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© Nuffield Foundation 2011 Nuffield Free- Standing Mathematics Activity Galileo’s projectil e model

© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

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Horizontal distance ( x metres) Height ( y metres) Time ( t seconds) Height ( y metres) Time ( t seconds) Horizontal distance ( x metres) Think about What assumptions are being made if the ball is modelled as a particle? Think about Which feature of a distance-time graph represents speed? Motion of a ball

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Page 1: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

© Nuffield Foundation 2011

Nuffield Free-Standing Mathematics Activity

Galileo’s projectile

model

Page 2: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Galileo’s projectile model

How far will the ski jumper travel before he lands?

How can you model the motion of the ski jumper?

Page 3: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Horizontal distance(x metres)

Height(y metres)

Time(t seconds)

Height(y metres)

Time(t seconds)

Horizontal distance(x metres)

Think about What assumptions are being made if the ball is modelled as a particle?

Think about Which feature of a distance-time graph represents speed?

Motion of a ball

Page 4: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Galileo’s projectile model

x

y

Horizontal direction – the motion has constant speed.

Vertical direction – projectile accelerates at 9.8 ms–2.

Vertical distance fallen is proportional to t2.

aob c d e

f

h

i

g

l

n

Think about What can you say about bc, cd, and de?What does this tell you about the horizontal velocity of the ball and the horizontal distance covered by the ball?How could you check that the vertical distances are proportional to t2?

Page 5: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

The modelling cycle

Defineproblem

Observe

Validate

Analyse

InterpretPredict

Real world MathematicsSet up a model

Page 6: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Experiment to validate Galileo’s model

Assumptions

• the ball is a particle

You need:

height h metres

range R metres

A B C

• air resistance is negligible

• the path of the projectile lies in a plane

Think about What modelling assumptions will be made?

Page 7: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Constants• the horizontal velocity of the

projectile after its launch from C

• the acceleration is g downwardsR

h

B C

Variables• the time, t seconds, measured from the instant of launch

• the height of the table h metres

• the distance, R metres, the ball lands from the foot of the table

Set up a model Think about What are the constants and variables?

Experiment to validate Galileo’s model

Page 8: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

AnalyseUse the equations for motion in a straight line with constant acceleration to predict :

• how long it will take the ball to fall to the ground• the horizontal distance, R metres, it will have travelled

Practical advice

To estimate the velocity of the ball at launch:

• assume the ball has constant velocity along BC.

• time the ball travelling a measured distance along BC.

• calculate the average velocity from distance travelledtime taken

Vary the release point A to vary the launch velocity

Use talcum powder or salt on paper to find where the ball lands

Experiment to validate Galileo’s model

Page 9: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Investigate how theoretical predictions compare with experimental results

Why might there be discrepancies between the two graphs?

Interpret

RangeR metres

Velocity of projection u ms–1

Graph based on analysis Graph of experimental results

RangeR metres

Velocity of projection u ms–1

Experiment to validate Galileo’s model

Page 10: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Experiment to validate Galileo’s model

Graph based on analysis

RangeR metres

Velocity of projection u ms-1

221atuts

Vertical motion downwards

gives 24.9th

4.9ht

Horizontal motion

221atuts

R = utgives

4.9huR

4.9hgradient

Analyse

Page 11: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Extension: projection at an angle to the horizontal

x

y

O uhoriz

uvert vhoriz

vvert

(x, y)

at time t

atuv

221atuts

Find equations for vhoriz, x, vvert and y in terms of uhoriz, uvert, t

In the horizontal direction, a = 0

In the vertical direction, a = –9.8

Sketch graphs of vhoriz, x, vvert and y against t

Page 12: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

x = uhoriz t

x

t0

vhoriz = uhoriz

uhoriz

vhoriz

t0

atuv

221atuts

In the horizontal direction, a = 0

Galileo’s projectile model

Page 13: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

vvert = uvert – 9.8t

0t

vvert

uvert

y = uvert t – 4.9t2

t

y

0

atuv

221atuts

In the vertical direction, a = –9.8

Galileo’s projectile model

Page 14: © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

Reflect on your work

• What are the advantages of Galileo’s projectile model?

• Do your experimental results validate the model?

• Suggest some examples of motion which could not be modelled very well as projectiles.