linear motion: kinematics and kinetics applied kinesiology 420:151

Linear Motion: Kinematics and Kinetics

Applied Kinesiology420:151

Agenda

Introduction to motion Linear kinematics Linear kinetics Analysis of linear motion

Introduction to Motion

Linear motion (translatory) Rectilinear Curvilinear Circular

Angular motion (rotary) General motion

Shoulder = angular or linear?

Figures 11.1, 2, 3, 4, Hamilton

Agenda

Introduction to motion Linear kinematics Linear kinetics Analysis of linear motion

Linear Kinematics

Linear: A point moving along a line Kinematics: The study of motion

(of a point moving along a line) in respect to displacement, velocity and acceleration

Displacement

Displacement: Change in position Vector quantity Magnitude and

direction Displacement vs. distance SI unit = m

Velocity

Velocity: Rate of displacement V = displacement/time

Vector quantity Velocity vs. speed SI unit = m/s

Average vs. Instantaneous Velocity

Average velocity = displacement/time Entire displacement start to finish

Instantaneous: Velocity at any particular instant within the entire displacement Still average velocity however time

periods much smaller therefore “essentially” instantaneous

s (m) tjohnson (s) tlewis (s) Vjohnson (m/s) Vlewis (m/s)

10 1.86 1.88 5.38 5.32

20 2.87 2.96 6.97 6.76

30 3.8 3.88 7.89 7.73

40 4.66 4.77 8.58 8.39

50 5.55 5.61 9.01 8.91

60 6.38 6.45 9.40 9.30

70 7.21 7.29 9.71 9.60

80 8.11 8.12 9.86 9.85

90 8.98 8.99 10.02 10.01

100 9.83 9.86 10.17 10.14

Instantaneous?

Velocity Figure - Johnson vs. Lewis (1988 Summer Olympics, Seoul Korea)

5.00

6.00

7.00

8.00

9.00

10.00

11.00

10 20 30 40 50 60 70 80 90 100

Meters (m)

Velo

city

(m/s

)

Johnson

Lewis

(m) Splits BJ (s) Splits CL (s) Vinst. BJ Vinst. CL

0 10 1.86 1.88 5.38 5.32

10 20 1.01 1.08 9.90 9.26

20 30 0.93 0.92 10.75 10.87

30 40 0.86 0.89 11.63 11.24

40 50 0.89 0.84 11.24 11.90

50 60 0.83 0.84 12.05 11.90

60 70 0.83 0.84 12.05 11.90

70 80 0.90 0.83 11.11 12.05

80 90 0.87 0.87 11.49 11.49

90 100 0.85 0.87 11.76 11.49

Instantaneous Velocity Figure - Johnson vs. Lewis (1988 Summer Olympics, Seoul Korea)

5.00

6.00

7.00

8.00

9.00

10.00

11.00

12.00

13.00

0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100

Meters (m)

Velo

city

(m/s

)

Johnson

Lew is

Acceleration

Acceleration: Rate of change of velocity A = vf – vi

Vector quantity SI unit = m/s/s or m/s2

Uniform acceleration Very rare Projectiles (more later)

Average vs. Instantaneous Acceleration

Average acceleration = Rate of change in velocity assumes uniform acceleration

Instantaneous: Acceleration between smaller time periods Provides more information Johnson vs. Lewis

Average acceleration for Ben Johnson?

A = (vf – vi) / t

A = (10.17 m/s – 0 m/s) / 9.83 s

A = (10.17 m/s) / 9.83 s

A = 1.03 m/s2

v BJ (m/s) v CL (m/s)

0 0

5.38 5.32

6.97 6.76

7.89 7.73

8.58 8.39

9.01 8.91

9.40 9.30

9.71 9.60

9.86 9.85

10.02 10.01

10.17 10.14

Average acceleration for Carl Lewis?

A = (vf – vi) / t

A = (10.14 m/s – 0 m/s) / 9.86 s

A = (10.14 m/s) / 9.86 s

A = 1.03 m/s2

Enough information?

s (m) t BJ (s) t CL (s) v BJ (m/s) v CL (m/s) a BJ (m/s2) a CL (m/s2)

0 0 0 0 0 0 0

10 1.86 1.88 5.38 5.32 2.89 2.83

20 2.87 2.96 6.97 6.76 0.55 0.49

30 3.8 3.88 7.89 7.73 0.24 0.25

40 4.66 4.77 8.58 8.39 0.15 0.14

50 5.55 5.61 9.01 8.91 0.08 0.09

60 6.38 6.45 9.40 9.30 0.06 0.06

70 7.21 7.29 9.71 9.60 0.04 0.04

80 8.11 8.12 9.86 9.85 0.02 0.03

90 8.98 8.99 10.02 10.01 0.02 0.02

100 9.83 9.86 10.17 10.14 0.02 0.01

Instantaneous?

Acceleration Figure - Johnson vs. Lewis (1988 Summer Olympics, Seoul Korea)

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 100

Distance (m)

Acce

lera

tion

(m/s

/s)

Johnson

Lewis

(m) Splits BJ (s) Splits CL (s) Vinst. BJ Vinst. CL a BJ (m/s2) a CL (m/s2)

0 10 1.86 1.88 5.38 5.32 2.89 2.83

10 20 1.01 1.08 9.90 9.26 4.48 3.65

20 30 0.93 0.92 10.75 10.87 0.92 1.75

30 40 0.86 0.89 11.63 11.24 1.02 0.41

40 50 0.89 0.84 11.24 11.90 -0.44 0.80

50 60 0.83 0.84 12.05 11.90 0.98 0.00

60 70 0.83 0.84 12.05 11.90 0.00 0.00

70 80 0.90 0.83 11.11 12.05 -1.04 0.17

80 90 0.87 0.87 11.49 11.49 0.44 -0.64

90 100 0.85 0.87 11.76 11.49 0.32 0.00

Instantaneous Acceleration Figure - Johnson vs. Lewis (1988 Summer Olympics, Seoul Korea)

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100

Meters (m)

Velo

city

(m/s

/s)

Johnson

Lew is

Positive and Negative Acceleration

Motion to the right or up is considered in the positive direction Velocities in these directions also

positive Motion to the left or down is

considered in the negative direction Velocities in these directions also

negative

Positive and Negative Acceleration Increasing speed in the positive

direction is + acceleration Decreasing speed in the positive

direction is – acceleration Increasing speed in the negative

direction is – acceleration Decreasing speed in the negative

direction is + acceleration

Use positive/negative acceleration as opposed to acceleration and deceleration

Uniform Acceleration Most common form of uniform

acceleration is projectile motion 9.8 m/s2

Laws of uniform acceleration Acceleration is constant Allows you to solve for:

Velocity: Initial or final Distance traveled by implement Time in the air

Uniform Acceleration Solve for final velocity when initial

velocity, time and acceleration are known

vf = vi + at Manipulate the formula

vi = vf – at t = vf – vi /a

What if initial or final velocity = 0?

What is the final velocity at 4 seconds if you know the velocity at 2 seconds?vf = vi + atVf = -19.6 m/s + (-9.8 m/s2)

(2s)Vf = -39.2 m/s

How long was it in the air if you know the initial and final velocities?t = -49 m/s – 0 m/s /-9.8 m/s2

t = -49 m/s / -9.8 m/s2

t = 5 s

What is the initial velocity if you know the velocity at 5 seconds?vi = vf – atVi = -49 m/s – (-9.8 m/s2)(5s)Vi = 0 m/s

Uniform Acceleration

Solve for velocity when initial velocity, distance and acceleration are known

vf2 = vi

2 + 2as Manipulate the formula

vi2 = vf

2 - 2as s = vf

2 - vi2 /2a

What if initial or final velocity = 0?

Page 288 of text book:

Assuming that a ball is thrown upward so that it reaches a height of 5 meters before starting to fall, what is its initial velocity? What is its final velocity?

Initial velocity:

vi2 = vf2 – 2as

vi2 = (0 m/s)2 - 2(-9.8 m/s2)(5m)

vi2 = 0 m2/s2 + 98 m2/s2

vi2 = 98 m2/s2 (sq. root both sides)

vi = 9.9 m/s

Final velocity:

vf2 = vi2 + 2as

vf2 = (0 m/s)2 + 2(-9.8 m/s2)(5m)

vf2 = 0 m2/s2 - 98 m2/s2

vf2 = -(98 m2/s2) (sq. root both sides)

vf = -9.9 m/sCheck work by solving for

distance

s = vf2 - vi2 / 2a

s = (-9.9)2 – (0)2 / 2(-9.8m/s2)

s = 98 m2/s2 / 19.6 m/s2

s = 5 m

Uniform Acceleration

Solve for distance when initial velocity, acceleration and time are known

s = vit + ½ at2

Manipulate the formula

How far will an object drop if you let if fall for 3 seconds?s = vit + ½ at2

s = 0 + ½ (-9.8 m/s2)(3 s)2

s = 0 + -4.9 m/s2 (9 s2)s = -44.1 m

How high will an object go in the air in 2 seconds if thrown with an initial velocity of 29.4 m/s?s = vit+ ½ at2

s = 29.4 m/s (2 s) + ½(-9.8 m/s2)(2 s)2

s = 58.8 m – 19.6 ms = 39.2 m

Vi = 0

Vf = -9.8 m/s

V = -4.9 m/s S = -4.9 m1 s

2 s

Vf = -19.6 m/s

V = -14.7 m/s

s = -14.7 m

3 s

Vf = -29.4 m/s

V = -24.5 m/s

s = -24.5 m

Total distance traveled:

(-4.9)+(-14.7)+(-24.5) = -44.1 m

1 s

Vi = 29.4 m/s

Vf = 19.6 m/s

V = 24.5 m/s S = 24.5 m

2 s

Vf = 9.8 m/s

V = 14.7 m/s S = 14.7 m

Total distance traveled:

14.7+24.5 =

39.2 m

Vacuum no air resistance

Unlimited constant acceleration

Real world air resistance

Terminal velocity

Continued acceleration until air resistance = gravity

Terminal velocity of skydiver = 120 mph

Agenda

Introduction to motion Linear kinematics Linear kinetics

Newton’s laws Types of force Work, power, energy

Analysis of linear motion

Linear Kinetics

What causes motion? Force Newton’s Three Laws of Motion:

Cannot be proved Universally accepted

1st Law: Law of Inertia

Law: A body continues in its state of rest/motion unless an unbalanced force acts upon it

Inertia: The resistance to motion proportionate to mass

Unbalanced forces during start? Unbalanced forces during stop?

2nd Law: Law of Acceleration

Law: The acceleration of an object is proportional to the force causing it and inversely proportional to its mass

F = ma a = F/m Impulse: The product of force and

time F = ma = m(vf – vi)/t Ft = m(vf – vi)

Impulse Max acceleration of an object = max

force and max time Hammer throw Baseball swing (quickness vs. velocity)

Manipulation of impulse Two basketball players of 100 kg Player A = 2500 N over 0.25 s Player B = 3000 N over 0.21 s Advantage?

Impulse and Momentum

Momentum = mv Ft = m(vf – vi) = mvf – mvi

Any change in momentum is equal to the impulse that produced it

Applications? Shot put: Effect of force, arm length? Catching a fastball

40 m/s or 90 mph

Weight = 5 oz.

Mass = 0.14 kg

Momentum = mv

Momentum = 40 m/s (0.14 kg)

Momentum = 5.6 kg*m/s

If you want to stop this pitch with your bare hands

(change mv from 5.6 kg*m/s to 0), this change in momentum will equal the impulse that creates the change in momentum!

If you try to stop the pitch in 0.05 seconds, it will require 112

N of force (5.6/0.05 = 112). This is approximately 25 lbs of

force!

Ft = mvf - mvi 112(0.05) = 5.6 - 0

Why does a 5 story fall typically result in

death?

Why wasn’t he able to hold on?

(15 July 1999, Alabama) A 25-year-old soldier died of injuries sustained from a 3-story fall, precipitated by his attempt to spit farther than his buddy. His plan was to hurl himself towards a metal guardrail while expectorating, in order to add momentum to his saliva. In a tragic miscalculation, his momentum carried him right over the railing, which he caught hold of for a few moments before his grip slipped, sending him plummeting 24 feet to the cement below. The military specialist had a blood alcohol content of 0.14%, impairing his judgment and paving the way for his opportunity to win a Darwin Award.

3rd Law: Law of Reaction Law: For every action there is an equal and

opposite reaction Objects at rest are in equilibrium the

weight of the object is balanced by the force of the surface pushing back against it

Ground reaction forces Cement vs. sand

Examples: Objects colliding? Conservation of momentum

Momentum lost by one body = momentum gained by other body

Unequal masses = unequal accelerations

Agenda

Introduction to motion Linear kinematics Linear kinetics

Newton’s laws Types of force Work, power, energy

Analysis of linear motion

Types of Forces

Gravity Ground reaction forces Friction Rebound Air resistance (drag)

Gravity “The force that causes objects to fall to

earth, the moon to orbit the earth and the planets to orbit the sun”

Weight: The attractive force of the earth 9.8 m/s/s Weight = mg 1 N = 0.2248 lbs

Gravity is a force vector Magnitude = weight Direction = straight down

Ground Reaction Force GRF: The reaction force

from a surface upon which one is moving

GRFs are force vectors Magnitude: Equal to amount

of force expressed into ground

Direction: Opposite direction from which force was expressed into ground

Energy can be transformed (ie sand)

Figure 12.18, Hamilton

Friction

Friction: the force that opposes efforts to slide or roll one body across/over another

Can work for us or against us More friction can be good Less friction can be good

Friction Friction depends on:

Nature of surfaces (coefficient of friction) Forces involved

PushFf

Coefficient of friction

Weight of book

Figure 12.14, Hamilton

Starting vs. sliding friction

Rebound Rebound: The force that causes objects

to rebound from each other after contact

Amount of rebound depends on several factors Elasticity Angle of rebound Spin Momentum

Rebound - Elasticity Objects that receive a stress will strain

(become distorted/deformed) Elasticity: Ability to return to original

shape once stress is removed Coefficient of restitution = stress/strain Greater elasticity = greater rebound

Rebound – Angle of Rebound Assume COR = 1.0 angle of incidence

= angle of reflection As COR decreases, so does angle of

reflection relative to angle of incidence

Figure 12.23, Hamilton

Rebound - Spin

Topspin: Increase Hv, decrease Vv Backspin: Increase Vv, decrease

Hv Sidespin: Rebound in direction of

spin

Rebound - Momentum

Momentum = mv As mass and/or velocity increase,

so does rebound

Air Resistance (Drag) Air resistance: The force that occurs as a result

of fluid pressure at the leading edge of an object and the backward pull created by turbulence on the trailing edge

Figure 12.25, Hamilton

Agenda

Introduction to motion Linear kinematics Linear kinetics

Newton’s laws Types of force Work, power, energy

Analysis of linear motion

Work, Power, Energy

Work: The product of force expended and distance through which force succeeds in overcoming resistance

W = Fd SI unit = J or N*m Distance measured vertically only

F

2 m

Force needed to overcome inertia of block = 20 N

Total vertical distance = 2 m

Work = 20 N * 2 m

Work = 40 J or 40 N*m

Figure 12.30, Hamilton

Work performed climbing stairs Work = Fd Force

Subject weight From mass, ie 65 kg

Displacement Height of each step

Typical 8 inches (20cm) Work per step

636 N x 0.2 m = 127 Nm Multiply by the number of steps

Work on a stair stepper

Work = Fd Force

Push on the step ????

Displacement Step Height

8 inches

Work per step ???N x .203 m = ???Nm

Work on a cycle ergometer

Work = Fd Force

belt friction on the flywheel mass ie 3 kg

Displacement revolution of the pedals

Monark: 6 m per rev

Work per revolution 3kg x 6 m = 18 kgm

Work, Power, Energy

Work can be positive or negative Work done in direction of force

application is positive work Concentric actions

Work done in opposite direction of force application is negative work Eccentric actions

Work, Power, Energy

Power: The rate at which work is performed

Power = Work/time or Fv SI unit = W or J/s or N*m/s

Calculate & compare power During the ascent phase of a rep of

the bench press, two lifters each exert an average vertical force of 1000 N against a barbell while the barbell moves 0.8 m upward

Lifter A: 0.50 seconds Lifter B: 0.75 seconds

Calculate & compare power

Lifter AF = 1000 Nd = 0.8 m

t = 0.50 s

Power = Fd/tPower = 1000 N*0.8 m /0.50 sPower = 800 J/ 0.50 sPower = 1600 W

Lifter B?

Power on a cycle ergometer

Power = Fd/tPower = Fd*rev/min

How much power output is there?Force = 3 kgd = 6 m / revolution60 revolutions / min

Power = 3kg*6m*60 rev/min

Power = 1080 kg*m/min

Note: 1 Watt = ~ 6 kg*m/min

How many Watts? 180 Watts

Other Ways to Calculate Power Treadmill sprinting

Work = weight*vertical displacement (angle of treadmill)

Time = length of sprint Vertical jumping (Lewis equation)

Work = weight*vj height Power is estimated with an equation

Stair stepper Work = resistance of step*displacement of step Time = time to move step

Stair running (Margaria-Kalaman test) Work = weight*vertical displacement*# steps Time = time it takes to get up the stairs

Work, Power, Energy Energy: The capacity to do work Amount of energy = work

accomplished Many types of energy

Heat, sound, light, electric, chemical, atomic

Mechanical energy Two main types Potential energy Kinetic energy

Mechanical Energy

Potential energy: Energy a body possesses due to its position PE = mgh

Kinetic energy: Energy a body possesses due to its motion KE = ½ mv2

PE = Maximum

KE = 0

PE = Decreasing

KE = Increasing

PE = 0

KE = Maximum

Energy can neither be created nor destroyed

Agenda

Introduction to motion Linear kinematics Linear kinetics

Newton’s laws Types of force Work, power, energy

Analysis of linear motion

General Analysis of Linear Motion Linear kinematics:

Linear displacement Linear velocity Linear acceleration

Linear kinetics: Effect of a force in an instant in time

F = ma Effect of a force applied over a period of time

Impulse and momentum Effect of a force applied over a distance

Work, power, energy

General Phases of Skill Performance

Ritual Phase Full of idiosyncrasies, useful for mental focus Does the ritual phase affect performance negatively,

positively? Preparation Phase

Wind up (force, velocity, accuracy, combination?) Is energy storage required?

Force Phase Is the force applied in the right direction? Is there enough force, too much?

Follow Through Phase Is there enough time to slow down body parts? Does the follow through promote correct application of force?