20/07/2012
Stephen Lee
(MEI)
Abstract
• Sporting examples can be used effectively to
motivate student learning in mathematics. They
can be particularly effective when used to
investigate and develop ideas in mechanics,
which is what this session will focus on.
Ball straight up in the air
Interesting
sporting
examples for
motivating
student
learning in
Mechanics
Ball straight up in the air
•
Image of a tennis player (removed as may be copyright)
Ball straight up in the air
• Likely wrong answer:
Correct answer:
1
20/07/2012
Ball through the air
• Projectile motion:
Ball through the air
• Mechanics in Action misconception
Ball through the air
• Likely wrong answer:
Correct answer:
Ball through the air - extensions
• Old MEI M2 coursework
• Available online: http://stem.org.uk/rx324
• Page 52/56 for these specific examples
Ball through the air - extensions
•
•
•
•
Modern technology – Hawkeye (tennis or cricket)
To review if a ball was in or out
To analyse serving statistics
TO MODEL!
•
Images of a tennis hawkeye shots (removed as may be copyright)
Interesting golf hole
• http://www.youtube.com/watch?v=b3_W_JryhR0
• Geogebra file to enable an investigation
2
20/07/2012
Shooting some hoops
•
Image of a basketball player (removed as may be copyright)
Shooting some hoops
• What picture best represents a basketball going
through a hoop?
A
Shooting some hoops
• What angle to the horizontal will enable a
basketball to go through a basketball hoop?
B
C
Shooting some hoops
• What angle to the horizontal will enable a
basketball to go through a basketball hoop?
– Ball diameter 24.6 cm
– Hoop diameter 45 cm
(from above)
Shooting some hoops
• What angle to the horizontal will enable a
basketball to go through a basketball hoop?
– Ball diameter 24.6 cm
– Hoop diameter 45 cm
(from above)
– Hoop diameter from an
angle is d = 45 sin 
Shooting some hoops
• What angle to the horizontal will enable a
basketball to go through a basketball hoop?
– Ball diameter 24.6 cm
– Hoop diameter 45 cm
(from above)
– Hoop diameter from an
angle is d = 45 sin 
– Will pass through if d > 24.6
i.e. 45 sin  > 24.6
  33.1o
3
20/07/2012
Shooting some hoops
• From ‘Mathematics in Sport’
by M. Stewart Townend
• Hand-out contains two further
(more difficult) examples:
– Free shots
– How high does a basketball reach?
Shooting some hoops – Free shots
• Nice investigation, though some more difficult
algebra
• Uses basic SUVAT equations (horzi/vert)
• Rearrange to give general equation of motion
• Turn into a quadratic and solve for the relevant
values [need to be careful, i.e. y is 0.9m (3.05m2.15m)]
• Eliminate, differentiate and solve
Press-ups!
• Figure 1 illustrates a model of a girl, Sarah, of mass 57
kg, in position to do a press-up on a horizontal floor.
– Her head is at A, her toes at B, her shoulders at C and her hands at D.
– Her body and legs, AB, and her arms, CD, are straight rigid rods. The centre of
mass of AB is at G.
– CD = 52 cm, CB = 140 cm and GB = 95 cm
Press-ups!
• Sarah now does press-ups on the stairs which are
shown in Figure 2 as a sloping plane at 35o to the
horizontal. Her arms are at 90o to the slope. The
distances CB and GB are the same as before .
• Calculate the force , F N, which Sarah’s arms must exert on her
body to hold herself up in this position.
• Calculate the force P N which her arms must exert on
her body to hold herself up in this position.
Press-ups!
• From ‘Revise for MEI Structured Mathematics
M2’
• Press-up example PT
Motorbike momentum
•
Image of a sphere (removed as may be copyright)
4
20/07/2012
Motorbike momentum
• A motorbike stuntman goes from town to town
riding his bike in a large sphere 6m in diameter
made from steel wire. The wire is so thick the
ball remains rigid while he rides inside it. See a
video here.
Motorbike momentum
• The motorbike and rider are about 2m tall together. We
might model this as a particle in the centre, so when the
bike & rider are in the sphere, this particle is 2m from the
centre of the sphere.
The wheels of the bike are 40cm in diameter. What is the
minimum angular speed the wheels must rotate if the
bike is to be able to go upside down over the top?
• http://nrich.maths.org/6643
Can you draw a graph of the reaction force that the
sphere exerts on the bike over time if the bike maintains
this speed?
Motorbike momentum
Skydiving
•
Image of a skydiver (removed as may be copyright)
Skydiving
Long shot
• Using a parachute, or not!
• From ‘Inspiring Maths in the classroom’ by
Alison Clark-Wilson and Adrian Oldknow
• Page 62:
http://education.ti.com/sites/UK/downloads/pdf/In
spiring_Maths_in_the_classroom.pdf
• August 1996
• In a UK poll conducted by Channel 4 in 2002,
the British public voted the goal by Beckham as
No.18 in the list of the 100 Greatest Sporting
Moments
• http://www.youtube.com/watch?v=DaSI2SiqNyA
5
20/07/2012
Long shot
• MEI M1 Paper June 2012
• “A football is kicked with speed 31 ms-1 at an
angle of 20o to the horizontal. It travels towards
the goal which is 50 m away. The height of the
crossbar of the goal is 2.44 m”
• Does the ball go over the top of the
crossbar? Justify your answer (6)
• State one assumption that you made (1)
Long shot
• Nice, as question is not leading through set
steps, the student has to work out the strategy:
– Calculate time to goal, via given horizontal distance,
then insert into SUVAT equation for vertical height at
that time
– OR Calculate equation for vertical height at time t,
then sub in s=2.44 to find time, then either
• insert into SUVAT equation for horizontal distance
• OR compare with time to goal via SUVAT equation with
horizontal distance
– OR Calculate equation of trajectory and sub in
horizontal distance to find vertical height
Prof. John Barrow articles
• John Barrow articles/lectures
• Maths in Sport series
via Gresham College
• http://www.gresham.ac.uk/category/lecturecategories/mathematics
• Let’s Twist Again: Throwing, Jumping, and
Spinning
• http://www.gresham.ac.uk/lectures-andevents/let%E2%80%99s-twist-again-throwingjumping-and-spinning
Ball bouncing down steps
• Advanced Mathematical Problems
(STEP- related)
by Stephen Siklos:
http://www.admissionstests.cambridgeassessment
.org.uk/adt/digitalAssets/110501_Advanced_Probl
ems_in_Mathematics.pdf
Ball bouncing down steps
In summary
•
•
•
•
Lots of different contexts
Lots of applications to real-life situations
Lots of opportunity to model
Lots of opportunity to investigate
• Chance to make mechanics interesting!
6