MEI Conference 2016

Preparing to teach

Projectiles

Kevin Lord Kevin.lord@mei.org.uk

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Session notes

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Horizontal Motion Vertical Motion

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Observing projectile motion

From Mechanics in Action M. Savage and J. Williams, 1990

downloaded from https://www.stem.org.uk/elibrary/resource/26065/mechanics-in-action

https://www.stem.org.uk/elibrary/resource/26065/mechanics-in-action

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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.

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.

(0,0)

P (x,y)

x

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Developing the projectile model

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Developing the projectile model

Initial displacement Displacement at time t

Horizontal

Vertical

Initial velocity Velocity at time t

Horizontal

Vertical

Initial acceleration Acceleration at time t

Horizontal

Vertical

U

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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.

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Misconceptions

From Mechanics in Action page 46

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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

Analysis

At time t the only force acting on the particle is its weight, mg.

Using Newton’s second law 𝐹 = 𝑚𝑎

Hence [0

−𝑚𝑔] = 𝑚 [

𝑎𝑥𝑎𝑦

]

Therefore horizontal acceleration = 0 and vertical acceleration = - g ms-2, which is constant.

Since 𝑎 = 𝑑𝑣

𝑑𝑡 and 𝑣 =

𝑑𝑟

𝑑𝑡 we can use integration to find the velocity and displacement of the

particle at time t.

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.

Velocity at time t 𝑣 = [𝑈𝑐𝑜𝑠𝛼

𝑈𝑠𝑖𝑛𝛼 − 𝑔𝑡]

Displacement at time t 𝑟 = [(𝑈𝑐𝑜𝑠𝛼)𝑡 + 𝑎

(𝑈𝑠𝑖𝑛𝛼)𝑡 −1

2𝑔𝑡2 + 𝑏

]

(where a and b are the initial displacement at t=0. If projected from the origin then a = b = 0)

U

This is a vector

equation as force and

acceleration are

vector quantities

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Some useful links

‘Mechanics in Action’ by M Savage and J Williams, 1990

http://www.nationalstemcentre.org.uk/dl/9c336158166118349649f6eca3e555fbcfdf9852/3635-

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.

GeoGebra file - Projectiles: Equation of trajectory

https://www.geogebra.org/material/simple/id/2545739

Projectiles applet

http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html

This is a simple free applet that is available online and useful for illustrating projectile motion.

Dan Meyer basketball activity

http://www.101qs.com/1195-will-it-hit-the-hoop

Some sporting examples

Human Cannonball – (The farthest distance for a human fired from a cannon is 59.05 m (193 ft 8.8 in) by

David Marvin Jr (USA) in Milan, Italy, on 10 March 2011.)

https://www.youtube.com/watch?v=0Y14kEY-6fk

Golf - McIlroy hole-in-one (16th Jan 2015)

http://www.bbc.co.uk/sport/0/golf/30858492

Basketball - Becky Hammon ‘Top 10’ (accessed 27th Jan 2016)

https://www.youtube.com/watch?v=0IxzescFEBk

American football - Odel Beckham catch (Nov 2014)

http://www.nfl.com/videos/nfl-cant-miss-plays/0ap3000000433422/Wk-12-Can-t-Miss-Play-Literally-

bending-like-Beckham

http://www.nationalstemcentre.org.uk/dl/9c336158166118349649f6eca3e555fbcfdf9852/3635-Mechanics%20in%20action.pdfhttp://www.nationalstemcentre.org.uk/dl/9c336158166118349649f6eca3e555fbcfdf9852/3635-Mechanics%20in%20action.pdfhttp://www.nationalstemcentre.org.uk/https://www.geogebra.org/material/simple/id/2545739http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.htmlhttp://www.101qs.com/1195-will-it-hit-the-hoophttp://www.101qs.com/1195-will-it-hit-the-hoophttps://www.youtube.com/watch?v=0Y14kEY-6fkhttp://www.bbc.co.uk/sport/0/golf/30858492http://www.bbc.co.uk/sport/0/golf/30858492https://www.youtube.com/watch?v=0IxzescFEBkhttps://www.youtube.com/watch?v=0IxzescFEBkhttp://www.nfl.com/videos/nfl-cant-miss-plays/0ap3000000433422/Wk-12-Can-t-Miss-Play-Literally-bending-like-Beckhamhttp://www.nfl.com/videos/nfl-cant-miss-plays/0ap3000000433422/Wk-12-Can-t-Miss-Play-Literally-bending-like-Beckhamhttp://www.nfl.com/videos/nfl-cant-miss-plays/0ap3000000433422/Wk-12-Can-t-Miss-Play-Literally-bending-like-Beckham

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Workshop reflections

Key points, interesting thoughts, things you want to explore further, ideas you now have…

Actions (with timescales if you dare!)

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OCR A Level Mathematics (2017) draft sample paper 3

12. 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 40o.

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]

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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 − 5𝑡2 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]

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Edexcel A Level Mathematics (2017) draft sample pure paper 1

16. An archer shoots an arrow. The height, H metres, of the arrow above the ground is modelled by the formula

H = 1.8 + 0.4d - 0.002d2, 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 1.8 + 0.4d - 0.002d2 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)

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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)

Preparing to

teach

projectiles

Kevin Lord

MEI

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.

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?

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?

2-D motion Imagine viewing the projectile motion from . . . .

behind the thrower above the thrower

Vertical motion

Horizontal motion

Practical work

Practical work

Trajectories, parametrics & GGB

+

+

Initial displacement Displacement at time t

Horizontal

Vertical

Initial velocity Velocity at time t

Horizontal

Vertical

Initial acceleration Acceleration at time t

Horizontal

Vertical

U

Misconceptions

Modelling the motion using

Newton’s 2nd Law

U

mg

Using Newton’s second law

𝐹 = 𝑚𝑎

Hence

0−𝑚𝑔

= 𝑚𝑎𝑥𝑎𝑦

𝑎𝑥 = 0 and 𝑎𝑦 = -g = -9.8 ms-2

Links to other topics

• Quadratic equations

• Trigonometry

• Calculus

• Parametric equations

• Vectors

• SUVAT equations

• Forces

• Newton’s 2nd Law

F-Preparing-teach-projectiles-klord.pdf (p.1-16)Kevinpowerpoint-F1.pdf (p.17-30)