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The Physics of Archery (1)

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Objectives To Understand the Basic Physical Principles of Archery Through Identifying: Energy Transfers Energy Storage Trajectories

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Bow Anatomy Limbs Riser/Handle String Grip

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Energy Transfer Procedure 1.Hold up bow and put arrow on string 2.Place fingers on string and pull string back 3.Anchor string and hand under the chin 4.Take aim 5.Release the string 6.Arrow hits target (hopefully!!!) TASK 1: Identify the stages in this energy transfer. Draw a Sankey diagram to show this.

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Energy Transfer (solution) Procedure 1.Hold up bow and put arrow on string 2.Place fingers on string and pull string back 3.Anchor string and hand under the chin 4.Take aim 5.Release the string 6.Arrow hits target (hopefully!!!) Main Energy transfer Chemical in arm to kinetic in arm, string & limbs Kinetic in arm & string to elastic potential in limbs Elastic potential in limbs to kinetic in string, limbs and arrow Kinetic in arrow and sound in limbs Kinetic in arrow to heat and sound in target

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Chemical in arm Energy Transfer (solution) Kinetic in arm, string, & limbs Sound in limbs; heat in arms; heat in limbs & string Elastic potential in limbs Kinetic in arrow Heat and sound in target Kinetic in string; sound in limbs and string; heat in limbs Heat and sound of arrow in flight Sankey Diagram Showing Losses

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70 Energy Storage Graph to show draw force against draw length Draw force (N) Draw length (cm) Work Done = Force (constant) X Displacement in direction of force Work Done = area under the line Task 2: Calculate the energy stored in 165N bow drawn to 70cm

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20 Energy Storage Graph to show force against distance Force (N) Distance (cm) Work Done = Force X Displacement in direction of force Consider an action that consists of two parts pushing a 20 kg block along for 20 cm pushing 2 20kg blocks along for 15 cm Area of rectangle = height X length Add the shaded boxes together! Task 2: Calculate the energy stored in 165N bow drawn to 70cm

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70 Energy Storage (solution) Graph to show draw force against draw length Draw force (N) Draw length (cm) Force = 165 N Distance = 70cm Work Done = ½ 165 * 0.7 Work Done = J

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Arrow Energy Task 3: Calculate the velocity of the arrow (mass 25g), assuming efficiency of energy transfer of limbs to arrow 0.70 Kinetic energy = ½ mass X velocity 2 Fletchings Nock Shaft Point

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Arrow Energy (solution) Kinetic energy = 0.70 X work done bow Kinetic energy = ½ mass X velocity 2 Therefore velocity = √(2 X kinetic energy/mass) Velocity = √(2 X / 0.025) = √ 3234 = ms -1 Fletchings Nock Shaft Point Mass = 25g Work done = 57.75J Efficiency = 0.7

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Trajectories Parabolic shape of arrow flight Can consider the vertical and horizontal components of the flight separately. Think SOH CAH TOA!!! v h = v cos θv v = v sin θ v = u + atv 2 = u 2 + 2as v = d / t θ Task 4: Split the components of the arrow velocity up and calculate the max range and the max height at that range. Assume air resistance is negligible. height t distance t height

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Trajectories (solution) Split the component into vertical & horizontal: v = ms -1 for maximum range, θ = 45 O v h = v cos θ= sin 45 O =40.21 ms -1 v v = v sin θ = cos 45 O = ms -1 Taking vertical component first up to highest point: u = ms -1 a = g = ms -2 v 2 = u 2 + 2as 0 = – 2 X 9.81 X s Therefore s = / (2 X 9.81) Maximum height = 82.4m θ height distance

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Trajectories (solution) v = u + at up 0 = X t up Therefore t up = / 9.81 = 4.10 s Therefore t flight = 8.20 s Taking the horizontal component: velocity = ms-1time = 8.20 s velocity = distance / time Therefore distance = velocity X time Max range = X 8.20 = m θ height distance

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Can humans dodge arrows? The human target would need to move outside of the area as shown Assume the archer is very accurate. Fastest human travels at 10ms -1 Time for the human to realise the arrow is incoming = 1 second Human response time 0.25 seconds Task 5: What is the minimum distance the target needs to be before they can successfully dodge an arrow? Top view Target 3 m 0.5 m

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Can humans dodge arrows? Time taken for human target to dodge: d 2 = d = 3.04 m t move = 3.04 / 10 = s t dodge = t realise + t react + t move = = s So we calculate the distance at which t flight = 1.554s t up = t flight / 2 = s v = u + at up u = v – at up = 0 + (9.81 X 0.777) = 7.62 ms -1 = v v v v = v sin θv = ms -1 sin θ = v v / v = 7.62 / = 0.13 therefore θ = v h = v cos θ = cos 7.7 = ms -1 v h = d / t flight s = v h X t flight = X = m So the human target would need to be at least m away from the archer in order to dodge the arrow.

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Safety Information Before taking part in archery you need to understand certain safety rules!!! Do not put the arrow on the string until you are standing on the shooting line Do not distract anyone who is shooting Once on the string, only ever point the arrows in the direction of the targets If you are not shooting stay well behind the shooting line If you see any possible hazard or danger (e.g. someone is walking behind the targets) then shout the word “FAST”. If you hear the word “FAST”, then do not shoot any arrows under any circumstances. One whistle means shooting can start, two whistles means that you can collect your arrows from the target Don’t draw and then release the bow without an arrow on it (this is called a “dry fire”) as this can damage the bows FOLLOW THE INSTRUCTIONS OF THE COACH AT ALL TIMES

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The Physics of Archery (2) Photos from the Archery Have A Go Here!!!

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Objectives To Reinforce our Understanding of the Basic Principles of Archery by: Looking at a real life application at the Battle of Agincourt Creating a poster and presenting on an area of what has been learnt

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Different Types of Bow Longbow Crossbow Recurve Compound

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The Battle of Agincourt England 6000 men 5000 archers Using longbows 50g 28” 12 arrows/min/archer 0.70 France 20,000-30,000 men 8000 archers Using crossbows 75g 16” 4 arrows/min/archer 0.60 Convert the units from imperial to metric Calculate the energy stored in each type of bow Calculate the speed of the arrow on release Calculate the maximum range Remember to note down any assumptions you have made Country: # of Men: # of Archers: Style of Bow: Mass an Arrow: Poundage: Release rate: Efficiency: 1415

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Conversion Rates & Useful Formulae 1 lb = 0.45 kg 1 inch (“) = m Area of triangle = ½(base X height) v = u + ats = ½ (v + u)t k.e. = ½ mv 2 v = d / t

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Energy stored in bow: Conversion: 150lb = 9.81 X 0.45 X 150 = N 28” = 0.7m Work done = ½ (662.2 X 0.7) = 231.8J Velocity of arrow on release: k.e. = ½ m v 2 v 2 = k.e. / ½ m = 0.7 X 231 / ½ 0.05 = v = 80.6 ms -1 Splitting the vertical and horizontal components: v v = v sin θ v h = v cos θ v v = 80.6 sin 45 0 = 57 ms -1 v h = 80.6 cos 45 0 = 57ms -1 English Longbow Range

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Time taken to reach highest point: v = u + at up t up = (v – u) / a = 57 / 9.81 = 5.81 s t flight = s Maximum range of longbow: v h = d / t flight d = v h X t flight = 57 X = 662.4m English Longbow Range

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Energy stored in bow: Conversion: 300lb = 9.81 X 0.45 X 300 = N 16” = 0.4m Work done = ½ ( X 0.4) = 264.9J Velocity of arrow on release: k.e. = ½ m v 2 v 2 = k.e. / ½ m = 0.6 X / ½ = v = 65.1 ms -1 Splitting the vertical and horizontal components: v v = v sin θ v h = v cos θ v v = 65.1 sin 45 0 = 46 ms -1 v h = 65.1 cos 45 0 = 46ms -1 French Crossbow Range

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Time taken to reach highest point: v = u + at up t up = (v – u) / a = 46 / 9.81 = 4.69 s t flight = 9.38 s Maximum range of crossbow: v h = d / t flight d = v h X t flight = 46 X 9.38 = m French Crossbow Range

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Terrain Sited in a narrowing valley Muddy rainy conditions Why did the English Win? Position English archers on flanks French multiple lines, archers behind front line Class/Tradition/Organisation French archers pushed backwards by nobility French disorganised Protection Armour Pikes in ground Timing Hours waiting Frequency of arrows Equipment Longer range Greater frequency

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Why did the English Win?

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Design a Poster. Presentations to be given at Friday’s lesson. Split into groups – each responsible for one area. Poster options: Physical A1 poster. PowerPoint poster. Web page poster. 1.Intro page 2.Equipment Anatomy & How to Shoot 3.Energy Transfers in Archery 4.Trajectories 5.The Battle of Agincourt Poster & Presentation

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