HSS 3843 Kinesiology (Biomechanics)

Textbook: Hall, SJ et al. (1999). Basic Biomechanics, 3rd Ed., St Louis: Mosby


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This chapter focuses on linear kinematics: the study of the form or sequencing of linear motion with respect to time. Linear kinematic quantities include the scaler quantities of distance and speed, as well as the vector quantities of displacement, velocity, and the acceleration. Depending on the motion being analyzed, either a vector quantity or its scalar equivalent may be of interest, and either an instantaneous or an average quantity may be of paramount importance.

The kinematics of projectile motion are introduced. A projectile is a body in free fall that is affected only by the forces of gravity and air resistance. The factors that determine the vertical and horizontal distances (height and range, respectively) that a projectile achieves are the projection angle, the projection speed, and the relative projection height, or simply release velocity and relative projection height.

Projectile motion is typically analyzed in terms of its horizontal and vertical components because the two components are independent of each other. Once the object is "airborne" only gravitational force and air resistance influences the vertical component. Only air resistance affects the horizontal component. To make it easier to understand the relationship between release parameters and flight of a projectile for analysis, air resistance is considered to be negligible. Although our subsequent discussions involving fluid dynamic parameters show that projectile movement patterns can be affected in air (a fluid). However, the assumption that air resistance effects are negligible allows us to apply the laws of constant acceleration to both the vertical and horizontal components of projectile motion.

Learning Hint: Frequently math resistant students will be terrified by the equations introduced in this chapter. It is important to keep the focus of the equations on what they show regarding prediction of skill performance based on a limited number of factors. For example, it is more important to recognize that how far and/or high the person jumps is determined by what the person does on the ground than being able to precisely calculate how far and/or high an individual jumps. The ultimate goal in learning biomechanics in this case, and other applications that we consider, is to have knowledge and be able to apply it regarding what the performer must do technique-wise to optimize these parameters.


kinematics: the form, pattern, or sequence of movement with respect to time

displacement: change in position in a specified direction

meter: most common unit of length in SI; equal to 3.281 feet

velocity: change in position with respect to time

acceleration: rate of change in velocity

projecile: a body in free fall that is subject only to the forces of gravity and air resistance

trajectory: flight path of a projectile

angle of projection: the angle of a body with respect to the horizontal at the time of release

projection speed: the magnitude of projection velocity

apex: the highest point in the trajectory of a projectile (also known as peak height). Note that the equations of constant acceleration in this chapter calculate apex relative to the height at release

range: the horizontal distance a projectile travels during the time it is in the air

relative projection height: the difference between projection height and landing height (positive if projection height > release height)

initial velocity: vector quantity describing rate of change of position at the moment of release; requires description of angle (direction) and speed (magnitude)

laws of constant acceleration: three formulas relating the kinematic quantities displacement, velocity and acceleration, and time during intervals when acceleration may be considered to be unchanging

average: occurring over a designated time interval

instantaneous: occurring during a designated time interval that is very short




An understanding of angular motion is an important part of the study of biomechanics because all volitional motion of the human body involves the rotation of bones around imaginary axes of rotation at the joints. As with like-named linear kinematics quantities, the angular kinematic quantities of angular distance and angular speed measure how far and how fast, while angular displacement and angular velocity measure how far and how fast while specifying a rotational direction of movement. Angular acceleration measures the time rate of change in angular velocity. Whenever acceleration is studied care must be taken when interpreting the effect of a positive or negative, or even constant, acceleration.

Both relative angles (formed by the longitudinal axes of two body segments articulating at a joint) and absolute angles (orientation of a single body segment with respect to a fixed reference line) are useful when analyzing movement. This chapter outlines the conventions for measuring angular kinematics and describes a variety of instruments used for direct measurement of angles on a human subject and for taking measures from video or film. The important relationship between angular and linear motion is outlined in this chapter. The previous chapter on linear kinematics showed the importance of release parameters in determining the trajectory of projectiles. This chapter shows how the linear velocity of a point on a rotating body, be it the contact point on a bat or foot during striking or the ball held in a hand during a throw, is related to the angular velocity of the rotating body.

Learning Hint: Understandably, while the angular kinematic quantities are closely related to the linear kinematic quantities, the greater implications of the angular measures when analyzing movement cannot be overemphasized. Observation of the angular kinematics of body segments, whether with the naked eye, with video assistance or some other method, provides insight to the magnitude and timing of force production by the performer. Inappropriate angular kinematics, "the performer does not look like she is supposed to," are symptoms of an underlying error in force production and may also help identify potential tissue overloads related to awkward posture or potential for injury. As such, the analyst needs to become familiar with evaluating angular position, displacement, and velocity so that the quantities become useful tools when diagnosing qualitatively the cause of less than optimal performance. Also, you are reminded to distinguish between angular and linear measures for range of motion of articulating joints.



instant center: the axis of rotation at a joint at a given instant in time; it varies throughout the ROM possible at a joint

relative angle: the angle formed by intersection of the longitudinal axes of two adjacent body segments

absolute angle: the angular orientation of a single body segment with respect to a fixed external line of reference

angular displacement: change in angular position with direction specified

radian: unit of angular measure used in angular-linear kinematic quantity conversions; equal to 57.3º (rad = 360º / 2p )

angular velocity: time rate of change in angular position

angular acceleration: time rate of change in angular velocity

right hand rule: procedure for identifying the direction of an angular motion vector; align fingers of the right hand with the direction of rotation, thumb will point in the direction of the representative vector

radius of rotation: distance from the axis of rotation to a given point of interest on a rotating body

tangential acceleration: component of angular acceleration along a tangent to the path of motion; causes a change in magnitude of the tangential linear velocity

radial acceleration: component of angular acceleration acting toward the center of rotation; causes a change in direction of the tangential linear velocity




This chapter serves as an introduction to linear kinetics – the study of the forces causing linear motion. The foundation for studying the relationships among the basic kinetic quantities is identified in the physical laws formulated by Sir Isaac Newton. Thus, the laws of inertia, acceleration, reaction, and gravitation are discussed.

The importance of friction to movement is made evident in the section of the chapter devoted to the mechanical behavior of bodies in contact. Friction is a force generated at the interface of two surfaces in contact when there is motion or a tendency for motion between the surfaces to occur; friction always opposes the sliding motion. Factors affecting friction magnitude include the nature of the surfaces in contact with each other and the normal force acting to hold the surfaces together. Examples of manipulating surfaces and normal force magnitude to optimize performance are given.

The chapter also introduces the quantity known as momentum. Momentum is the product of a body’s mass and its velocity. As described in the impulse-momentum relationship (Ft = Δmv), change in momentum results from an impulse, an external force acting over a time interval. Momentum is important in determining the outcome of a collision between two bodies (since each body will impart an equal in magnitude but oppositely directed impulse on the other, in accordance with Newton’s laws). Another factor affecting the behavior of two bodies during a collision is elasticity, which governs the amount of velocity present in the system after the impact. The relative elasticity of two impacting bodies is represented by the coefficient of restitution.

The concepts of mechanical work, power, and energy are presented, and the interrelationships among these quantities are explained. Mechanical energy has two major forms: kinetic and potential. When gravity is the only acting external force, the sum of the kinetic and potential energies possessed by a given body remains constant. Changes in a body’s energy are equal to the mechanical work done by an external force.

Learning Hint: Newton’s laws provide a valuable starting point for the study of kinetics because they clarify the cause-effect relationship between force and motion: the net force acting on a body causes a change in linear motion in the direction of the net force. However, the concept of impulse is more valuable as an evaluative tool because it more realistically describes skill performance: we translate (move) by exerting forces of changing magnitude and direction on the ground for a period of time. Reactive impulses are imposed on us, and they change our linear momentum in the direction they act. We change the linear momentum of implements and balls by exerting linear impulses on them. If the subsequent momentum of our body or the implement is not correct (and if mass is constant, the incorrect momentum reflects too much or too little velocity in a particular direction), the cause of the error is an inappropriately applied impulse. The force magnitude may have been wrong (too much or too little force), and/or it may have been applied in the wrong direction, and/or the duration of force application may have been wrong. By learning the "how and why" of analyzing movement based on the impulse-momentum relationship may be one of the most valuable contributions in understanding the contribution of biomechanics to the world around us.



friction : a force acting at the point of contact between two surfaces in the direction opposite that of sliding or intended sliding

maximum static friction: the maximum amount of friction that can be generated between two static surfaces

kinematic friction: the constant friction generated between two surfaces in contact during motion

coefficient of friction: a number that serves as an index of the interaction between two surfaces in contact

normal reaction force: the net force acting perpendicular to two surfaces in contact

momentum: the product of a body’s mass and its velocity

impulse: the product of a force and the time interval over which the force acts

impact: collision characterized by the exchange of a large force during a small time interval

perfectly elastic impact: impact during which the velocity of the system is conserved

perfectly plastic impact: impact resulting in the total loss of system velocity

coefficient of restitution: a number that serves as an index of the elasticity of a collision

work: the expression of mechanical energy; calculated as force multiplied by the distance the resistance is moved

power: the rate of work production; calculated as work divided by the time during which the work was done

kinematic energy: energy of motion; calculated as ½mv2

potential energy: stored energy; calculated as a body’s weight multiplied by its height above a set reference point

strain energy: a form of potential energy stored when a body is deformed

net force: the sum of all positive and negative forces acting in a given direction




The similarities between quantities describing linear and angular motion are further examined in this chapter. Particular emphasis is placed upon the factors that affect the rotation of a body. The causal relationships of kinetics and kinematics is again emphasized, with torque (rotational force) causing a change in angular motion.

While a body’s resistance to linear acceleration is inversely proportional to its mass (a = F / m), resistance to angular acceleration is inversely related to the moment of inertia (I). This relationship is explained by Newton’s Second Law for angular or rotational motion (a = T / I). Noting the angular analogues for: force is torque; mass is inertia; acceleration is the rate of angular rotation or velocity. Thus, when studying rotational movement I of a body reflects not only the mass, but how that mass is distributed with respect to a specific axis of rotation. Similarly, while linear momentum is the product of the mass and linear velocity, angular momentum is the product of moment of inertia and angular velocity. In the absence of external torques, angular momentum is conserved (Newton’s First Law). An airborne human performer may alter total body angular velocity by manipulating their moment of inertia, that is, by changing body configuration relative to the axis around which rotation is occurring. Skilled performers are also able to alter the axis of rotation and to initiate rotation when no angular momentum is present while airborne. Newton’s Third Law (action-reaction) also has an angular analog. It is stated as that for every torque exerted by one body on another, there is an equal and opposite torque exerted by the second body on the first.

Two linear forces that act on rotating bodies are the centripetal force and the tangential force. These forces are related to the linear velocity of the rotating body. The centripetal, or center-seeking force, is always directed toward the center of rotation, and it causes the change in direction of the tangential velocity. The magnitude of centripetal force is dependent upon the mass, speed, and radius of rotation of the rotating body. The tangential force causes the change in magnitude (speed) of the tangential velocity. Centrifugal force, a term used frequently to describe angular motion, is actually a reaction force equal in magnitude and opposite in direction to the centripetal force.


Learning Hint: The cause-and-effect relationship in angular motion is one of the most poorly understood concepts among movement analysts. It is critical to understand that, for most skills, the angular momentum is constant while in the air. The important implication is that the success of a rotary move is primarily determined while on the ground, when torques act to produce angular momentum. Once in the air, a performer can manipulate I to increase or decrease angular velocity, but only within the constraints of a constant angular momentum. The symptoms of errors that a coach typically focuses on (i.e., a young athlete landing a front flip in a tucked position) are often reflective of errors in force and/or torque production while the athlete was on the ground. If the athlete leaves the ground with inadequate angular momentum, the athlete will stay tucked (i.e., decrease I) to increase angular velocity to complete the trick and not land on their head. "Expert" commentators on television broadcasts of gymnastics and figure skating typically provide fine examples of focusing on the symptom of the error and not the cause. It is unusual for the take-off, or torque generation phase, to be included in the slow motion replay of a faltered trick, while the commentator focuses on the improper kinematics adopted in an attempt to make up for poor kinetics.



moment of inertia: the inertial property for rotating bodies; increases with both mass and the distance that the mass is distributed from the axis of rotation

radius of gyration: distance from the axis of rotation to a point where the body’s mass could be concentrated without altering its rotational characteristics

principal axes: the three mutually perpendicular axes passing through the center of gravity; referred to respectively as the transverse, anteroposterior, and longitudinal axes

principal moment of inertia: total body moment of inertia relative to one of the principal axes

angular momentum: the quantity of angular motion possessed by a body; equal to the product of moment of inertia and angular velocity

angular impulse: produces a change in angular momentum; equal to the product of torque and the time interval over which the torque acts

centripetal force: force directed toward the center of rotation of any rotating body; calculated as mvt2/r

centrifugal force: the reaction force equal in magnitude and opposite in direction to centripetal force