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LINEAR KINEMATICS
KIN3323: Biomechanics
TYPES OF MOTION
Description of the form, pattern, or sequencing of movement with respect to time
– Involves: position, velocity and acceleration of a body without concern for the forces which cause motion
– Does not involve force
Kinematics
1. Linear Motion – Translation: all points on an object move the same
distance, in the same direction, and at the same time’ – 2 Types
a) Rectilinear: Movement in straight line path b) Curvilinear: Movement in straight curved path
Types of Motion
1a. Rectilinear Translation (Linear Motion)
Example • Figure skater gliding across the ice in a static position
1a. Rectilinear Translation (Linear Motion)
Example • Figure skater gliding across the ice in a static position
1b. Curvilinear Translation (Linear Motion)
Example • Skateboarder in air holding a static position • Ski jumper
2. Angular Motion – Rotation: all points on an object moves in circles
about the same fixed axis
Types of Motion
2. Angular Motion
Example • Elbow flexion • Figure skater spinning
3. General Motion – Combination of translation and rotation – Most common type of motion in sports and human
movement
Types of Motion
DESCRIPTION OF MOTION
• Position • Distance / displacement • Speed / velocity • Acceleration
Description of motion
• The location of a point, with respect to the origin, within a spatial reference frame
• Reference frames – Cartesian coordinate system
• Fixed point (origin) with axes that are perpendicular to each other
• 2-D or 3-D system
Description of motion: Position
Origin = (0,0)
+y
-x
-y
+x
2-D Coordinate System
Used when motion is primarily in one plane
• x (horizontal) • y (vertical)
(0,0) means that the point is located at x=0 & y=0 ex. (2,5) describes the point located at x=2 & y=5
Example of reference frames in Biomechanics
Origin
+y
-x
-y
+x
Origin
+y
-x
-y
+x
Global reference frame Relative to gravity
Anatomical reference frame Relative to body segment
(ex. forearm relative to arm)
Distance • Length of path
followed by object from initial to final position
• Scalar quantity – Only magnitude
Displacement • Straight line distance
in a specific direction from initial to final position
• Vector quantity – Has magnitude and
direction
Distance vs. Displacement
x
y
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1
2
3
4
5
6
7
8
9
10
11
12
(0,0)
(2,7)
(11,10)
Distance = length of path
1m
1.5m
3m
4m
1m
2m
3.8m
2m
Distance = 1+1.5+3+4+1+2+3.8+2 = 17.8
x
y
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1
2
3
4
5
6
7
8
9
10
11
12
(0,0)
(2,7)
(11,10)
Displacement = Straight line distance
Δx
Δy
Displacement2 = Δx2 + Δy2
Displacement = Δx2 + Δy2 = 92 + 32 = 9.5
Displacement = Straight line distance
(2,7)
(11,10)
Δx = 11 – 2 = 9
Δy = 10 – 7 = 3
Which is more relevant: Distance or displacement?
400m Race
Which is more relevant: Distance or displacement?
400m Race
Start End
Displacement? Distance?
Which is more relevant: Distance or displacement?
Javelin throw
Which is more relevant: Distance or displacement?
Javelin throw
Distance = 200m
Distance = 200m
Displacement = 20m
Displacement = 80m
Speed • A distance traveled
over time • Scalar quantity
– Magnitude only
Velocity • A displacement
achieved over time • Vector quantity
– Has magnitude and direction
Description of motion: Speed vs. Velocity
10 20 30 40 50 10 20 30 40
10 20 30 40 50 10 20 30 40
Speed vs. Velocity
A football player caught a ball at 0 yd line. He ran 52 yd and got tackled after gaining 40 yds and moving 20 yds to the left. The play lasted 5 sec.
Speed = distance / Δ time
Speed = distance traveled over time
Distance = 55 yd ΔTime = 5 sec
Speed = distance / Δ time
= 55 yd / 5 sec
He ran 55 yds over 5 sec = 11 yd / sec
Velocity = displacement achieved over time
40yds
20yds
Velocity = displacement / Δ time
Displacement = 402 + 202
ΔTime = 5 sec
Speed = distance / Δ time
= 44.7 yd / 5 sec
= 8.9 yd / sec
= 44.7 yd
Velocity = displacement achieved over time
20yd/5sec = 4yr/sec
Alternative solution
Velocity = (8yr/sec)2 + (4yr/sec)2
= 8.9 yd / sec
40yd/5sec = 8yr/sec
Average • Speed or velocity
averaged over time – ex. Running pace
Instantaneous • Speed or velocity at a
specific instant – ex. Speedometer reading
• Used when interested in knowing – Max/min values – Values at specific instant
Average vs. instantaneous speed / velocity
“10 minute mile” = 1mile / 10min = 0.1 mile / min = 6 mile / hour = 6 mph
Instantaneous velocity
9m/s
2m/s
At take off, the high jumper’s vertical velocity was 9m/s and her horizontal velocity was 2m/s. Calculate the take off velocity and angle.
? m/s
?°
Instantaneous velocity
Vx
90mph
At the instant of ball release, instantaneous velocity of the ball was 90 mph at a direction 10° above horizontal. What was the vertical and horizontal
velocity of the ball?
Vy
10°
• Rate of change in velocity – ex. “0 to 60mph in 3 seconds” – Vector quantity
Description of motion: Acceleration
Acceleration = Rate of change in velocity
Acceleration = Δ velocity / Δ time
= (Vf – Vi) / Δ time
Where: Vf = final velocity Vi = initial velocity
Acceleration = Rate of change in velocity
Sean is running a 100m dash. When the starter’s pistol fires, he leaves the starting block and
continues speeding up until he reaches his top speed of 11m/s 6s into the race. He holds this
speed for 2s and then gradually slows down until he crosses the finish line at 9m/s.
What was Sean’s acceleration during:
- First 6s of the race - From 6-8s into the race - Last 3s
Acceleration = Rate of change in velocity
First 6s: Vi = 0m/s Vf = 11m/s
6-8s:
Vi = 11m/s Vf = 11m/s
Last 3s:
Vi = 11m/s Vf = 9m/s
(+) and (-) Acceleration
• Positive acceleration indicates that the object is speeding up – Acceleration
• Negative acceleration indicates that the object is slowing down – Deceleration
Acceleration due to gravity
• The rate of change in velocity caused by the force of gravity – 9.81m/s2 downward
Summary of Kinematic Descriptors
Scalar Vector
Distance Displacement
Speed Velocity
“Length of path” “Straight line distance”
“Distance over Fme” “Displacement over Fme”
Accelera=on “Change in velocity over Fme” “A rate of change in velocity”
CHARACTERISTICS OF PROJECTILE MOTION
A rifle is shot in a perfectly horizontal plane. At the same instant a bullet is dropped from the
same height. Which bullet hits the ground first, the one shot from the rifle or the one dropped
next to the rifle?
Projectile Motion
Don’t say it, think about it....
• Projectile is an object that has been projected into the air or dropped and is only acted on by the forces of gravity and air resistance – In this unit, we consider air resistance negligible
• Examples: – Soccer ball after impact – Diver, long jumper, and high jumper after a take off – Ball dropped from a top of the building
Projectile Motion
Apex
Trajectory (Parabolic)
Release
Hei
ght
Projectile Motion
Landing Distance
• The only force acting on projectiles is the gravitational force (ignoring air resistance)
• Gravitational force only affects vertical velocity – Vertical and horizontal velocity are independent!!
Effects of gravity on projectile motion
• Gravity causes 9.81m/s2 acceleration in vertical (downward) direction – Vertical velocity of the projectile decrease by 9.81m/s
every second
Effects of gravity on vertical velocity
Apex
Hei
ght (
m)
Projectile Motion
0 1 2 3 4 5 6 7 8
Distance (m)
Velo
city
(m/s
)
0
Vertical velocity
39.2m/s -‐9.81 -‐9.81
-‐9.81 -‐9.81
-‐9.81 -‐9.81
-‐9.81 -‐9.81
• Gravity does not cause acceleration in horizontal direction – Gravity has no influence on horizontal velocity – Horizontal velocity of the projectile does not change (=
stays constant)
Effects of gravity on horizontal velocity
Apex
Posi
tion
(m)
Projectile Motion
0 1 2 3 4 5 6 7 8
Time (sec)
Velo
city
(m/s
)
0
Horizontal velocity
20.0m/s
• Trajectory of the center of mass (COM) of the projectile is parabolic – Symmetric about the apex – Time up = time down
• Vertical velocity – Decreases by 9.81m/s every second during up phase – 0m/s at apex
• Horizontal velocity – Is constant (ignoring air resistance)
Summary of the characteristics of projectile motion
FACTORS INFLUENCING PROJECTILE MOTION
• For the analysis of projectile motion, velocity is often resolved into horizontal (Vx) and vertical (Vy) component
Horizontal and vertical velocity
Velocity
θ
Vx
Vy
• Increasing projection angle will: – Decrease horizontal velocity – Increase vertical velocity
20m/s
20m/s
θ=70° θ=30°
18.8m/s
6.8m/s 17.3m/s
10.0m/s
Changing projecFon angle will change the raFo between horizontal and verFcal velocity
Smaller projec0on angle = Smaller Vy and greater Vx
Greater projec0on angle = greater Vy and smaller Vx
Effects of projection angle on horizontal and vertical velocity
Milder projection angle = flatter parabola
Steeper projection angle = taller parabola
Effects of projection angle on horizontal and vertical velocity
• Projection speed influences the shape of projectile’s trajectory
• Increasing projection speed will proportionally increase horizontal and vertical velocity
Effects of projection speed on horizontal and vertical velocity
20m/s
10m/s
θ=45° θ=45°
7.1m/s
7.1m
/s
14.1m/s
14.1m/s
Increasing projecFon speed will proporFonally increase both horizontal and verFcal velocity
• Projection speed influences the size of projectile’s trajectory
Smaller projection speed = smaller parabola
Greater projection speed = greater parabola
Effects of projection speed on horizontal and vertical velocity
• To increase vertical velocity, you can: 1. Increase projection speed 2. Increase projection angle
• To increase horizontal velocity, you can: 1. Increase projection speed 2. Decrease projection angle
Vertical and horizontal velocity
1. Maximum height • How high does the object travel?
2. Flight time • How long does the object stay in air?
3. Flight distance • How far does the object travel?
Variables of interest in projectile motion
• Maximum height is affected by 2 factors – Vertical velocity
– Projection height
Factors Influencing Maximum Height
Increasing verFcal velocity by increasing projec=on velocity will increase maximum height
0 1 2 3 4 5 6 7 8 0
Distance (m)
Hei
ght (
m)
• Maximum height is affected by 2 factors – Vertical velocity
– Projection height
Factors Influencing Maximum Height
Increasing verFcal velocity by increasing projec=on angle will increase maximum height
Hei
ght (
m)
0 1 2 3 4 5 6 7 8 0
Distance (m)
• Maximum height is affected by 2 factors – Vertical velocity
– Projection height
Factors Influencing Maximum Height
Increasing projecFon height will increase maximum height
0 1 2 3 4 5 6 7 8 0
Distance (m)
Hei
ght (
m)
1. Maximum height • How high does the object travel?
2. Flight time • How long does the object stay in air?
3. Flight distance • How far does the object travel?
Variables of interest in projectile motion
• Flight time is affected by 2 factors – Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing – Vertical velocity
Factors Influencing Flight Time
Shorter flight time Longer flight time
PosiFve relaFve height will decrease the flight Fme
NegaFve relaFve height will increase the flight Fme
• Flight time is affected by 2 factors – Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing – Vertical velocity
Factors Influencing Flight Time
RelaFve height only affects 0me down
Increasing verFcal velocity by increasing projec=on velocity will increase flight Fme
0 1 2 3 4 5 6 7 8 0
Distance (sec)
Hei
ght (
m)
• Flight time is affected by 2 factors – Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing – Vertical velocity
Factors Influencing Flight Time
Increasing verFcal velocity by increasing projec=on angle will increase flight Fme
Hei
ght (
m)
0 1 2 3 4 5 6 7 8 0
Distance (sec)
• Flight time is affected by 2 factors – Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing – Vertical velocity
Factors Influencing Flight Time
1. Maximum height • How high does the object travel?
2. Flight time • How long does the object stay in air?
3. Flight distance • How far does the object travel?
Variables of interest in projectile motion
• Flight distance is affected by 2 factors – Flight time
• Given the horizontal velocity, longer the object is in air, the longer the flight distance
– Horizontal velocity • Given the flight time, greater the horizontal velocity, the longer
the flight distance
Factors Influencing Flight Distance
Speed = Distance
Time Distance = Speed x Time
• Flight distance is affected by 2 factors – Flight time
• Given the horizontal velocity, longer the object is in air, the longer the flight distance
– Horizontal velocity • Given the flight time, greater the horizontal velocity, the longer
the flight distance
Factors Influencing Flight Distance
Flight =me Horizontal velocity Increase projecFon speed Increase * Increase RelaFve height of release Increase no effect Increase projecFon angle Increase * Decrease Decrease projecFon angle Decrease ** Increase
* by increasing verFcal velocity
** by decreasing verFcal velocity
• Goal of the task – High jump vs. long jump
• Projection height – Release Ht = Landing Ht è Optimal projection θ = 45 °
• Ex: kick a ball for max horizontal displacement – Release Ht > Landing Ht è Optimal projection θ < 45 °
• Ex: throw a ball for max horizontal displacement – Release Ht < Landing Ht è Optimal projection θ > 45 °
• Ex: throw a ball onto elevated surface
Optimal angle of release depends on:
Summary
Variable Determined by Increased horizontal velocity Increased projecFon speed
Decreased projecFon angle Increased verFcal velocity Increased projecFon speed
Increased projecFon angle Increased maximum height Increased verFcal velocity
Increased projecFon height Increased flight Fme Increased verFcal velocity
Increased projecFon height Decreased relaFve height
Increased flight distance Increased horizontal velocity Increased flight Fme