## Linear Motion

Motion is change in position with respect to some frame of reference. In mathematical terms, linear motion is simple to define: final position minus initial position. The simplest linear motion variable is a scalar called distance (l). The use of the symbol l may be easy to remember if you associate it with the length an object travels irrespec tive of direction. Typical units of distance are meters and feet. Imagine an outdoor adventurer leaves base camp and climbs for 4 hours through rough terrain along the path illustrated in Figure 5.1. If her final position traced a 1.3-km climb measured relative to the base camp (0 km) with a pedometer, the distance she climbed was 1.3 km (final position - initial position). Note that 1.3 km (kilometers) is equal to 1300 meters. The odometer in your car works in a similar fashion, counting the revolutions (angular motion) of the tires to generate a measurement of the distance (linear variable) the car travels. Because distance is a scalar, your odometer does not tell you in what direction you are driving on the one-way street!

The corresponding vector quantity to distance is displacement (d). Linear displacements are usually defined relative to right-angle directions, which are convenient for the purpose of the analysis. For most two-dimensional (2D) analyses of human movements, like in Figure 5.1, the directions used are horizontal and vertical, so displacements are calculated as final position minus initial position in that particular direction. The usual convention is that motions to the right on the x-axis and upward along the y-axis are positive, with motion in the opposite directions negative. Since displacement is a vector quantity, if motion upward and to the right is defined as positive, motion downward and motion to the left is a negative displacement. Recall that the sign of a number in mechanics refers to direction.

Assuming Figure 5.1 is drawn to the scale shown, it looks like the climber had a positive 0.8 km of horizontal displacement and 0.7 km of vertical displacement. This eyeballing of the horizontal and vertical components of the hike will be fairly accurate because displacement is a vector. Vectors can be conveniently represented by combinations of right-angle components, like the horizontal and vertical displacements in this example. If our adventurer were stranded in a blizzard and a helicopter had to lower a rescuer from a height of 0.71 km above base camp, what would be the rescuer's vertical displacement to the climber? The vertical displacement of the rescuer would be -0.01 km or 100 meters (final vertical position minus initial vertical position or 0.7 km - 0.71 km).

Biomechanists most often use measures of displacement rather than distance because they carry directional information that is crucial to calculation of other kinematic and kinetic variables. There are a couple of subtleties to these examples. First, the analysis is a simple 2D model of truly 3D reality. Second, the human body is modeled as a point mass. In other words, we know nothing about the orientation of the body or body segment motions; we just confine the analysis to the whole body mass acting at one point in space. Finally, an absolute frame of reference was used, when we are interested in the displacement relative to a moving objectâ€”like the helicopter, a relative frame of reference can be used.

In other biomechanical studies of human motion the models and frames of ref erence can get quite complicated. The analysis might not be focused on whole-body movement but how much a muscle is shortening between two attachments. A three-dimensional (3D) analysis of the small accessory gliding motions of the knee joint motion would likely measure along anatomically relevant axes like proximal-distal, medio-lateral, and antero-posterior. Three-dimensional kinematic measurements in biomechanics require considerable numbers of markers, spatial calibration, and mathematical complexity for completion. Degrees of freedom represent the kinematic complexity of a biomechanical model. The degrees of freedom (dof) correspond to the number of kinematic measurements needed to completely describe the position of an object. A 2D point mass model has only 2 dof, so the motion of the object can be described with an x (horizontal) and a y (vertical) coordinate.

The 3D motion of a body segment has 6 dof, because there are three linear coordinates (x, y, z) and three angles (to define the orientation of the segment) that must be specified. For example, physical therapy likes to describe the 6 dof for the lower leg at the knee joint using using the terms arthrokinematics (three anatomical rotations) and osteokinematics (three small gliding or linear motions between the two joint surfaces). The mathematical complexity of 3D kinematics is much greater than the 2D kinematics illustrated in this text. Good sources for a more detailed description of kinematics in biomechanics are available (Allard, Stokes, & Blanchi, 1995; Zatsiorsky, 1998). The field of biomechanics is striving to develop standards for reporting joint kinematics so that data can be exchanged and easily applied in various professional settings (Wu & Cavanagh, 1995).

The concept of frame of reference is, in essence, where you are measuring or observing the motion from. Reference frames in biomechanics are either absolute or rela tive. An absolute or global frame of reference is essentially motionless, like the apparent horizontal and vertical motion we experience relative to the earth and its gravitational field (as in Figure 5.1). A relative frame of reference is measuring from a point that is also free to move, like the motion of the foot relative to the hip or the plant foot relative to the soccer ball. There is no one frame of reference that is best, because the biomechanical description that is most relevant depends on the purpose of the analysis.

This point of motion being relative to your frame of reference is important for several reasons. First, the appearance and amount of motion depends on where the motion is observed or measured from. You could always answer a question about a distance as some arbitrary number from an "unknown point of reference," but the accuracy of that answer may be good for only partial credit. Second, the many ways to describe the motion is much like the different anatomical terms that are sometimes used for the identical motion. Finally, this is a metaphor for an intellectually mature ki-nesiology professional who knows there is not one single way of seeing or measuring human motion because your frame of reference affects what you see. The next section will examine higher-order kinematic variables that are associated with the rates of change of an object's motion. It will be important to understand that these new variables are also dependent on the model and frame of reference used for their calculation.

### Speed and Velocity

Speed is how fast an object is moving without regard to direction. Speed is a scalar quantity like distance, and most people have an accurate intuitive understanding of speed. Speed (s) is defined as the rate of change of distance (s = l/t), so typical units are m/s, ft/s, km/hr, or miles/hr. It is very important to note that our algebraic shorthand for speed (Z/t), and other kinematic variables to come, means "the change in the numerator divided by the change in the denominator." This means that the calculated speed is an average value for the time interval used for the calculation. If you went jogging across town (5 miles) and arrived at the turn-around point in 30 minutes, your average speed would be (5 miles / 0.5 hours), or an average speed of 10 miles per hour. You likely had intervals where you ran faster or slower than 10 mph, so we will see how representative or accurate the kinematic calculation is depends on the size of time interval and the accuracy of your linear measurements.

Since biomechanical studies have used both the English and metric systems of measurements, students need to be able to convert speeds from one system to the other. Speeds reported in m/s can be converted to speeds that make sense to American drivers (mph) by essentially doubling them (mph = m/s â€¢ 2.23). Speeds in ft/s can be converted to m/s by multiplying by 0.30, and km/hour can be converted to m/s by multiplying by 0.278. Other conversion fac tors can be found in Appendix B. Table 5.1 lists some typical speeds in sports and other human movements that have been reported in the biomechanics literature. Examine Table 5.1 to get a feel for some of the typical peak speeds of human movement activities.

Be sure to remember that speed is also relative to frame of reference. The motion of

Table 5.1

Typical Peak Speeds in Human Movement

Speed

Table 5.1

Typical Peak Speeds in Human Movement

Speed

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