Biomechanics of Sports and Exercise EXSS 323 Linear Kinematics - Chapter 10

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Kinematics
deals with the description of motion
(position, velocity, acceleration)

To describe the kinematics of an object, we need a system of reference
The most commonly used system is the Cartesian (rectangular) system
Named for French mathematician René Descartes (1596-1650)

The 3-D coordinate systems above are "right-handed"
Right-handed systems are the most commonly used
We will deal with two-dimensional kinematics in this class

Some key concepts in kinematics

Scalar - a quantity that is fully described by its magnitude
example: mass, time, distance, speed, temperature

Vector - a quantity that possesses both magnitude and direction
example: displacement, velocity, acceleration, force

Assumptions about vectors (in 2-D)
Positive (+) = Up, Right
Negative (-) = Down, Left

Vector Mathematics

1 + 1 = 2
Right?
Not necessarily

Adding Vectors: Graphic

Adding Vectors: Trigonometry

• First, find Horizontal and Vertical Components of each vector
• Add the Horizontal Components Together
• Add the Vertical Components Together
• Overall Sum is the Resultant of Horizontal and Vertical Sums
• Add these vectors using trigonometry:
•     Vector A: 10 N at 30 degrees
•     Vector B: 20 N at 60 degrees

    Sum?

Linear Kinematic Quantities

Position: an object's location in a coordinate system using 1, 2, or 3 numeric coordinates. An object's position depends on time.

Instantaneous Position: an object's location at a certain instant in time.

Displacement:  Change of position during a time interval; vector quantity having both magnitude and direction

Compare Distance vs Displacement for Boston Marathon

Sprinting Speed

SLOPE of Position-Time Graph shows Velocity

 

What is the difference between speed and velocity?

• Example using Speed:
  
  What was a marathon runner's average speed if he/she finished
  the 42.2 km race in 2 hours 10 minutes?

• Example using Velocity:
  
  A rowing shell was progressing across stream in the Willamette River
  at 8 mph relative to the water, while the river was flowing at 6 mph.
  
  What was the velocity of the shell?

• What about when velocity is changing?
  Return look at Sprint Speed

 

Acceleration is a measure of how velocity is changing over time:

Acceleration is a vector quantity with magnitude and direction.

Note the definition of average acceleration is of similar form to the definition of average velocity. Just as velocity corresponds with slope on position-time graph, acceleration corresponds with slope on a velocity-time graph.

• Example Comparing Position, Speed, Velocity, Acceleration:
  Describe the motion of a shotput tossed up with a vertical velocity of 10 m/s:
  • Position-Time Data
  • Velocity-Time Data

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Linear Kinematic Quantities

Quantity	Symbol	SI Unit	Abbreviation
Position	d, h, s, x, y, z	meter	m
Speed	-	meters per second	m/s
Velocity	v	meters per second	m/s
Acceleration	a	meters per second per second	m/s2

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So far, we've talked about AVERAGE velocities and accelerations

There are situations, however, when we want
to know INSTANTANEOUS velocities and accelerations

 

Two ways to do it

Shorten the time period

However, this causes a problem

Solution

First Central Differencing

As the time interval gets smaller, the calculated (average)
velocity/acceleration approaches the INSTANTANEOUS value

First Central Difference Formulas
Velocity	vn = (sn+1 - sn-1)/2Dt
Acceleration	an = (sn+1 - 2sn + sn-1)/Dt2
n = point in time | n+1 = next point in time | n-1 = prior point in time Dt = time difference between SUCCESSIVE time points

 

Calculus
(is not a Nasty Word)

Slope of position-time = velocity
Slope of velocity-time = acceleration

That's Differential Calculus

Linear Kinematics Lab

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Kinematics of Running
Terminology and Definitions

Stride Length = distance from one footfall       to next footfall of same foot Stride Time = time to take one stride Stride Rate = 1 / (Stride Time) Stride Length = 2 * Step Length Stride vs. Step

 

Locomotion Equations

Velocity = (Displacement)/(Time)
= (Stride Length)/(Stride Time)
= (Stride Length)*(Stride Rate)

How do Stride Length & Rate vary?

Speed Control in Running

Running Kinematics Lab

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Algebraic manipulation of speed & velocity equations

Distance = Speed * Time
Displacement = Velocity * Time

yields formulas for constant speed/velocity situations

BUT what about accelerated motion?

Displacement starting from rest under constant acceleration Displacement with an initial velocity under constant acceleration

 

Acceleration of Bobsled
20 m/s to 30 m/s in 5 seconds
a = 2 m/s²

Displacement of Bobsled
d = (20 m/s)(5 s) + (1/2)(2 m/s²)(5 s)²
d = 125 m

 

How about when acceleration is changing?

We need Calculus
(its not a nasty word)

Area under velocity-time = change of position
Area under acceleration-time = change of velocity

That's Integral Calculus

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Acceleration is a vector -- it has magnitude & direction.
So, what does NEGATIVE acceleration mean, anyway?

Summary of Velocity & Acceleration

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Kinematics of Projectiles

A Projectile is an object that is subject to no external forces other than gravity; any body that has been set on its path by some force and continues in motion by its own inertia.

Gravity subjects projectiles to constant vertical acceleration; vertical velocity constantly changes during flight.

Air resistance, for short duration, low velocity flights, is negligible; horizontal velocity remains constant during flight

Examples of Projectiles include: Bullet, Basketball, Shotput, Human Body

Understanding Projectiles
Let's start simple with Vertical Motion from rest.

Now, vertical Motion with Initial Horizontal Velocity
If Initial Horizontal Velocity is +10 m/s,
where is the projectile 2 seconds later?
Motion of Projectiles

Vertical & Horizontal Motion are Independent

Vertical motion is motion with constant acceleration,
while horizontal motion is motion with constant velocity

Kinematic equations for velocity & position can be used

Vertical Motion	Horizontal Motion
vyf = vyi + gt hf = hi + vyit + ˝gt2	x = xi + vxt if we assume that xi = 0, then x = vxt
Combination of the above equations yields a 3rd equation vyf2 = vyi2 + 2g(hf - hi)
g = -9.81 m/s2

 

Characteristics of Trajectory

Parabolic - apex is highest point
v_y before apex > 0  |  v_y @ apex = 0  |  v_y after apex is < 0
v_x constant

How high and how far do projectiles travel?

Height depends only on initial vertical velocity
Time of Flight depends only on initial vertical velocity
Horizontal Motion depends on horizontal velocity and time of flight

 

Implications for Sport: | High Jump | LongJump |

 

Practice Questions:

vx = 5 m/s vyi = 10 m/s

Flight Time = Maximum Height = Horizontal Displacement =

v = 20 m/s @ 45 degrees

Flight Time = Maximum Height = Horizontal Displacement =

Projectile Homework Problem

Projectile Lab

Angular Kinematics