MEI Conference Preparing to teach Projectiles. Kevin Lord.

Transcription

1 MEI Conference 2016 Preparing to teach Projectiles Kevin Lord

2 Session notes 2

3 Horizontal Motion Vertical Motion 3

4 Observing projectile motion From Mechanics in Action M. Savage and J. Williams, 1990 downloaded from 4

5 Mechanics in Action - Worksheet 28 guidance A projectile problem (1) Once you have the equipment set up so that the ball is following a consistent path, dampen the ball and then trace over the path with a marker pen. P (x,y) x (0,0) Let the start of the path be the origin, (0,0). Mark on the paper vertically lines at regular intervals and record the horizontal and vertical displacements (x,y) of points on the path. Roll the ball along the path recording the time to reach the points. This should be repeated to ensure reliability and consistency. x y t Using the data, plot graphs of x against y, x against t and y against t. Interpret your graphs Find suitable functions, y=f(x), x=f(t) and y=f(t) which fit your data. 5

6 Developing the projectile model 6

7 Developing the projectile model U Initial displacement Displacement at time t Horizontal Vertical Initial velocity Velocity at time t Horizontal Vertical Initial acceleration Acceleration at time t Horizontal Vertical 7

8 Projectiles experiment - The Skateboard/Scooter This experiment enables a key misconception of projectile motion to be explored by asking students to predict where a ball will land when released with horizontal velocity. Equipment: - Skateboard or scooter - Masking Tape or post-it - Ball Overview: A person will travel along on the skateboard/scooter carrying the ball. When they pass over the masking tape marker they will release the ball and others will see where it lands. Instructions: 1. Stick a piece of masking tape on the floor to serve as the point at which the ball will be released. 2. Each member of the group takes a piece of masking tape/post-it and writes their name on it and sticks it at the point where they think the ball will first hit the floor after being released. 3. A volunteer needs to be pushed (or propel themselves) on the skateboard at a constant speed. 4. When the skateboard passes over the marker they must release the ball by dropping it. 5. Other members of the group can identify where the ball first hit the floor and mark this point. This can process can be repeated a couple of times to check the accuracy of the experiment. 6. Discuss who made the best guess and find out why people chose the point they did. Questions: - Does the point where the ball lands change as the speed increases or decreases? - Draw a diagram showing the path of the ball. Why does this happen? - What are the limitations of the experiment? Note: An alternative approach if no skateboard or scooter is available could be for a person to run along the corridor and drop the ball as they run past the marker. 8

9 Misconceptions From Mechanics in Action page 46 9

10 Modelling projectile motion using Newton s Laws Set up the model Assume the projectile is a particle of mass m kg Intital velocity is U ms -1 Angle of projection is α x and y are the horizontal and vertical displacement of the particle from the origin (point of projection) at time t U Analysis At time t the only force acting on the particle is its weight, mg. Using Newton s second law Hence [ F = ma 0 mg ] = m [a x ] y This is a vector equation as force and acceleration are vector quantities Therefore horizontal acceleration = 0 and vertical acceleration = - g ms -2, which is constant. Since a = dv dr and v = dt particle at time t. dt we can use integration to find the velocity and displacement of the Integration of the horizontal and vertical components of acceleration will lead to the suvat equations for the horizontal and vertical components of velocity and displacement. Ucosα Velocity at time t v = [ Usinα gt ] (Ucosα)t + a Displacement at time t r = [ (Usinα)t 1 2 gt2 + b ] (where a and b are the initial displacement at t=0. If projected from the origin then a = b = 0) 10

11 Some useful links GeoGebra file - Projectiles: Equation of trajectory Projectiles applet This is a simple free applet that is available online and useful for illustrating projectile motion. Dan Meyer basketball activity Some sporting examples Human Cannonball (The farthest distance for a human fired from a cannon is m (193 ft 8.8 in) by Mechanics in Action by M Savage and J Williams, Mechanics%20in%20action.pdf A fantastic resource that is available electronically from the National STEM Centre It discusses modelling in mechanics and gives a number of practical experiments that can be done with students. It includes photocopiable handouts and details the theory behind the experiments as well as talking through solutions and results. David Marvin Jr (USA) in Milan, Italy, on 10 March 2011.) Golf - McIlroy hole-in-one (16 th Jan 2015) Basketball - Becky Hammon Top 10 (accessed 27 th Jan 2016) American football - Odel Beckham catch (Nov 2014) 11

12 Workshop reflections Key points, interesting thoughts, things you want to explore further, ideas you now have Actions (with timescales if you dare!) 12

13 OCR A Level Mathematics (2017) draft sample paper A girl is practising netball. She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop. The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows. The initial velocity of the ball has magnitude U ms 1. The angle of projection is 40 o. The ball is modelled as a particle. The hoop is modelled as a point. This is shown on the diagram below. (i) For U = 10, find (a) the greatest height above the ground reached by the ball, [5] (b) the distance between the ball and the hoop when the ball is vertically above the hoop. [4] (ii) Calculate the value of U which allows her to hit the hoop. [3] (iii) Suggest two improvements that might be made to this model. [2] 13

14 MEI A Level Mathematics (2017) draft sample paper 1 7. In this question take g = 10. A small stone is projected from a point O with a speed of 26 ms 1 at an angle θ above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7. You are given that sin θ = 12/13. After t seconds the horizontal and vertical displacements of the stone from O are x metres and y metres. (i) Using the standard model for projectile motion, show that y = 24t 5t 2 and find an expression for x in terms of t. [4] The stone passes through a point A which is 16m above the level of O. (ii) Find the two possible horizontal distances of A from O. [4] Suppose that a toy balloon is projected from O with the same initial velocity as the small stone. (iii) Give one way in which the model should be adapted in order to find expressions for the horizontal and vertical displacements of the balloon. [1] 14

15 Edexcel A Level Mathematics (2017) draft sample pure paper An archer shoots an arrow. The height, H metres, of the arrow above the ground is modelled by the formula H = d d 2, d 0 where d is the horizontal distance of the arrow from the archer, measured in metres. Given that the arrow travels in a vertical plane until it hits the ground, (a) find the horizontal distance travelled by the arrow, as given by this model. (3) (b) With reference to the model, interpret the significance of the constant 1.8 in the formula. (1) (c) Write d d 2 in the form A- B(d -C) 2 where A, B and C are constants to be determined. (3) (d) Hence, or otherwise, state the maximum height of the arrow above the ground. (1) 15

16 Edexcel A Level Mathematics (2017) draft sample mechanics and statistics paper 3 [In this question use g = 10 m s-2] A boy throws a stone with speed U ms -1 from a point O at the top of a vertical cliff. The point O is 18 m above sea level. The stone is thrown at an angle α above the horizontal, where tanα = ¾. The stone hits the sea at the point S which is at a horizontal distance of 36 m from the foot of the cliff, as shown in Figure 2. The stone is modelled as a particle moving freely under gravity. Find (a) the value of U, (6) (b) the time taken by the stone to travel from O to S, (2) (c) the speed of the stone when it is 10.8 m above sea level, giving your answer to 2 significant figures. (5) 16

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18 Preparing to teach projectiles Kevin Lord MEI

19 Session aims: - To develop and strengthen teachers understanding of projectile motion; and - To demonstrate effective teaching ideas including the use of IT and practical work and show how these can be used to improve students understanding.

20 Getting started... What is a projectile? What is it not? Observing projectile motion. What does it look like? How does it differ from linear motion? How is the distance/displacement, speed/velocity and acceleration of the projectile changing? Can we model the motion? Why does a projectile move in this way?

21 Kinematics How do you know where it will land? How can you vary the path followed? What is the same/different about the motion of the objects? As the projectile moves what is changing and what is staying the same?

22 2-D motion Imagine viewing the projectile motion from.... behind the thrower above the thrower

23 Vertical motion Horizontal motion

24 Practical work

25 Practical work

26 Trajectories, parametrics & GGB

27 U + + Initial displacement Displacement at time t Horizontal Vertical Initial velocity Velocity at time t Horizontal Vertical Initial acceleration Acceleration at time t Horizontal Vertical

28 Misconceptions

29 Modelling the motion using Newton s 2 nd Law Using Newton s second law Hence 0 mg F = ma = m a x a y U mg a x = 0 and a y = -g = -9.8 ms -2

30 Links to other topics Quadratic equations Trigonometry Calculus Parametric equations Vectors SUVAT equations Forces Newton s 2 nd Law