Kinematics = description of movement 

A. With respect to time.

B. Without regard for the forces that cause movement.

Linear kinematics focuses on linear movement rather than angular or rotational movement. Linear motion occurs when all particles of the moving object follow parallel paths, covering the same distance in the same amount of time. Pure linear movement rarely occurs in human movement.  

Although human movement is rarely linear, we often judge movement performance in terms of a singular linear quantity. What exactly are we measuring then? The center of gravity of the body or system.

Concept 1. Distance and Displacement (How far a systems COG goes)

• Distance is an absolute change in length along the path of motion of the COG (scalar = magnitude, but no direction (mass, volume, length, speed))

• Displacement is the relative in length of the COG from initial to final position (vector = magnitude and direction(velocity, acceleration))

 These terms are often interchanged and interchangeable, but there are distinct differences.

• When we want to look at how much distance is covered by an object in a certain amount of time, we observe SPEED

Speed = how fast a body is moving = distance/change in time (m/s) 

• When we want to look at how much an object is displaced in a certain period of time, we observe VELOCITY 

Velocity = speed and direction of a body = displacement/change in time (m/s)

Concept 3. Acceleration (How much the system speeds up/slows down)

Acceleration is the change in velocity over a certain amount of time.

Acceleration = change in velocity/change in time (m/s² ) = positive (gaining speed) or negative (losing speed)

 EXAMPLES:

1. A sports car goes from 0 to 50 km/hr in 5 sec. Acceleration?

2. A runner has a velocity of 7.5 m/sec after the first 10 meters of the 100-meter dash. What is his acceleration if he completed 10 m in 10 seconds?

Basic concepts:

• Angular kinematics differ from linear in that they consider rotational components of movement. There is some Afixed@ point about with an object rotates.
• Linear motion is often a result of angular motion.
• The relationships between displacement, velocity and acceleration are similar.

Angular displacement and distance.

• The angular distance through which a rotating body moves is the angle between its initial and final position measured following the path followed by the body.
• The angular displacement through which a rotating body moves is equal to the smaller of the two angles between its initial and final position (difference in initial and final position).

Note 1: Clockwise is negative and counterclockwise is positive.

Note 2: angles are measured in degrees or radians. 1 radian = 57.3E.

 Average angular speed = angular distance/time taken (scalar) s = f/t

 Average angular velocity = Angular displacement/time taken (vector) w = q/t

Average angular acceleration = change in angular velocity/time taken 

= Final ang. velocity - initial ang. velocity

Time taken

a = wf - wi

t

 Part III. Relationship between linear and angular kinematics

Linear and angular displacement. Determining the relationship between linear and angular motion. In these equations, radians must be used (because they are unitless) (divide by 57.3)

We can calculate how far some point on a rotating system travels long a curved path. This relationship is defined by the equation:

  d = rq

Where: d = linear displacement (m),

r = radius of rotation (m)

q = Angular displacement (rads)

  Linear and angular velocity

 We can calculate how fast a point on a rotating system travels along a curved path. This relationship is defined by the equation:

  v = rw

where: v = linear velocity (m/s)

r = radius of rotation (m)

w = angular velocity (rad/sec)

The greater the angular velocity and/or radius of rotation, the greater the rotating point=s linear speed or velocity. Instantaneous linear velocity = tangential

 This relationship is defined by the following sets of equations:

 1. Tangential acceleration and velocity: Change in linear speed for a body travelling on a curved path.

a_t = v2-v1

t

 where: v1 = tangential linear velocity at time 1

v2 = tangential linear velocity at time 2

t = change in time

 2. Linear and angular acceleration

a_t = ra

 where: a_t = tangential acceleration

r = radius of rotation

a = angular acceleration

Radial acceleration. This is the rate of change in the direction of a body in angular motion.

a_r = v²

r

 where: a_r = radial acceleration

v = tangential linear velocity

r = radius of rotation

Some extra problems! 

1. What distinguishes angular speed from angular velocity? Give an example for each.

 2. Which segment has the greatest angular acceleration: (a) an arm that gains an angular velocity of 300E/sec in 0.5 seconds or (b) an arm that gains an angular velocity of 200E/sec in 0.2 seconds?

 3. What is the instantaneous release velocity of a softball released by an upper extremity, 0.5 m long, that is rotation at 573E/sec at the time of release?

 4. A 1.2 m golf club is swung in a planar motion by a right-handed golfer with an arm length of 0.76 m. If the initial velocity of the golf ball is 35 m/s, what was the angular velocity of the left shoulder at the point of ball contact? (Assume that the left arm and the club form a strainght line and that the initial velocity of the ball is the same as the linear velocity of the club head at impact).

 5. A cyclist enters a curve of 30 m radius at a speed of 12 m/s. As the brakes are applied, speed is decreased at a constant rate of 0.5 m/s². What are the magnitudes of the cyclist=s radial and tangential accelerations when his speed is 10 m/s?