Linear Motion – Learning Outcomes Use the units of mass, length, and time.

Define displacement, velocity, and acceleration.

Use the units of displacement, velocity, and

acceleration.

Measure velocity and acceleration.

Use distance-time and velocity-time graphs.

Discuss linear motion in the context of sports.

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Linear Motion – Learning Outcomes Derive the equations of motion:

𝑣 = 𝑢 + 𝑎𝑡

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣2 = 𝑢2 + 2𝑎𝑠

Solve problems using the equations of motion.

Measure g.

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Use the Units of Mass, Length, and Time Unit of mass – kilogram (kg)

Derived units of mass – gram (1 g = 0.001 kg)

Unit of length – metre (m)

Derived units of length – millimetre (1 mm = 0.001 m),

centimetre (1 cm = 0.01 m), kilometre (1 km = 1000 m)

Unit of time – second (s)

Derived units of time – millisecond (1 ms = 0.001 s),

minute (1 min = 60 s), hour – (1 hr = 3600 s)

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Define Displacement Displacement, 𝑠 is distance in a given direction.

Like length and distance, it is measured in metres.

Displacement is a vector – only size and direction

matter: the path taken is irrelevant.

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More on vectors later.

Define Velocity Velocity, 𝑣 or 𝑢, is the rate of change of displacement

with respect to time.

Formula: 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑡𝑖𝑚𝑒or 𝑣𝑎𝑣𝑔 =

𝑠

𝑡

Velocity is the vector version of speed – magnitude and

direction both matter.

Like speed, it is measured in metres per second, m∙s-1.

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Solve Problems About VelocityWhen this carousel is in motion, its passengers move in a

circle at a steady 10 m∙s-1. Which of the following

statements are true?

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A. Only speed is

constant.

B. Only velocity is

constant.

C. Speed and velocity

are constant.

D. Neither speed nor

velocity are

constant.

by A

nd

rea

s P

rae

fcke

–C

C-B

Y-S

A-3

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Solve Problems About Velocity e.g. In 12 seconds, Grace travels along a curved path 68

m. The overall displacement she undergoes is 40 m NE.

Calculate her average speed and average velocity for

the journey.

e.g. Suzanne drives south alone a straight road with a

constant velocity of 30 m∙s-1. Find the displacement she

undergoes in 10 s.

e.g. Azhraa walks 3 m east, then 4 m north, with a total

journey time of 5 s. Calculate her displacement and

average velocity.

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Use Distance-Time Graphs Distance-time graphs display distance travelled on the

y-axis and time taken on the x-axis.

The slope of this graph is the speed.

e.g. the table below shows the distance travelled by a

runner.

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Distance (m) 0 2 4 6 8 10 12 14

Time (s) 0 1 2 3 4 5 6 7

Draw a graph representing this motion.

Use Distance-Time Graphs

1. Describe the motion shown in the table.

2. Plot a distance-time graph to represent the data.

3. When had Sorcha travelled 25 m?

4. How far had Sorcha travelled after 21 s?

5. Find the slope of the graph.

6. What was Sorcha’s speed?

Distance (m) 0 10 20 30 40 50 60 70

Time (s) 0 4 8 12 16 20 24 28

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The table below shows the distance travelled by Sorcha

on a bicycle over time.

Use Distance-Time GraphsWhich of these graphs shows an object:

a) moving away b) stopped c) moving closer

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Define Acceleration Acceleration, 𝑎 is the rate of change of velocity with

respect to time.

Formula: 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 =𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑡𝑖𝑚𝑒or 𝑎 =

𝑣−𝑢

𝑡

Acceleration is a vector – both magnitude and direction

matter.

Acceleration measures how quickly velocity changes so

it is measured in metres per second per second, m∙s-2.

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Solve Problems About Acceleration1. Bronagh accelerates in a car from 10 m∙s-1 to 30 m∙s-1 in

5 seconds. What is her acceleration?

2. Laura lands an aircraft at 60 m∙s-1. It takes her two

minutes to come to a stop. Calculate her acceleration

while she slows down.

3. Rachel is out for a run at 3 m∙s-1. She realises she has

forgotten her media player and turns around, reaching

5 m∙s-1 in 4 seconds. What is her acceleration as she

turns around?

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Use Velocity-Time Graphs Velocity-time graphs display velocity on the y-axis and

time on the x-axis.

The slope of this graph is the acceleration.

The area under the graph is the distance travelled.

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Use Velocity-Time Graphs A cheetah can go from rest up to a velocity of 28 m∙s-1 in

just 4 seconds and stay running at this velocity for a

further 10 seconds.

1. Sketch a velocity−time graph to show the variation of

velocity with time for the cheetah during these 14

seconds.

2. Calculate the acceleration of the cheetah during the

first 4 seconds.

3. What is the total distance travelled by this cheetah?

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Use Velocity-Time Graphs In a pole-vaulting competition, Michelle sprints from rest

and reaches a maximum velocity of 9.2 m s–1 after 3

seconds. She maintains this velocity for 2 seconds before

jumping.

1. Draw a velocity-time graph to illustrate Michelle’s

horizontal motion.

2. Use the graph to calculate her acceleration for the first

3 seconds.

3. Use your graph to calculate the distance she travelled

before jumping.

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Use Velocity-Time Graphs Amy is driving a speedboat. She starts from rest and

reaches a velocity of 20 m∙s-1 in 10 seconds. She

continues at this velocity for a further 5 seconds. She

then comes to a stop in the next 4 seconds.

1. Draw a velocity-time graph to show the variation of

velocity of the boat during its journey.

2. Use your graph to estimate the velocity of the

speedboat after 6 seconds.

3. Calculate the acceleration of the boat during the first

10 seconds.

4. What was the distance travelled by the boat when it

was moving at a constant velocity?

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Discuss Sports Linear motion comes into a number of sporting areas,

particularly in athletics, for example:

Running,

Swimming,

Drag racing,

Curling.

Note that most ball sports, circuit-races, hurdles etc. do

not fall under linear motion, as the motion takes place in

more than one dimension.

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Derive Equations of Motion The following equations of motion appear on pg 50 of

the Formula and Tables:

𝑣 = 𝑢 + 𝑎𝑡

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣2 = 𝑢2 + 2𝑎𝑠

These are the first three of eleven derivations on our

course.

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Derive 𝑣 = 𝑢 + 𝑎𝑡

𝑎 =𝑣 − 𝑢

𝑡Definition of acceleration

⇒ 𝑣 − 𝑢 = 𝑎𝑡 Multiplying both sides by 𝑡

⇒ 𝑣 = 𝑢 + 𝑎𝑡 Adding 𝑢 to both sides

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Derive 𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣𝑎𝑣𝑔 =𝑢 + 𝑣

2Definition of average

⇒𝑠

𝑡=𝑢 + 𝑣

2substituting 𝑣𝑎𝑣𝑔 =

𝑠

𝑡

⇒𝑠

𝑡=𝑢 + 𝑢 + 𝑎𝑡

2

substituting 𝑣 = 𝑢 + 𝑎𝑡

⇒𝑠

𝑡=2𝑢 + 𝑎𝑡

2

Combining like terms

⇒ 𝑠 =2𝑢𝑡 + 𝑎𝑡2

2

Multiplying both sides by 𝑡

⇒ 𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

Operating on the division by 2.

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Derive 𝑣2 = 𝑢2 + 2𝑎𝑠𝑣 = 𝑢 + 𝑎𝑡 From previous derivation

⇒ 𝑣2 = 𝑢 + 𝑎𝑡 2 Squaring both sides

⇒ 𝑣2 = 𝑢2 + 2𝑢𝑎𝑡 + 𝑎2𝑡2 Distributing the bracket

⇒ 𝑣2 = 𝑢2 + 2𝑎 𝑢𝑡 +1

2𝑎𝑡2

Factoring out 2𝑎

⇒ 𝑣2 = 𝑢2 + 2𝑎𝑠 Since 𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

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Solve Problems About Motion There are usually five quantities in linear motion

problems:

𝑢 – initial velocity

𝑣 – final velocity

𝑎 – acceleration

𝑠 – displacement

𝑡 – time

Each formula uses four of these quantities.

To solve problems, figure out which formula has the

quantities given / asked for in the question.

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Solve Problems About Motion A car starting from rest has an acceleration of 4 m∙s-1.

Find:

1. its velocity after 5 seconds,

2. the distance it travels in 5 seconds,

3. the time at which the car is travelling at 24 m∙s-1.

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Solve Problems About Motion1. A car with velocity 4 m∙s-1 accelerates at 2 m∙s-2. How

long does it take to reach a velocity of 30 m∙s-1?

2. A stone is dropped from the top of a building 40 m high.

The acceleration due to gravity is 9.8 m∙s-2.

i. With what speed does the stone hit the ground?

ii. How long does it take the stone to hit the ground?

3. An object is thrown upwards with an initial velocity of

100 m∙s-1. Find:

i. the greatest height reached,

ii. the time taken to reach the ground again.

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Solve Problems About Motion1. A bicycle passes a point on a road with velocity 4 m∙s-1

and an acceleration of 2 m∙s-2. Four seconds later, a car

passes the same point with velocity 2 m∙s-1 and

acceleration 4 m∙s-2. When and how far from the point

do the bicycle and car meet?

2. A stone is thrown vertically upwards from a point on the

ground with an initial speed of 50 m∙s-1. Three seconds

later, another stone is thrown vertically up from the

same point with a speed of 70 m∙s-1. How long after the

first stone leaves the ground do the two stones meet?

What is their height above the ground at this time?

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