Stephen Lee (MEI) Interesting sporting examples for motivating student learning in Mechanics 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 Image of a tennis player (removed as may be copyright) Ball straight up in the air Ball straight up in the air Likely wrong answer: Correct answer: 1
Ball through the air Ball through the air Projectile motion: Likely wrong answer: Correct answer: Ball through the air Mechanics in Action misconception 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! Interesting golf hole http://www.youtube.com/watch?v=b3_w_jryhr0 Geogebra file to enable an investigation Images of a tennis hawkeye shots (removed as may be copyright) 2
Image of a basketball player (removed as may be copyright) What picture best represents a basketball going through a hoop? A B C Ball diameter 24.6 cm Hoop diameter 45 cm (from above) Ball diameter 24.6 cm Hoop diameter 45 cm (from above) Hoop diameter from an angle is d = 45 sin 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 33.1 o 3
From Mathematics in Sport by M. Stewart Townend Hand-out contains two further (more difficult) examples: Free shots How high does a basketball reach? 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 (3.05m- 2.15m)] 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 35 o to the horizontal. Her arms are at 90 o 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 Image of a sphere (removed as may be copyright) 4
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. http://nrich.maths.org/6643 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? Can you draw a graph of the reaction force that the sphere exerts on the bike over time if the bike maintains this speed? Skydiving Image of a skydiver (removed as may be copyright) Skydiving 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 Long shot 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
Long shot MEI M1 Paper June 2012 A football is kicked with speed 31 ms -1 at an angle of 20 o 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