5. KINEMATICS. Figure 5.1 Types of planar motions

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1 5. KINEMATICS Suddenly there is no more acceleration and no more pressure on my body. It is like being at the top of a roller coaster and my whole body is in free-fall. Roberta Bondar, Touching the Earth. p.22, 1994 As mentioned previously, kinematics is the study of motion without regard to its causes. It is concerned with motion description and motion quantification. For example, kinematics is concerned with how far a body travels, how fast it travels, its direction of motion, its rotation and its pattern of motion. Kinematic quantities include such variables as position, speed, acceleration and their rotational counterparts. To distinguish among motion patterns all movements can be classified as either translational or rotational or a combination of both. Translational motion occurs when the particles in a body follow parallel paths. Thus, any straight line within the body will have a constant orientation with the respect to an inertial frame of reference. The motion can be either rectilinear or curvilinear. With rectilinear translation, also called, linear motion, all motion is in a straight line; with curvilinear translation the paths are not linear (see figure 5.1). Rotational motion, also called angular motion, occurs when particles within the body are considered to have rotated with respect to the centre of the body or an inertial frame of reference. General plane motion is any motion that Figure 5.1 Types of planar motions occurs in a single plane and is a combination of both translational and rotational motions. It is rare for any human motion to be only translational or rotational almost all movements involve some combination of both. In this chapter, kinematics will be divided into linear and angular kinematics with a special section on projectile motion. Linear kinematics is used to describe rectilinear translational motions while angular kinematics is used to describe rotational motion. At the end of the chapter general plane motion will be considered by examining the relationships between linear and angular motion.

2 82 Linear Kinematics 5.1 Linear Kinematics Displacement versus Distance In mechanics the word distance takes on a special meaning; it is the length of the path travelled by a body. Since the direction of motion along a path may not be in a single direction, distance cannot have an associated direction. Thus, distance must be a scalar quantity. Displacement. Displacement is the term used to describe the straight line that connects a point s position from one instant in time to another. It must include both the length and direction of this line. Thus, displacements are vector quantities because the straight line has both a magnitude and an associated direction. As vectors, displacements do not add algebraically as do distances, unless all the displacements happen to be along the same straight line. Instead, displacements add according to the Parallelogram law. The question arises: which quantity best describes human movements, distance or displacement? The answer depends upon the motion that is being analyzed and the question that is being asked. It is always better and safest to quantify the total movement of an object or person by distance since distance measures the actual path that the person travelled. For example, in a marathon race where the route travelled is often quite circuitous and can actually start and end at the same point, distance would be the only Figure 5.2 Distance versus displacement appropriate measure (figure 5.2). The distance would be some amount greater than 26 miles 385 yards depending upon the exact route taken by the particular runner. The displacement could be ridiculously low, considering only the difference between the runner s starting and finishing positions. Displacement, based upon the previous example, seems to be a relatively useless measure. It plays an important role, however, when used in the proper situations. Displacement should only be used when the amount of movement from one position to another is relatively short. It is the better measurement in these situations since it will show the direction of travel and not just the amount. It is often difficult to quantify the exact path (i.e., distance) that a person travels from one instant to another. On the other hand, the displacement can be easily determined knowing only the initial and final positions. Displacement measurements lend themselves well to the quantification of human movement by cinematography (motion picture film) or videography (television). Both media record movements at discrete points in time. For th television systems the rate of recording is every 30 of a second (in the Americas, th th th or 25 second elsewhere) and for cinecameras can range from a 24 to a 500 of a second or even faster. Specialized television, CCD or infrared video systems can reach speeds of 2000 frames per second and faster.

3 Kinematics 83 These speeds should not be confused with the shutter speeds on modern video cameras. Shutter speeds indicate how long light is allowed to pass through the camera s shutter. A shutter speed of 1/1000 will freeze a rapid motion that would otherwise be a blur. The camera, however, only records these snapshots at 30 per second. Actually video systems operate in an interlaced fashion. That is, they record at the rate of 60 frames per second but only on half of the picture s scan lines. It requires two sweeps to produce a full picture image and therefore it takes (2 times 1/60) 1/30 of a second to record. Video systems that biomechanics use to digitize a motion sequence use this feature to double the sampling rate to 60 frames per second by separating the two frames that make up a single television image. This process yields a sampling rate that is just fast enough to record accurately most human motions. Higher frame rates, however, are needed for recording athletic motions, such as, running, jumping and throwing skills. High-speed video-based camera systems are available for this purpose but are very expensive. Speed versus Velocity Speed. Speed is defined as the rate of change of distance over time and is therefore a scalar quantity, as is distance. No direction need be associated with speed because the movement may not be in a straight line. To compute average speed, divide the distance travelled by the travel time. That is, (5.1) The line over the word, speed, indicates that the speed is an average value rather than an instantaneous one. (The symbols within the square brackets in this equation and all subsequent equations are the accepted SI abbreviations for the units of the answer.) The standard SI unit for speed is metres per second (m/s) but kilometres per hour (km/h) is acceptable for some purposes, such as automobile speeds. Speed is rarely used in biomechanics research because it gives no information about the direction of motion. A speedometer on a car for example reads the same speed no matter which direction you are travelling. Furthermore, speed or more specifically average speed can lead to misleading conclusions. A good example occurred in the 1996 Olympics when Donovan Bailey set a world record in the 100-m sprint (9.84). Later, Michael Johnson also set a world record in the 200 m (19.32). Usually, the winner of the 100-m sprint is declared the fastest human but many in the American media calculated that Michael Johnson was faster because by dividing his 200-m time in half produced two 100-m times that were less than Donovan Bailey s 100-m time. By the same logic all four members of the world record m relay team ran faster than both Bailey and Johnson with times of ( =) 9.35 s! Why were these assessments incorrect? The reason, obviously, is that Johnson and three members of the relay team had running starts. If they had run from a motionless start their 100-m times would have been longer than Bailey s time. Of course, we do not know what would have happened if the 100-m race had been extended to 200 metres. Undoubtedly, Bailey would have had to slow

4 84 Linear Kinematics down due to local muscle fatigue. If we extrapolate (and we should not) Bailey would have run the second 100-m in 8.33 seconds assuming that he maintained his maximum velocity of, approximately, 12 m/s. This would have produced a 200-m time of ( =) s, 1.15 s faster than Johnson! Presumably, no one can run full out for 200 metres. Each race demands a different running strategy. But these examples point out the problem of applying equations to inappropriate situations or of extrapolating results. One should always remember the limitations of each equation. In this case, an equation which assumes constant speed was used to estimate the time taken to run a race which is not run at constant speed. Velocity. Velocity is defined as the rate of change of displacement. As such, it must be a vector quantity since displacement is a vector. Therefore, a velocity will always have an associated direction that indicates the direction of movement of the person or object. Velocities are measured in the same units as speed. For a linear movement in an arbitrary direction, S, the average velocity,, of an object may be determined by dividing the change in displacement over the duration of time. I.e., (5.2) where s f and s i represent the final and initial displacements (in metres) of the person or object and t represents the time elapsed in seconds. Literally, the symbol (Greek capital delta) means change in some quantity, in this case, displacement and time. This relation is also called the slope of the displacement history since it represents the rise in displacement over the run in time. Figure 5.3 shows this concept graphically. The direction of the velocity will be along the S-axis. If the velocity is positive the direction will be in the positive S-direction; if negative it will move in the negative S- direction. If the duration of time is large the instantaneous velocity of the object may vary considerably compared to its average velocity. It can be proven Figure 5.3 Average velocity as slope of displacement history mathematically, however, that at least once during the interval the object travelled at the average velocity. To have an instantaneous measure of the object s velocity it is necessary to reduce the change in time. We can reduce this duration as small as we want until it is infinitesimal (see figure 5.4). This process is called differentiation. This mathematical operation is part of the discipline called,

5 Kinematics 85 differential calculus. It was originally developed, simultaneously and independently, by Isaac Newton and Gottfried Wilhelm Leibnitz in the late 1600s. In mathematical terms the instantaneous velocity, v, may be written: (5.3) The limit as delta t approaches zero means that we are concerned with the rate of change of displacement at an instant in time. The terms, ds and dt, represent very small changes or differentials of displacement and time, respectively. The terms, s and t, represent discrete, relatively large, changes in displacement and time, respectively. The delta terms are used more often in biomechanics because human motion is usually measured at discrete times by cine or video Figure 5.4 Instantaneous velocity as cameras. In which case a simpler form tangent to displacement history of differential calculus is used, called finite difference calculus. This calculus deals with rates of change at discrete th th time intervals (e.g., 1/100 or 1/30 of a second) instead of instantaneous points in time. Since human movements are relatively slow, as long as we sample the movement quickly enough, the average velocity between adjacent video or motion picture fields will be approximately equal to the instantaneous velocity. For planar motion the situation is similar excepting you must apply the velocity equation twice, once for each direction (X and Y). That is, (5.4) You should notice that we are using the term displacement differently here. Earlier we defined displacement to mean the vector which connects two positions in space. Here we will use displacement to mean the vector that connects the origin of our axis system to the position of the object. Thus, the instantaneous displacement of an object (from the origin) may be written, s, or (s, s ) or (x, y). x y

6 86 Linear Kinematics Similarly, the velocity vector may be written in its rectangular coordinates,, or using polar coordinates,, where (5.5) The direction v, will be the same Figure 5.5 Velocity in rectangular and direction as the line that connects the polar coordinates initial displacement to the final displacement. In other words, it will be in the direction of the line of motion. Acceleration Acceleration is defined as the rate of change of velocity (equation 5.6). It may also be defined as the rate of change, of the rate of change, of displacement. Since displacements and velocities are vectors, accelerations are vectors and will always have associated directions. Instantaneous linear acceleration (a) is defined: (5.6) where v is the change in velocity and t is the time duration. For linear motion analyses in an arbitrary direction, S, when an object s velocity changes from an initial velocity of v i to a final velocity of v f, the equation for the magnitude of the average acceleration vector will be: (5.7) where, t, represents the time taken to make the change in velocity. The direction of the acceleration in the S-direction will depend upon the magnitudes of the two velocities. If the magnitude of is positive then the acceleration will be in the positive S-direction; if negative then the acceleration will be toward the negative S-direction. Be aware that the direction of the acceleration may be different from those of the two velocities. For example, if a body moving in the negative direction slows from m/s to 8.00 m/s in 1 second the acceleration will be m/s (i.e., [ 8.00 ( 10.00)]/1 = +2.00). Thus, the acceleration is positive while the velocities are negative. This can be called deceleration in the negative direction a double negative but it is the same as a positive acceleration in the positive direction of a body that increases from 5.00 m/s to 7.00 m/s in 1.000

7 Kinematics 87 second. The two motions will look different but the two accelerations will be the same. Of course, if the first motion (i.e., slowing from m/s to 8.00 m/s) had been in the positive direction, the acceleration would have been negative, in other words, deceleration. Clearly, one must be concerned with both the direction of motion (as indicated by the sign of the velocity) and the direction (+/ ) of the acceleration that a body experiences to describe precisely its motion. For planar motion analyses, acceleration is computed by separating the velocity of a body into its X-Y components as follows: (5.8) where the numerators represent the changes from an initial velocity to a final velocity for the X- and Y-directions, respectively. Acceleration is a difficult concept yet it is the most important kinematic quantity because it is directly related to the cause of motion force as defined by Newton s Second Law (to be defined more precisely in chapter six). This law states that whenever an object undergoes acceleration, a force must be present to cause the acceleration. Determining when an object accelerates, however, is difficult to do by observation alone. The human mind is incapable of performing the required mathematical operations while the movement is taking place. Therefore, it is necessary to record the motion in some way and then apply the appropriate operations to compute the acceleration pattern. The direction of acceleration of an object, which is the same as the direction of the total applied force, must not be confused with the direction of movement of the object. The direction of movement is equivalent to the direction of the object s velocity and, as mentioned earlier, can be the same or quite different from that of the object s acceleration. For example, if a ball is thrown straight up its acceleration and velocity will both be directed upwards while in the person s hand. After being released from the hand, the ball will continue for a time with an upward velocity until the top of its flight. Then, for an instant, it will have zero velocity, followed by an increasing downward velocity until it strikes the ground. In contrast, its acceleration will remain constant throughout its flight and be directed Figure 5.6 Trajectory of a thrown ball always straight down towards the ground or, more precisely, the centre of the earth (ignoring any air resistance). Take another example the motion of a sprinter. After the starting gun is fired and a brief delay due to human reaction time, a sprinter will start accelerating forward, gradually increasing velocity in the same direction. At some

8 88 Linear Kinematics point in the middle of the race the sprinter will reach maximum forward velocity. For a time the sprinter will maintain this velocity and, therefore, have an acceleration of zero. As local muscle fatigue sets in and reduces the ability to create adequate force, the sprinter s velocity may begin to reduce and deceleration will occur. Yet the sprinter s velocity will continue to be directed forward. It may appear to an observer watching from the side that a particular sprinter is accelerating past other sprinters at the end of a race. What may be happening, however, is that the other runners are decelerating. In a study of Ben Johnson and Angella Issajenko (Lemaire & Robertson, 1989), both former world record holders, it was found that they accelerated up to a maximum velocity of approximately 12 and 10 m/s, respectively, during the first 60 to 70 metres of a 100 m race. After this acceleration phase, they essentially ran at constant velocity until the finish line. Of course, less welltrained athletes may begin decelerating before the finish. Many coaches had assumed that all sprinters decelerated, presumably due to fatigue, somewhere Figure 5.7 Horizontal displacement histories of three sprinter for the last 60 metres of a competitive 100 m sprint race before the end of the race. Accurate kinematic measurements have shown this presumption false. Why this error can be made was exhibited by analyzing another athlete, Desai Williams, running in the same race as Johnson (see figure 5.7). To observers, it appeared that this athlete was decelerating at the end of the race. The data showed, however, that he was also running at a constant speed but at a slower rate (of 11 m/s). Thus, relatively speaking, Williams was decelerating with respect to Johnson; in absolute terms, however, he had a lower constant velocity that was 1.0 m/s slower than Johnson s.

9 Kinematics 89 Constant Linear Velocity Whenever the resultant force acting on a moving body is zero, the body will continue to keep moving at constant velocity along the same straight line. By knowing its velocity and line of motion it is possible to determine the position of the body at any point in time until the resultant force becomes nonzero. By rearranging equation (5.2) we can determine the final displacement of a body, s f, by knowing the body s initial displacement, s i, the initial velocity, v i (which remains constant), and the amount of time elapsed, t. We will drop the use of the delta t ( t) in the following equations for simplicity, although it should be understood that t refers to a period of time and not a particular point in time. (5.9) Example For example, if a person is on a train that is moving at the constant velocity of km/h after 2.00 hours the person will be (assuming an initial displacement of zero kilometres). Example A shot putter releases a shot with a horizontal velocity of 3.50 m/s. If it stays airborne for 3.00 seconds and was released 40.0 cm past the front edge of the ring, how far will the shot travel from the ring? Example How long will it take an arrow to strike a target 20.0 metres away if it is released at a velocity of km/h? First convert the velocity to m/s. Then rearrange equation 5.9 to compute the flight time. Note the initial position, s, is assumed to be zero. i

10 90 Linear Kinematics Example The speed of sound in air at 20 C is 343 m/s (see table F.5 in Appendix F). How much delay will there be between when the starter s gun fires and the athletes hear the sound if they are separated by m? Thus, 3 hundredths of a second will be added to the athlete s race time. This is why starting blocks at major competitions have audio speakers behind each athlete. Constant Linear Acceleration Constant linear acceleration occurs whenever the resultant force acting on the body is a constant. In this special case the position and velocity of the body at any instant may be predicted exactly by knowing the initial position and initial velocity of the object. The following series of equations may be used to determine position, velocity or movement time given the acceleration of the body plus some of the initial conditions of the body. The first equation is a rearrangement of the definition of average acceleration (equation 5.7). This equation may be used to determine the velocity of a body after accelerating for a particular duration (t). The second equation is based on the equation used to predict the final position of a body that is moving at constant linear velocity (equation 5.9) plus the additional displacement produced by the constant acceleration, a. Since the velocity is increasing linearly the displacement will increase exponentially. Graphically, we can show that the increase in position is 2 equal to ½at (figure 5.8). First, the increase in velocity from v i to v f will be equal to at based on equation The additional displacement caused by the acceleration will be equal to the area of the triangle (cross-hatched area in figure 5.8) Figure 8 Velocity versus time with constant acceleration (5.10)

11 Kinematics 91 formed by the duration of the acceleration, t, times one half the increase in velocity, at. (The area of a triangle is one half base times height.) That is, (5.11) where s f and s i are the final and initial displacements, respectively. The third equation permits the kinematic description of motion where the duration of the motion is unknown or of no relevance to the problem. It can be derived from equation 5.11 by substituting t with the expression: (which is a rearrangement of equation 5.10). Simplifying produces the following: Table 5.1 Equations for constant acceleration and their relationships Number Equation si vi sf vf t a I v f =v i + at II s f =s i + v i t+ ½at III v f =v i + 2a(s f s i) IV s f =s i + ½(v i + v f) t (5.12) Using the above equation, the final velocity may be computed given the initial and final positions of the body and its initial velocity or the final position of the body may be predicted given the initial position and the initial and final velocities. Table 5.1 presents the relationships among the various variables and equations 5.10, 5.11 and 5.12 plus a fourth equation (IV in table 5.1) that may be derived from the others. As an exercise demonstrate how it (Equation IV) is derived. Example If a person accelerates at the rate of m/s from rest (i.e., zero velocity), what will be the person s velocity after m? First, select an appropriate equation (from table 5.1), that is, an equation that involves the quantity that you want to compute (i.e., v f). Equations I, III and IV all satisfy this condition. Second, exclude any equations with variables that are unknown; thus, equations I and IV can be eliminated since they include the variable, t, which is unknown. Third, solve the remaining equation, III, for the unknown variable. Since the initial position, s i, is not given we can set it equal to zero. That is,

12 92 Linear Kinematics Thus, the final velocity of the person will be 5.48 m/s. The negative velocity is an extraneous solution. Example How far will it take for a sprinter to stop if she crosses the finish line with a horizontal speed of m/s and decelerates at the rate of 2.00 m/s 2? Select Equation III since it involves the unknown variable, s f, and the given information, v, and a. The final velocity will be zero since the sprinter is stopped. i Thus, it will take the sprinter 25.0 m to decelerate at this rate. Example How long will it take for a curling stone to stop if it is released at a velocity of 5.00 m/s and decelerates at the rate 2 of m/s? How far will it travel? Select Equation I since it involves time and both initial and final velocity. Thus, it will take 6.67 seconds for the stone to come to rest. Use Equation II to compute the distance travelled. The rock will travel metres down the rink. Note, this problem assumes that the stone does not curl and that there is no sweeping.

13 Kinematics 93 Example How far will a puck slide down the ice in seconds if 2 it decelerates at the rate 5.00 m/s and it starts with an initial velocity of 35.0 m/s? Thus, the puck will slide 46.9 metres, unless it hits the end of the rink. Example If a person skis down a 220 m hill in 25.0 seconds what will be the velocity at the end of the run assuming a constant rate of acceleration and an initial velocity of zero? Rearranging Equation IV from table 5.1 and substituting for the variables, we have: Thus, the skier will finish the 220 metres with a velocity of m/s. Kinematic analysis of fastball pitching. Body is modelled as a series of bones defined by the paths of markers attached at joint centres.