2. linear kinematics i

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