A stroller at $12$ kilometres per hour: rejected — three times the strolling pace. The bus: $10$ km in $30$ min is $20$ kilometres per hour — exactly city-bus pace, accepted. The train: $75$ km in $15$ min is $300$ kilometres per hour — top-of-the-board but plausible for a [high-speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) train: accepted.

**Exercise 40.9 ★★.**

The dozing passenger of [Remark 40.3](#rem-g6-describing-motion-relative): describe their motion — [trajectory](#def-g6-describing-motion-trajectory) and pace — first for the passenger opposite, then for the cow in the field. Why are both descriptions honest?

**Solution of Exercise 40.9.**

For the passenger opposite: no motion at all — no [trajectory](#def-g6-describing-motion-trajectory), pace zero. For the cow: a long [straight trajectory](#def-g6-describing-motion-trajectory) at the train’s full pace. Both honest: each describes the dozer compared to their own viewpoint, and motion is always motion compared to something.

**Exercise 40.10 ★★.**

A metro line runs $9\,\mathrm{km}$ end to end. The timetable allows $18$ minutes including six one-minute station stops. During the actual riding time, is the metro’s average pace above or below $45$ kilometres per hour? (Find the riding time first, then use a table.)

**Solution of Exercise 40.10.**

Riding time: $18 - 6 = 12$ minutes for $9\,\mathrm{km}$. Table: $12$ min $\to$ $9$ km, so $60$ min $\to$ $45$ km — exactly $45$ kilometres per hour: neither above nor below, but spot on.

**Exercise 40.11 ★★.**

Design a dotted film to settle a family argument: does the dog run at a steady pace when fetching, or sprint-and-coast? List your steps, what you will [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring), and what verdict each pattern of dots would give.

**Solution of Exercise 40.11.**

Film the fetch with a still camera; pause every half-second; mark the dog’s positions against fence posts; [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) the gaps. Even gaps throughout: steady pace. Gaps that spread, then shrink (or alternate wide and narrow): sprint-and-coast. The dots deliver the verdict, whatever the family shouted.

**Exercise 40.12 ★★★.**

A ball rolls off a table’s edge. Sketch its [trajectory](#def-g6-describing-motion-trajectory) from table-edge to floor as seen by someone standing in the room. Then argue — viewpoints again! — what [trajectory](#def-g6-describing-motion-trajectory) a tiny observer riding a cart that rolls under the falling ball at just the ball’s forward pace would see. (One of the two sees a curve; the other, something far simpler. This puzzle returns, solved, in the High School volume.)

**Solution of Exercise 40.12.**

From the room: a curve — the ball arches forward and down from the table edge to the floor. From the little cart rolling at the ball’s forward pace: the ball keeps no lead and no lag — it appears to fall *straight down*. Same fall, two viewpoints, and the simpler view belongs to the moving observer: a seed of a great idea.

## 40.5 Problem: The Dotted Film of Line 7

**Problem 40.1.**

Weekend problem — a transit engineer reads the dotted films of a tram line; diagnosing the ride, timing the line, catching the timetable’s lie

A transit engineer has filmed tram runs with a roadside camera that marks the tram’s position at every tick (one tick $=$ $10$ seconds), and hands you the films.

**Part I — Reading the films.** Film A, between two stations, shows gaps (in [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)): $50, 50, 50, 50, 50$.

1. Diagnose the motion on film A.
2. How far did the tram travel during film A’s $50$ seconds?
3. Film B, leaving a station, shows gaps $10, 25, 40, 50, 50$ . Tell the ride’s story in words.
4. On film B, during which tick-intervals is the motion (nearly) uniform?

**Part II — The line’s true pace.**

5. On its uniform stretch, the tram covers $50\,\mathrm{m}$ every $10$ seconds. Build the table up to one minute: how far per minute?
6. Continue to the hour: what is the tram’s cruising [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) in kilometres per hour?
7. The whole line is $12\,\mathrm{km}$ . At cruising [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) without stops, how many minutes end to end?
8. The timetable allows $50$ minutes end to end. How many minutes does the real line lose to stations, red [lights](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) and slow zones altogether?

**Part III — The advertisement on trial.** The company’s poster boasts: “Line 7 — clear across town at $18$ kilometres per hour, twice a walker’s best!”

9. Check the poster’s arithmetic against the timetable: $12\,\mathrm{km}$ in $50$ minutes — build the table toward one hour. Is “ $18$ kilometres per hour” honest for the *whole* journey?
10. A pedant objects: “No number of that kind is ever on the speedometer — the tram is either cruising at $18$ or standing at $0$ !” Explain what a whole-journey [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) really describes, and why it is still the fair number for a journey of many paces.
11. The engineer proposes skipping two stations to save four minutes. What would the new end-to-end time be, and — table again — the new whole-journey [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) in kilometres per hour? (Round sensibly.)
12. Write the engineer’s one-sentence report: which number should commuters trust for planning, the cruising [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) or the whole-journey [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) , and why?

**Solution of Problem 40.1.**

**1.** Equal gaps of $50\,\mathrm{m}$: [uniform motion](#def-g6-describing-motion-uniform). **2.** $5 \times 50 = 250\,\mathrm{m}$. **3.** The tram pulls away from rest: short gaps growing — speeding up over the first three ticks — then settling to steady $50\,\mathrm{m}$ gaps: cruising. **4.** The last two intervals, at $50\,\mathrm{m}$ each. **5.** $10$ s $\to$ $50\,\mathrm{m}$, so $60$ s $\to$ $300\,\mathrm{m}$ per minute. **6.** $60$ min $\to$ $60 \times 300 = 18\,000\,\mathrm{m}$: cruising [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) $18$ kilometres per hour. **7.** Table: $18\,\mathrm{km}$ per $60$ min, so $12\,\mathrm{km}$ in $40$ minutes. **8.** $50 - 40 = 10$ minutes lost to stops and slow zones. **9.** $50$ min $\to$ $12\,\mathrm{km}$, so $60$ min $\to$ $12 \times 60 \div 50 = 14.4\,\mathrm{km}$: the whole journey runs at about $14$ kilometres per hour — the poster’s $18$ is honest only for the cruising stretches, not “clear across town”. **10.** A whole-journey [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) describes the trip *as if* it were uniform: the one steady pace that would cover the same $12\,\mathrm{km}$ in the same $50$ minutes. No speedometer shows it, yet it is the fair number for planning, because it already contains every stop and slow zone. **11.** $50 - 4 = 46$ minutes; $12 \times 60 \div 46 \approx
15.7\,\mathrm{km}$ per hour — call it about $16$. **12.** For example: “Commuters should plan by the whole-journey [speed](https://one-course.com/books/physics/1/en/chapter/35-speed-distance-and-time#def-g5-speed-distance-time-speed) — about $14$ kilometres per hour today — because it is the only number that includes the waiting; the cruising $18$ flatters the line and misses every stop.”