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

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


400m Race

Start End

Displacement? Distance?


Javelin throw


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


= 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


= 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


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


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)




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)




Increasing projecFon height will increase maximum height

0 1 2 3 4 5 6 7 8 0

Distance (m)

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

Shorter flight time Longer flight time

PosiFve relaFve height will decrease the flight Fme

NegaFve relaFve height will increase the flight Fme




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)




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 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

2. linear kinematics i

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