Speed and Velocity – Uniform Rectilinear Motion (Senior 1 Physics) | Ganzaa

Uniform Rectilinear Motion

Lessons

Senior 1 · Physics

Uniform Rectilinear Motion

5 lessons

Lesson 2 of 5

Speed and Velocity

25 min

How fast is "fast"?

Buses in Rwanda carry a speed governor, a device that stops the engine pushing the bus faster than a set speed, and roadside speed cameras in Kigali photograph vehicles that break the limit. Both machines measure the same thing: how much distance is covered in each unit of time. That quantity is speed, and in this lesson you will measure it on your own school field.

Definition

The average speed of an object is the total distance it travels divided by the total time taken:

v = \frac{d}{t}

where $d$ is the distance and $t$ the time. Speed is a scalar. SI unit: metre per second, written $\mathrm{m\,s^{-1}}$ . On roads, speeds are usually given in kilometres per hour, $\mathrm{km\,h^{-1}}$ .

Experiment: how fast do you walk and run?

Figure 2.1: Measuring average speed over a marked 20 m course.

Measure a straight 20.0 m course on flat ground with a tape measure, and mark each end with a cone.
A starter stands at the first cone. The runner starts a few metres behind it, so that she is already moving at a steady pace when she passes the cone.
The starter drops a flag as the runner passes the first cone; the timekeeper starts the stopwatch when the flag drops and stops it as the runner passes the second cone.
Record the time, then repeat for walking, jogging and running.

Pupil	Pace	Distance (m)	Time (s)	Average speed ( $\mathrm{m\,s^{-1}}$ )
Aline	walking	20.0	14.2	1.41
Keza	jogging	20.0	7.6	2.63
Eric	running	20.0	4.1	4.88

Each speed is the distance divided by the time: for Aline, $v = \frac{20.0\,\mathrm{m}}{14.2\,\mathrm{s}} = 1.41\,\mathrm{m\,s^{-1}}$ .

Important

Safety. Use flat ground free of stones and holes, run in one direction only, leave several metres beyond the finish cone to slow down, and keep spectators off the course.

Using the speed equation

The equation $v = \frac{d}{t}$ can be rearranged to find a distance or a time:

d = v \times t \qquad\qquad t = \frac{d}{v}

Changing units

$1\,\mathrm{km\,h^{-1}} = \frac{1000\,\mathrm{m}}{3600\,\mathrm{s}} = \frac{1}{3.6}\,\mathrm{m\,s^{-1}}$ . So:

to change $\mathrm{km\,h^{-1}}$ to $\mathrm{m\,s^{-1}}$ , divide by 3.6;
to change $\mathrm{m\,s^{-1}}$ to $\mathrm{km\,h^{-1}}$ , multiply by 3.6.

Moving thing	Typical speed ( $\mathrm{m\,s^{-1}}$ )	Typical speed ( $\mathrm{km\,h^{-1}}$ )
a person walking	1.4	5
a cyclist	5	18
a moto in a town street	11	40
a bus on a main road	17	60
a passenger jet cruising	250	900
sound in air	340	1224

Worked example 1: the average speed of a bus

A bus covers the road from Kigali to Musanze, about 90 km, in 2.0 h. Find its average speed in $\mathrm{km\,h^{-1}}$ and in $\mathrm{m\,s^{-1}}$ .

v = \frac{d}{t} = \frac{90\,\mathrm{km}}{2.0\,\mathrm{h}} = 45\,\mathrm{km\,h^{-1}}

45\,\mathrm{km\,h^{-1}} = \frac{45}{3.6}\,\mathrm{m\,s^{-1}} = 12.5\,\mathrm{m\,s^{-1}}

On the climbs and in the towns it went slower than this, and on the straight stretches faster: the average hides these changes.

Worked example 2: time for a journey

A car travels at $72\,\mathrm{km\,h^{-1}}$ . How long does it take to cover 1.5 km?

First convert the speed: $72 \div 3.6 = 20\,\mathrm{m\,s^{-1}}$ . Then, with $d = 1500\,\mathrm{m}$ :

t = \frac{d}{v} = \frac{1500\,\mathrm{m}}{20\,\mathrm{m\,s^{-1}}} = 75\,\mathrm{s}

Definition

Velocity is speed in a stated direction. Average velocity is the displacement divided by the time taken:

\bar{v} = \frac{\Delta x}{t}

Velocity is a vector; its SI unit is also $\mathrm{m\,s^{-1}}$ . Example: $8.0\,\mathrm{m\,s^{-1}}$ east.

Instantaneous speed is the speed at one particular moment. It is what a speedometer shows.

Worked example 3: speed and velocity are not the same

A runner sprints 100 m east in 12.5 s, then jogs 100 m back west to the start in 17.5 s. Find (a) her velocity on the way out, (b) her average speed for the whole run, (c) her average velocity for the whole run.

(a) $\bar{v} = \frac{100\,\mathrm{m}}{12.5\,\mathrm{s}} = 8.0\,\mathrm{m\,s^{-1}}$ east.

(b) Total distance 200 m in 30 s: $v = \frac{200\,\mathrm{m}}{30\,\mathrm{s}} = 6.7\,\mathrm{m\,s^{-1}}$ .

Warning

The average-speed trap. A car goes 30 km at $60\,\mathrm{km\,h^{-1}}$ and then 30 km at $30\,\mathrm{km\,h^{-1}}$ . Its average speed is NOT $45\,\mathrm{km\,h^{-1}}$ . The first part takes 0.5 h and the second 1.0 h, so

v = \frac{\text{total distance}}{\text{total time}} = \frac{60\,\mathrm{km}}{1.5\,\mathrm{h}} = 40\,\mathrm{km\,h^{-1}}

Always divide total distance by total time. Also convert units BEFORE you calculate: never divide kilometres by seconds.

Tip

Draw the triangle with $d$ on top and $v$ and $t$ side by side underneath. Cover what you want: covering $d$ leaves $v \times t$ ; covering $v$ leaves $\frac{d}{t}$ ; covering $t$ leaves $\frac{d}{v}$ .

Speed on Rwanda's roads

Speed limits in Rwanda are set in $\mathrm{km\,h^{-1}}$ and are lower in towns, near schools and on busy streets than on the open road. A speed camera measures a vehicle's instantaneous speed as it passes. A speed governor stops a bus or lorry from going over a set speed at any moment. Both matter because a vehicle's speed decides how far it travels in the second or so before its driver can react, as you will see in Lesson 5.

What Are Speed and Velocity? (GPB Education, Physics in Motion). Watch for the difference between average and instantaneous speed, and why velocity needs a direction.

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Practice

A cyclist rides 18 km in 1.5 h. Find her average speed in $\mathrm{km\,h^{-1}}$ and in $\mathrm{m\,s^{-1}}$ .
Convert (a) $90\,\mathrm{km\,h^{-1}}$ to $\mathrm{m\,s^{-1}}$ ; (b) $15\,\mathrm{m\,s^{-1}}$ to $\mathrm{km\,h^{-1}}$ .
How long does a bus travelling at an average of $60\,\mathrm{km\,h^{-1}}$ take to cover 45 km? Give your answer in minutes.
A moto travels at a steady $12\,\mathrm{m\,s^{-1}}$ for 2.5 minutes. How far does it go?
Sound travels at $340\,\mathrm{m\,s^{-1}}$ . How long does a shout take to cross a valley 1.7 km wide?
A girl walks 400 m east to a borehole in 5 minutes and returns to her starting point in another 5 minutes. Find her average speed and her average velocity.
A bus's speedometer reads $50\,\mathrm{km\,h^{-1}}$ as it passes a school, but its average speed for the whole journey was $38\,\mathrm{km\,h^{-1}}$ . Explain how both can be true.

Show answersHide answers

$v = \frac{18}{1.5} = 12\,\mathrm{km\,h^{-1}} = \frac{12}{3.6} = 3.3\,\mathrm{m\,s^{-1}}$
(a) $90 \div 3.6 = 25\,\mathrm{m\,s^{-1}}$ ; (b) $15 \times 3.6 = 54\,\mathrm{km\,h^{-1}}$
$t = \frac{45}{60} = 0.75\,\mathrm{h} = 45\,\mathrm{min}$
$t = 2.5 \times 60 = 150\,\mathrm{s}$ ; $d = 12 \times 150 = 1800\,\mathrm{m} = 1.8\,\mathrm{km}$
$t = \frac{1700\,\mathrm{m}}{340\,\mathrm{m\,s^{-1}}} = 5.0\,\mathrm{s}$
Average speed $= \frac{800\,\mathrm{m}}{600\,\mathrm{s}} = 1.3\,\mathrm{m\,s^{-1}}$ . Average velocity $= 0$ , because her displacement is zero.
The speedometer shows the instantaneous speed at that moment. The average speed is the total distance divided by the total time, which includes slower parts, stops at junctions and climbs.

Key points

Average speed $v = \frac{d}{t}$ ; rearranged, $d = v t$ and $t = \frac{d}{v}$ . SI unit $\mathrm{m\,s^{-1}}$ .
Divide by 3.6 to change $\mathrm{km\,h^{-1}}$ to $\mathrm{m\,s^{-1}}$ ; multiply by 3.6 to go back.
Velocity is speed with a direction (a vector); average velocity is displacement divided by time.
A speedometer shows instantaneous speed; average speed is always total distance over total time.

Sources and credits

2.3: Time, Velocity, and Speed, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Analogue Speedometer+Odometer - Huret.jpg by Illustratedjc, Wikimedia Commons, licensed under CC BY-SA 4.0.
Video: What Are Speed and Velocity? | Physics in Motion, GPB Education, embedded from YouTube.

The worked examples, diagrams, Rwandan applications and practice exercises are written by Ganzaa.

1How fast is "fast"?

2Experiment: how fast do you walk and run?

3Using the speed equation

Changing units

4Speed on Rwanda's roads

How fast is "fast"?

Experiment: how fast do you walk and run?

Using the speed equation

Speed on Rwanda's roads