Biomechanics Notes: Linear and Angular Kinematics

Linear Kinematics of Human Movement

• Kinematics: The study of the form, pattern, or sequencing of movement with respect to time, without considering the forces causing the motion.
  • Focuses on describing the appearance of motion (form or technique).
• Kinetics: The study of forces associated with motion.
• Kinematics can be:
  • Qualitative: Observing movement and identifying joint actions (e.g., hip flexion, knee extension, dorsiflexion during a soccer kick).
  • Quantitative: Measuring precise sequencing, timing, and range of motion of body segments.

Distance vs. Displacement

• Distance: Measured along the path of motion (scalar quantity).
  • Example: Running one time around a 400m track covers a distance of 400m.
• Linear Displacement: Measured in a straight line from the initial position to the final position (vector quantity).
• Unit: The most commonly used unit for both distance and displacement is the meter.

Scalar vs Vector Quantities

• Distance is a scalar quantity; displacement is a vector quantity.
  • Scalar quantities: Described by magnitude alone (numerical value), such as miles or meters run.
  • Vector quantities: Include both magnitude and direction.
  • Example: A yacht sailing 900 m due south has a displacement of 900 m to the south.
  • Direction can be identified using:
    • North/south/east/west
    • Up/down
    • Positive/negative

Displacement and Distance Relationship

• Displacement and distance can be identical when motion is linear.
  • Example: A cross-country skier traveling down a straight path.
• If the path of motion is not linear, the distance traveled and the magnitude of the displacement will differ.
• Displacement will never be greater than distance.

Speed

• Speed: A scalar quantity defined as the distance covered divided by the time taken to cover it.
  • Measured in meters per second.

Linear Velocity

• Velocity: A vector quantity defined as the change in position (displacement) over a given period of time.
  • Represents the rate of change in location.
  • Measured in meters per second.
  • Includes both direction and magnitude.
  • If displacement is positive, velocity is positive.

Velocity Calculation Example

• A swimmer crosses a lake 0.9 km wide in 30 minutes (0.5 hours).
• Velocity = Displacement / Time = $0.9 \text{ km} / 0.5 \text{ hr} = 1.8 \text{ km/hr}$
• Speed cannot be calculated without knowing the actual distance traveled.
• Modified problem: A swimmer crosses a lake by swimming 1.3 km in 30 min
• Speed = Distance / Time = $1.3 \text{ km} / 0.5 \text{ hr} = 2.6 \text{ km/hr}$

Resultant Velocity

• When multiple velocities act, the overall velocity and direction are called the resultant velocity.
• The resultant velocity of a swimmer in a river is the vector sum of the swimmer's velocity and the current's velocity.

Factors Influencing Human Gait Speed

• Speed is the product of stride length and stride frequency.
• Kinematic variables during running are influenced by:
  • Muscle fiber composition
  • Footwear
  • Level of fatigue
  • Injury history
  • Inclination and stiffness of the running surface
  • Running biomechanics/training

Linear Acceleration

• Linear acceleration: The rate of change in velocity or the change in velocity over a given time interval.
  • Unit: meters per second squared ($\text{m/s}^2$
• Example:
  • A car increasing its velocity by 1 km/hr each second has an acceleration of 1 km/hr/s.
  • A skier increasing velocity by 1 m/s each second has an acceleration of 1 m/s/s or 1 m/s^2.
• Represents how velocity changes with respect to time.
  • A body accelerating at $2 \text{ m/s}^2$ increases its velocity by 2 m/s each second.
  • If the initial velocity is 0, after one second it's $2 \text{ m/s}$, then $4 \text{ m/s}$, then $6 \text{ m/s}$.

Positive and Negative Acceleration

• In general usage, acceleration means speeding up or increasing velocity.
  • If $v<em>2 > v</em>1$, (final velocity > initial velocity), acceleration is positive, and the body speeds up.
• Acceleration can be negative, indicating the body is slowing down.

Negative Acceleration Example

• Negative acceleration indicates slowing down.
• Example: A base runner sliding to a stop.
  • Initial velocity ($v_1$) = 4 m/s
  • Final velocity ($v_2$) = 0 m/s
  • Time (t) = 0.5 s
  • Acceleration = $(v<em>2 - v</em>1) / t = (0 - 4 \text{ m/s}) / 0.5 \text{ s} = -8 \text{ m/s}^2$
• Whenever $v<em>1 > v</em>2$, acceleration will be negative

Direction and Acceleration

• Acceleration may be positive or negative based on the direction of motion and the change in velocity.
  • Positive acceleration can mean speeding up, and negative acceleration means slowing down.
  • Positive and negative acceleration can also indicate direction.
    • A ball falling towards the ground has a negative direction.
    • Right is positive, and left is negative.

Constant Velocity and Implications of Acceleration

• Acceleration is zero when velocity is constant (when $v<em>1$ and $v</em>2$ are the same).
• Instantaneous velocity: Velocity during a small interval of time.
• Average velocity: Velocity over a designated time interval.
• Acceleration and deceleration impact the human body, because changing velocity results from the application of force.
• Injuries often occur during rapid deceleration or quick changes in direction.

Kinematics of Projectile Motion

• Projectile: A body in free fall subject only to gravity and air resistance (e.g., basketball, discus, high jumper, skydiver).
• Different kinematic quantities are relevant depending on the projectile.
• Airplanes and rockets are not considered projectiles because they are influenced by factors other than gravity and air resistance.

Analyzing Projectile Motion Components

• Horizontal and vertical components of projectile motion are analyzed separately because:
  • The vertical component is influenced by gravity, while the horizontal component is not.
  • The horizontal component relates to the distance traveled, and the vertical component relates to the maximum height achieved.

Projectile Motion Example

• Two balls, one dropped and one projected horizontally from the same height, land at the same time because gravity affects their vertical velocities equally.
• The horizontally projected ball also has horizontal displacement.

Influence of Gravity on Projectile Motion

• Gravity produces a constant acceleration of $-9.81 \text{ m/s}^2$ on bodies near the Earth's surface.
• This acceleration is constant regardless of the projectile's size, shape, or weight.

Vertical Velocity During Projectile Motion

• A ball thrown upward has an initial vertical velocity.
• As the ball rises, its velocity decreases due to gravity (negative acceleration).
• At the peak of its flight, vertical velocity is 0 m/s.
• As the ball falls, its speed increases due to gravity.

Factors Influencing Projectile Motion

• Air resistance affects the horizontal component of projectile velocity.
• A ball thrown outside will travel farther with a tailwind.
• Without air resistance, the horizontal speed of a projectile remains constant throughout its trajectory.
• Factors that influence the trajectory of a projectile:
  • Angle of Projection
  • Projection Speed
  • Relative Height of Projection

Projection Angle

• Projection Angle: The direction at which a body is projected with respect to the horizontal.
• Trajectory shape is solely dependent on projection angle.
• Trajectory shapes:
  • Perfectly vertical projection angle: Perfectly vertical trajectory.
  • Oblique projection angle (between 0° and 90°): Parabolic trajectory.
  • Horizontal projection angle (0°): Trajectory resembling one-half of a parabola.

Projection Angle in Sports

• Projection angle is important in sports such as basketball.
• A common error is shooting the ball with too flat a trajectory.

Projection Speed

• Projection speed: The magnitude of projection velocity.
• When projection angle and other factors are constant, the projection speed determines the length or size of a projectile’s trajectory.
• When a body is projected vertically upward, the projectile’s initial speed determines the height of the trajectory’s apex
• When a body is projected at an oblique angle, the speed of projection determines both the height and horizontal length of the trajectory

Relative Projection Height

• Relative projection height: The difference between projection height and landing height.
• A projectile’s flight time is increased by increasing the vertical component of projection velocity or by increasing the relative projection height.

Angular Kinematics of Human Movement

• Angle: Composed of two sides that intersect at a vertex.
• A protractor can be used to make hand measurements of angles for quantitative kinematic analysis.
• Joint centers form the vertices of angles between adjacent body segments.
• Modern kinematic analysis uses multiple computer-linked cameras that automatically track reflective markers.

Measuring Joint Angles

• Joint angles are measured as segment movement away from anatomical position.
• In anatomical position, all joint angles are at 0 degrees.
• The straight, fully extended position at a joint is regarded as zero degrees.

Body Segment Orientation

• Angular orientation: Orientation of a body segment with respect to a fixed line of reference (usually horizontal or vertical).

Instant Center of Rotation

• Instant Center of Rotation: Precisely located center of rotation at a joint at a given instant in time.
• Throughout the range of motion, the joint center will move due to the underlying anatomy of the joint.

Angular Distance and Displacement

• Similar concept to linear distance and displacement.
• Distance: total amount covered during a specific time
• Displacement: the difference between the starting and ending points
• Example: Bicep Curl
  • Start with arm fully extended (0°), bend elbow and move into full elbow flexion (150°) then return to full extended (0°)
  • What is the angular distance?
  • 0 to 150 = 150
  • 150 to 0 = 150
  • 150 + 150 = 300° of distance

Angular Displacement

• Angular Displacement: Change in the angular position or orientation of a line segment.
• Assessed as the difference in the initial and final angular positions of a moving body.
• Defined by both magnitude and direction.
• Measured in units of degrees, radians, or revolutions.
• Example: Biceps curl
• As in the previous example, begin at 0°, move up to 150 ° and end at 0 °
• Angular Displacement = 0 ° since you began and ended at the same point

Direction of Rotation

• Rotation occurs in either clockwise or counterclockwise directions
• The counterclockwise direction is generally designated as positive
• The clockwise direction is generally designated as negative
• With the human body, it is also appropriate to indicate the direction of the angular displacement with joint related terminology such as flexion or abduction

Units of Angular Measurement

• Three units of measure are commonly used to represent angular distance and angular displacement::
  • Degrees
  • Radians
  • Revolutions

Radians

• Radian: unit of angular measure used to quantify conversions; equal to 57.3°
• Is defined as the size of the angle subtended at the center of a circle by an arc equal in length to the radius of the circle
• One complete circle is an arc of 2π radians or 360°
• 2 x 3.14 x 57.3 = 360

Revolutions

• Revolutions: an arc equal to a circle (360°)

Angular Speed and Velocity

• Angular Speed: the angular distance covered divided by the time over which the motion occurred
  • Angular Speed = Angular Distance / Change in Time
• Angular Velocity: angular displacement that occurs during a given period of time
  • Angular Velocity = Angular Displacement / Change in Time
• Angular velocity must include the direction (clockwise/counterclockwise, positive/negative) in which the displacement occurs
• Measured in units of degrees per second, radians per second, revolutions per second, and revolutions per minute.

Angular Velocity Example

• Major League Baseball pitchers have been reported to reach 2320 deg/s in elbow extension and 7240 deg/s in shoulder internal rotation when pitching

Angular Acceleration

• Angular acceleration: The rate of change in angular velocity
  • Angular acceleration equals change in angular velocity divided by change in time. Angular Acceleration = change in angular velocity / change in time
• Examples of units of angular acceleration are degrees per second squared, radians per second squared, and revolutions per second squared

Relationships Between Linear and Angular Motion

• The greater the radius is between a given point on a rotating body and the axis of rotation, the greater is the linear distance traveled by that point during an angular motion.
• Radius of Rotation: distance from the axis of rotation to a point of interest on a rotating body

Radius of Rotation and Linear Velocity

• With all other factors held constant, the greater radius of rotation at which a swinging implement hits a ball, the greater the linear velocity imparted to the ball
• Longer golf clubs are designed to hit the ball farther
• In theory, longer bats will hit the ball farther, as long as they don’t weight too much
• Heavier bats will slow the swing speed