In this section we will…

…develop our understanding of using numbers and equations to describe motion.

What equations to describe motion have you come across before?

What do we understand about 'acceleration'?

v ua

t

acceleration (m s–2) is

v ua

t

rate of change of velocity per unit time

Rearrange this to give v =…

v = u + at

There are three equations which together are known as the equations of motion.

when acceleration is constant (uniform)

and

motion is in a straight line

You need to be able to: select the correct formula identify symbols and units carry out calculations to solve

the problems of real-life motion.

You need to be able to:

carry out experiments to verify the equations of motion

To do this fully, you might find it an interesting challenge to…

understand where the equations come from.

Label the formula using correct symbols and units

v = u + at

Describe the motion of this object

How can we determine the displacement of the object?

Area under the graph = 1500 + 4500 = 6000 m

Area 1 = 5 × 300 = 1500 m

Area 2 = ½bh = ½ × (35 –- 5) × 300

= 4500 m

Area 1 = ut

Area 2 = ½bh = ½ × (v – u) × tSince v = u + atand v – u = at

v

u t

Area 1 = ut

Area 2 = ½bh = ½ × (v – u) × t

= ½ × at × t= ½ × at2

v

u t

Area under the graph = displacement s

Area 1 = ut

Area 2 = ½bh = ½ × (v – u) × t

= ½ × at × t= ½ × at2

v

u t

s = ut + ½at2

v = u + at

Start with Equation 1

v2 = (u + at)2

and square it

v2 = u2 + 2uat + a2t2

v2 = u2 + 2a(ut+ ½at2)

v2 = u2 + 2as

What do we need to think about when using the equations of motion?

What do the following quantities have in common?

velocity displacement

acceleration

The sign convention

When dealing with vector quantities we

must have both magnitude and

direction.

When dealing with one-dimensional

kinematics (ie motion in straight lines) we

use + and – to indicate travel in opposite

directions.

The sign convention

Normally we use the following convention:

positive + negative –

positive + negative –

The sign conventionNormally we use the following convention

positive + negative –

positive + negative –

Take care – in some questions the sign convention is reversed

v = u + atWhat does a positive value of acceleration mean?

Using the normal sign convention

–ve +ve

Christine Arron is a 100-m sprint athlete.

Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s.

Her maximum velocity is 10.49 m s–1.

Calculate her acceleration over the first 50 m of the race, showing full working.

–ve +ve

Her acceleration is 2.52 m s–2.

In this case, acceleration is a rate of change of velocity with time, with which we are familiar.

A positive value means, in this case, increasing velocity with time.

What else might it mean?

–ve +ve

As she passes the finish line, Christine begins to slow down.

She comes to rest in 8.20 s from a velocity of 9.73 m s–1.

Calculate her acceleration, showing full working.

–ve +ve

Her acceleration is a = –1.19 m s–2.

Notice that the acceleration has a negative value.

Explain this.

–ve +ve

Now consider Christine running in the opposite direction.

Notice that the sign convention remains the same.

–ve +ve

Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s.

Her maximum velocity is –10.49 m s–1 (why is it negative?).

Calculate her acceleration over the first 50 m of the race, showing full working.

–ve +ve

Her acceleration is –2.52 m s–2.

What does the negative mean?

–ve +ve

As she passes the finish line, Christine begins to slow down.

She comes to rest in 8.20 s from a velocity of –9.73 m s–1.

Calculate her acceleration, showing full working.

–ve +ve

Her acceleration is a = 1.19 m s–2.

Notice that the acceleration has a negative value.

Explain this.

–ve +ve

Equation 1 and the sign convention

A positive value means gaining speed while moving in the positive direction.

–ve +ve

Initial velocity

0 m s–1

Final velocity Acceleration

+10.49 m s–1210.49 0

2.52 4.16

v ua m s

t

–ve +ve

0 m s–1–9.73 m s–1 20 ( 9.73)1.19

8.20

v ua m s

t

OR

A positive value means the object is losing speed while moving in the negative direction.

Initial velocity Final velocity Acceleration

–ve +ve

In summary:

A negative value means the object is gaining speed while moving in the negative direction.

–10.49 m s–10 m s–1 210.49 02.52

4.16

v ua m s

t

Initial velocity Final velocity Acceleration

OR

A negative value means the object is losing speed while moving in the positive direction.

–ve +ve

Initial velocity

0 m s–1

Final velocity Acceleration

+10.49 m s–120 10.49

2.52 4.16

v ua m s

t

Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist.

Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s–1 in 1.75 s. Calculate his displacement in this time.

–ve +ve

Step 1: Write down the sign convention.

Step 2: Write down what you know (think suvat).

s displacementu initial velocityv final velocitya accelerationt time

Step 3: Any other information, eg acceleration due to force of gravity?

Step 4: Select formula – use data sheet.

Step 5: Substitute values then rearrange formula.

Step 6: Write the answer clearly, including magnitude and direction, and units.

Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist.

Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s-1 in 1.75 s. Calculate his displacement in this time.

–ve +ve

s = ? mu = 0 m s–1

v = 8.70 m s–1

a = ?t = 1.75 s

–ve +ve

2

2 2

1

2

2

v u at

s ut at

v u as

s = ? mu = 0 m s–1

v = 8.70 m s–1

a = ?t = 1.75 s

-ve +ve

2

2

1

21

(0 1.75) ( 1.75 )2

s ut at

s a

s = ? mu = 0 m s–1

v = 8.70 m s–1

a = ?t = 1.75 s

-ve +ve

–2

8.70 0 ( 1.75)

8.70 1.75

8.704.97m s

1.75

v u at

a

a

a

s = ? mu = 0 m s–1

v = 8.70 m s–1

a = ?t = 1.75 s

-ve +ve

2

2

2

1

21

(0 1.75) ( 1.75 )2

10 4.97 1.75

27.61m

s ut at

s a

s x

s

In the previous section we developed…

…our understanding of using graphs to describe motion

…our skills in interpreting graphs of motion

…our skills in describing motion using physics terms correctly.

In this section we planned to…

…develop our understanding of using numbers and equations to describe motion.

Next we will bring all of this together and use…

…our understanding of using graphs to describe motion

…our skills in interpreting graphs of motion

…our skills in describing motion using physics terms correctly

… our understanding of using numbers and equations to describe motion

for vertical motion

Everyday acceleration

What sort of accelerations do you

experience in everyday life?

How can this be investigated?

Everyday acceleration

Do you experience

accelerations only in the

horizontal?

Everyday acceleration

An accelerometer (a device which measures acceleration in three dimensions) can be used

to investigate accelerations which you experience in everyday life.

A stationary tennis ball

Describe its motion.

A tennis ball travelling vertically upwards

Film the ball as it is thrown upwardsand use tracker.jar to analyse its motion.

Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement.

Describe its motion.

A tennis ball dropped from a height

Film the ball as it falls and use tracker.jar to analyse its motion.

Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement.

Describe its motion.

Two tennis ballsdropped from a height

Predict the motion.

Observe.

Explain!

Two tennis ballsdropped from a height

Was your initial prediction that the two identical tennis balls dropped from the same height would hit the ground at exactly the same time?

Two tennis ballsdropped from a height

What did you think when you discovered that one ball had a significantly greater mass than the other?

What do you think should have happened?What did you observe?

The mass does not matter!

Both balls will hit the ground at the same time when dropped from the same height.

If you do not believe this, tracker.jar will allow you to analyse the motion.

© Erich Schrempp / Science Photo Library

© Nicola Jones

Suppose we can

switch off air

resistance.

Which will hit

the ground first?

The elephant and the feather in free-fall

The force of gravity near

the Earth’s surface gives

all objects the same

acceleration.

So why doesn’t the

feather reach the ground

at the same time as the

elephant?

The elephant and the feather with air resistance

We commonly use

a negative to

indicate

downward motion.

Dropping an elephant…

but be warned – you may come across questions in which the sign convention is reversed.

Dropping an elephant in the absence of air resistance.

Dropping an elephant in the absence of air resistance

Calculate speed, velocity, distance and displacement at 1-s intervals.

Time

(s)

Speed

(m s–1)

Velocity

(m s–1)

Distance fallen (m) Displacement (m)

0 0 0 0 0

1

2

3

4

5

Sketch graphs to show how the speed, distance, velocity and displacement vary

with time during the free-fall.

Time

Speed

Time

Distance

Time

Velocity

Time

Displacement

Sketch graphs to show how the speed, distance, velocity and displacement vary

with time during free-fall.

Time

Speed

Time

Distance

Time

Velocity

Time

Displacement

What is missing from the graph

sketches?

Time (s)

Speed (m s–1)

Displacement (m)Velocity (m s–1)

Distance ( m)

Time (s)

Time (s) Time (s)

0 0

0 0

A tennis ball dropped from a height and allowed

to bounce

Consider the ball being dropped, allowed to bounce and return to its original height.

Sketch your predictions for speed–time, velocity–time and acceleration–time graphs.

Compare to the results from this simulation.

http://www.helpmyphysics.co.uk/bouncing-ball.html

–1speed (m s )

time (s)

Describe the motion.

0

–1velocity m s

time (s)

When dropped, the ball gains speed in the negative direction hence the –ve sign for acceleration.

The ball then loses speed in the positive direction, coming to rest at the original height.

Does this happen in real life? Explain!

0

0

a ( m s–2)

Time (s)

Consider a tennis ball thrown upwards and allowed to fall back to

its starting position.

A tennis ball thrown upwards then allowed to fall

back to its starting position

Sketch the velocity, speed and acceleration graphs to describe its motion until it returns to its starting position.

Virtual Higher Experiments

→ Higher Physics → Mechanics

and Properties of

Matter → Activity 5b.

What is the force acting on a tennis ball thrown upwards?

Estimate the initial acceleration of a jumping popper.

What assumptions are you making?

How could your calculation be improved?

Calculate the initial acceleration of a jumping popper.

Observe what happens when the groaning tube is dropped.

Explain!

In the previous section we developed…

…our understanding of using graphs to describe motion

…our skills in interpreting graphs of motion

…our skills in describing motion using physics terms correctly.

In this section we…

…developed our understanding of using numbers and equations to describe motion.

Review your progress!