The Physics of Archery (1)
Objectives
To Understand the Basic Physical Principles of Archery
Through Identifying:
• Energy Transfers
• Energy Storage
• Trajectories
Bow Anatomy
Riser/Handle
Limbs
Grip
String
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.
Energy Transfer (solution)
Procedure
Main Energy transfer
1.
Hold up bow and put arrow
on string
Chemical in arm to kinetic in
arm, string & limbs
2.
Place fingers on string and
pull string back
Kinetic in arm & string to elastic
potential in limbs
3.
Anchor string and hand
under the chin
4.
Take aim
Elastic potential in limbs to
kinetic in string, limbs and
arrow
5.
Release the string
6.
Arrow hits target
(hopefully!!!)
Kinetic in arrow and sound in
limbs
Kinetic in arrow to heat and
sound in target
Energy Transfer (solution)
Sankey Diagram Showing Losses
Kinetic in string; sound in
limbs and string; heat in limbs
Chemical in arm
Kinetic in arm,
string, & limbs
Elastic potential
in limbs
Kinetic in arrow
Heat and sound in
target
Heat and sound of arrow
in flight
Sound in limbs; heat in
arms; heat in limbs & string
Energy Storage
Draw
force
(N)
165
Graph to show draw force against draw length
Work Done = Force (constant)
X Displacement in
direction of force
Work Done = area under the line
0
70
Draw
length
(cm)
Task 2: Calculate the energy stored in 165N bow drawn to 70cm
Energy Storage
Graph to show force against distance
Work Done = Force X Displacement
in direction of force
Force
(N)
400
Consider an action that consists of
two parts
• pushing a 20 kg block along for 20
cm
200
• pushing 2 20kg blocks along for
15 cm
0
20
35
Distance
(cm)
Area of rectangle = height X length
Add the shaded boxes together!
Task 2: Calculate the energy stored in 165N bow drawn to 70cm
Energy Storage (solution)
Draw
force
(N)
165
Graph to show draw force against draw length
Force = 165 N
Distance = 70cm
Work Done = ½ 165 * 0.7
0
70
Draw
length
(cm)
Work Done = 57.75 J
Arrow Energy
Nock
Shaft
Fletchings
Point
Kinetic energy = ½ mass X velocity2
Task 3: Calculate the velocity of the arrow (mass 25g), assuming
efficiency of energy transfer of limbs to arrow 0.70
Arrow Energy (solution)
Nock
Shaft
Fletchings
Point
Mass = 25g
Kinetic energy = 0.70 X work donebow
Work done = 57.75J
Kinetic energy = ½ mass X velocity2
Efficiency = 0.7
Therefore velocity = √(2 X kinetic energy/mass)
Velocity = √(2 X 40.425 / 0.025) = √ 3234 = 56.87 ms-1
Trajectories
Parabolic shape of arrow flight
Can consider the vertical and horizontal components of
the flight separately. Think SOH CAH TOA!!!
height
vh = v cos θ
vv = v sin θ
v = u + at
v=d/t
v2 = u2 + 2as
t
distance
height
t
θ
distance
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.
Trajectories (solution)
Split the component into vertical & horizontal:
v = 56.87 ms-1
for maximum range, θ = 45O
vh = v cos θ = 56.87 sin 45O = 40.21 ms-1
vv = v sin θ = 56.87 cos 45O = 40.21 ms-1
Taking vertical component first up to highest point:
u = 40.21 ms-1
a = g = -9.81 ms-2
v2 = u2 + 2as
0 = 40.212 – 2 X 9.81 X s
Therefore s = 40.212 / (2 X 9.81)
Maximum height = 82.4m
height
θ
distance
Trajectories (solution)
v = u + atup
0 = 40.21 - 9.81 X tup
Therefore tup = 40.21 / 9.81 = 4.10 s
Therefore tflight = 8.20 s
Taking the horizontal component:
velocity = 40.21 ms-1
time = 8.20 s
velocity = distance / time
Therefore distance = velocity X time
Max range = 40.21 X 8.20 = 329.7 m
height
θ
distance
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
3m
Time for the human to realise the arrow is
incoming = 1 second
0.5 m
Human response time 0.25 seconds
Target
Top view
Task 5: What is the minimum distance the
target needs to be before they can
successfully dodge an arrow?
Can humans dodge arrows?
Time taken for human target to dodge:
d2 = 32 + 0.52
d = 3.04 m
tmove = 3.04 / 10 = 0.304 s
tdodge = trealise + treact + tmove = 1 + 0.25 + 0.304 = 1.554 s
So we calculate the distance at which tflight = 1.554s
tup = tflight / 2 = 0.777 s
v = u + atup
u = v – atup = 0 + (9.81 X 0.777) = 7.62 ms-1 = vv
vv = v sin θ
v = 56.87 ms-1
sin θ = vv / v = 7.62 / 56.87 = 0.13 therefore θ = 7.7 0
vh = v cos θ = 56.87 cos 7.7 = 56.36 ms-1
vh = d / tflight
s = vh X tflight = 56.36 X 1.554 = 87.58 m
So the human target would need to be at least 87.58 m away from the archer
in order to dodge the arrow.
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
The Physics of Archery (2)
Photos from the Archery Have A Go Here!!!
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
Different Types of Bow
Longbow
Crossbow
Recurve
Compound
The Battle of Agincourt
1415
Country:
England
France
# of Men:
6000 men
20,000-30,000 men
# of Archers:
5000 archers
8000 archers
Style of Bow:
Using longbows
Using crossbows
Mass an Arrow:
50g
75g
Poundage:
150lbs @ 28”
300lbs @ 16”
Release rate:
12 arrows/min/archer
4 arrows/min/archer
Efficiency:
0.70
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
Conversion Rates & Useful Formulae
1 lb = 0.45 kg
1 inch (“) = 0.025 m
Area of triangle = ½(base X height)
v = u + at
s = ½ (v + u)t
k.e. = ½ mv2
v=d/t
English Longbow Range
Energy stored in bow:
Conversion:
150lb = 9.81 X 0.45 X 150 = 662.2 N
28” = 0.7m
Work done = ½ (662.2 X 0.7) = 231.8J
Velocity of arrow on release:
k.e. = ½ m v2
v2 = k.e. / ½ m = 0.7 X 231 / ½ 0.05 = 6489.3
v = 80.6 ms-1
Splitting the vertical and horizontal components:
vv = v sin θ
vh = v cos θ
vv = 80.6 sin 450 = 57 ms-1
vh = 80.6 cos 450 = 57ms-1
English Longbow Range
Time taken to reach highest point:
v = u + atup
tup = (v – u) / a = 57 / 9.81 = 5.81 s
tflight = 11.62 s
Maximum range of longbow:
vh = d / tflight
d = vh X tflight = 57 X 11.62 = 662.4m
French Crossbow Range
Energy stored in bow:
Conversion:
300lb = 9.81 X 0.45 X 300 = 1324.35 N
16” = 0.4m
Work done = ½ (1324.35 X 0.4) = 264.9J
Velocity of arrow on release:
k.e. = ½ m v2
v2 = k.e. / ½ m = 0.6 X 264.9 / ½ 0.075 = 4237.9
v = 65.1 ms-1
Splitting the vertical and horizontal components:
vv = v sin θ
vh = v cos θ
vv = 65.1 sin 450 = 46 ms-1
vh = 65.1 cos 450 = 46ms-1
French Crossbow Range
Time taken to reach highest point:
v = u + atup
tup = (v – u) / a = 46 / 9.81 = 4.69 s
tflight = 9.38 s
Maximum range of crossbow:
vh = d / tflight
d = vh X tflight = 46 X 9.38 = 431.5 m
Why did the English Win?
Terrain
Timing
Sited in a narrowing valley
Hours waiting
Muddy rainy conditions
Frequency of arrows
Position
Class/Tradition/Organisation
English archers on flanks
French archers pushed backwards by
nobility
French multiple lines, archers behind
front line
French disorganised
Equipment
Protection
Longer range
Armour
Greater frequency
Pikes in ground
Why did the English Win?
Poster & Presentation
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