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Application: Speed

One of the most important athletic abilities in many sports is speed. Coaches often quip that "luck follows speed" because a fast athlete can arrive in a crucial situation before their opponent. Coaches that often have a good understanding of speed are in cross-country and track.The careful timing of various intervals of a race, commonly referred to as pace, can be easily converted average speeds over that interval. Pace (time to run a specific distance) can be a good kinematic variable to know, but its value depends on the duration of the interval, how the athlete's speed changes over that interval, and the accuracy of the timing. How accurate do you think stopwatch measures of time and, consequently, speed are in track? What would a two-tenths-of-a-second error mean in walking (200 s), jogging (100 s), or sprinting (30 s) a lap on a 220 m track? Average speeds for these events are 1.1, 2.2, and 7.3 m/s, with potential errors of 0.1, 0.2, and 0.7%. Less than 1%! That sounds good, but what about shorter events like a 100-m dash or the hang time of a punt in football? American football has long used the 40-yard dash as a measure of speed, ability, and potential for athletes despite little proof of its value (see Maisel, 1998). If you measured time by freezing and counting frames of video (30 Hz), how much more accurate would a 40-meter dash timing be? If the current world record for 100 m is 9.79 seconds and elite runners cover 40 m (43.7 yards) out of the starting blocks in about 4.7 seconds, should you believe media guides that say that a certain freshman recruit at Biomechanical State University ran the forty (40-yard dash) in 4.3 seconds?

Figure 5.2. The speed of runner A depends on the frame of reference of the measurement. Runner A can be described as moving at 8 m/s relative to the track or -1.5 m/s relative to runner C.

the runner in Figure 5.2 can correctly be described as 8 meters per second (m/s), 1 m/s, or -1.5 m/s. Runner A is moving 8 m/s relative to the starting line, 1 m/s faster than runner B, and 1.5 m/s slower than runner C. All are correct kinematic descriptions of the speed of runner A.

Velocity is the vector corresponding to speed. The vector nature of velocity makes it more complicated than speed, so many people incorrectly use the words interchangeably and have incorrect notions about velocity. Velocity is essentially the speed of an object, in a particular direction. Velocity is the rate of change of displacement (V = d/t), so its units are the same as speed, and are usually qualified by a directional adjective (i.e., horizontal, vertical, resultant). Note that when the adjective "angular" is not used, the term velocity refers to linear velocity. If you hear a coach say a pitcher has "good velocity," the coach is not using biomechanical terminology correctly. A good question to ask in this situation is: "That's interesting. When and in what direction was the pitch velocity so good?"

The phrase "rate of change" is very important because velocity defines how quickly position is changing in the specified direction (displacement). Most students might recognize this phrase as the same one used to describe the derivative or the slope of a graph (like the hand dynamometer example in Figure 2.4).

Remember to think about velocity as a speed, but in a particular direction. A simple example of the velocity of human movement is illustrated by the path (dotted line in Figure 5.3) of a physical education student in a horizontal plane as he changes exercise stations in a circuit-training program. The directions used in this analysis are a fixed reference frame that is relevant to young students: the equipment axis and water axis.

The student's movement from his initial position (I) to the final position (F) can be vectorially represented by displacements along the equipment axis (dE) and along the water axis (dW). Note that the definition of these axes is arbitrary since the student must combine displacements in both directions to arrive at the basketballs or a drink. The net displacements for this student's movement are positive, because the final position measurements are larger than the initial positions. Let's assume that dE = 8 m and dW = 2 m and that the time it took this student to change stations was 10 seconds. The average velocity along the

Figure 5.3. The horizontal plane path (dashed line) of a physical education student changing practice stations. The linear displacements can be measured along a water fountain axis and equipment axis.

water axis would be VW = dW/t = 2/10 = 0.2 m/s. Note that the motion of most interest to the student is the negative displacement (-dW), which permits a quick trip to the water fountain. The average velocity along the equipment axis would be 0.8 m/s (VE = dL/t = 8/10). Right-angle trigonometry can then be used to calculate the magnitude and direction of the resultant displacement (dR) and then the average velocity of the student. We will use right-angle trigonometry in chapter 6 to analyze the effect of force vectors. By the way, if your right-angle trigonometry is a little rusty, check out appendix D for a refresher.

Calculations of speed and velocity using algebra are average velocities over the time interval used. It is important to realize that the smaller the time interval the greater the potential accuracy of kinematic calcula tions. In the previous example, for instance, smaller time intervals of measurement would have detected the negative velocity (to get a drink) and positive velocity of the student in the water direction. Biome-chanics research often uses high-speed film or video imaging (Gruen, 1997) to make kinematic measurements over very small time intervals (200 or thousands of pictures per second). The use of calculus allows for kinematic calculations (v = dd/dt) to be made to instantaneous values for any point in time of interest. If kinematic calculations are based over a too large time interval, you may not be getting information much better than the time or pace of a whole race, or you may even get the unusual result of zero velocity because the race finished where it started.

Graphs of kinematic variables versus time are extremely useful in showing a pattern within the data. Because human movement occurs across time, biokinematic variables like displacement, velocity, and acceleration are usually plotted versus time, although there are other graphs that are of value. Figure 5.4 illustrates the horizontal displacement and velocity graphs for an elite male sprinter in a 100-m dash. Graphs of the speed over a longer race precisely document how the athlete runs the race. Notice that the athlete first approaches top speed at about the 40- to 50-meter mark. You can compare your velocity profile to Figure 5.4 and to those of other sprinters in Lab Activity 5.

Acceleration

The second derivative with respect to time, or the rate of change of velocity, is acceleration. Acceleration is how quickly velocity is changing. Remember that velocity changes when speed or direction change. This vec tor nature of velocity and acceleration means that it is important to think of acceleration as an unbalanced force in a particular direction. The acceleration of an object can speed it up, slow it down, or change its direction. It is incorrect to assume that "acceleration" means an object is speeding up. The use of the term "deceleration" should be avoided because it implies that the object is slowing down and does not take into account changes in direction.

Let's look at an example that illustrates why it is not good to assume the direction of motion when studying acceleration. Imagine a person is swimming laps, as illustrated in Figure 5.5. Motion to the right is designated positive, and the swimmer has a relatively constant velocity (zero horizontal acceleration) in the middle of the pool and as she approaches the wall. As her hand touches the wall there is a negative acceleration that first slows her down and then speeds her up in the negative direction to begin swimming again. Thinking of the acceleration at the wall as a push in the negative direction is correct throughout the

Figure 5.4. The displacement-time (dashed curve) and velocity-time (solid curve) graphs for the 100-m dash of an elite male sprinter.

Figure 5.5. The motion and accelerations of swimmers as they change direction in lap swimming. If motion to the right is designated positive, the swimmer experiences a negative acceleration as they make the turn at the pool wall. The negative acceleration first slows positive velocity, and then begins to build negative velocity to start swimming in the negative direction. It is important to associate signs and accelerations with directions.

Figure 5.5. The motion and accelerations of swimmers as they change direction in lap swimming. If motion to the right is designated positive, the swimmer experiences a negative acceleration as they make the turn at the pool wall. The negative acceleration first slows positive velocity, and then begins to build negative velocity to start swimming in the negative direction. It is important to associate signs and accelerations with directions.

turn. As the swimmer touches the other wall there is a positive acceleration that decreases her negative velocity, and if she keeps pushing (hasn't had enough exercise) will increase her velocity in the positive direction back into the pool.

The algebraic definition of acceleration (a) is V/t, so typical units of acceleration are m/s2 and ft/s2. Another convenient way to express acceleration is in units of gravitational acceleration (g's). When you jump off a box you experience (in flight) one g of acceleration, which is about -9.81 m/s/s or -32.2 ft/s/s. This means that, in the absence of significant air resistance, your vertical velocity will change 9.81 m/s every second in the negative direction. Note that this means you slow down 9.81 m/s every second on the way up and speed up 9.81 m/s every second on the way down. G's are used for large acceleration events like a big change of direction on a roller coaster (4 g's), the shockwaves in the lower leg following heel strike in running

(5 g's), a tennis shot (50 g's), or head acceleration in a football tackle (40-200 g's). When a person is put under sustained (several seconds instead of an instant, like the previous examples) high-level acceleration like in jet fighters (5-9 g's), pilots must a wear pressure suit and perform whole-body isometric muscle actions to prevent blacking out from the blood shifting in their body.

Acceleration due to gravity always acts in the same direction (toward the center of the earth) and may cause speeding up or slowing down depending on the direction of motion. Remember to think of acceleration as a push in a direction or a tendency to change velocity, not as speed or velocity. The vertical acceleration of a ball at peak flight in the toss of a tennis serve is 1 g, not zero. The vertical velocity may be instantaneously zero, but the constant pull of gravity is what prevents it from staying up there.

Let's see how big the horizontal acceleration of a sprinter is in getting out of the blocks. This is an easy example because the rules require that the sprinter have an initial horizontal velocity of zero. If video measurements of the sprinter showed that they passed the 10-m point at 1.9 seconds with a horizontal velocity of 7 m/s, what would be the runner's acceleration? The sprinter's change in velocity was 7 m/s (7 -0), so the sprinter's acceleration was: a = V/t = 7/1.9 = 3.7 m/s/s. If the sprinter could maintain this acceleration for three seconds, how fast would he be running?

Close examination of the displacement, velocity, and acceleration graphs of an object's motion is an excellent exercise in qualitative understanding of linear kinematics. Examine the pattern of horizontal acceleration in the 100-m sprint mentioned earlier (see Figure 5.4). Note that there are essentially three phases of acceleration in this race that roughly correspond to the slope of the velocity graph. There is a positive acceleration phase, a phase of near zero acceleration, and a negative acceleration phase. Most sprinters struggle to prevent running speed from declining at the end of a race. Elite female sprinters have similar velocity graphs in 100-m races. What physiological factors might account for the inability of people to maintain peak speed in sprinting?

Note that the largest accelerations (largest rates of change of velocity) do not occur at the largest or peak velocities. Peak velocity must occur when acceleration is zero. Coaches often refer to quickness as the ability to react and move fast over short distances, while speed is the ability to cover moderate distances in a very short time. Based on the velocity graph in Figure 5.4, how might you design running tests to differentiate speed and quickness?

Acceleration is the kinematic (motion description) variable that is closest to a kinetic variable (explanation of motion). Kinesiology professionals need to remember that the pushes (forces) that create accelerations precede the peak speeds they eventually create. This delay in the devel opment of motion is beyond the Fore-Time Principle mentioned earlier. Coaches observing movement cannot see acceleration, but they can perceive changes in speed or direction that can be interpreted as acceleration. Just remember that by the time the coach perceives the acceleration the muscular and body actions which created those forces occurred just before the motion changes you are able to see.

Uniformly Accelerated Motion

In rare instances the forces acting on an object are constant and therefore create a constant acceleration in the direction of the resultant force. The best example of this special condition is the force of earth's gravity acting on projectiles. A projectile is an object launched into the air that has no self-propelled propelling force capability (Figure 5.6). Many human projectile movements have vertical velocities that are sufficiently small so that the effects of air resistance in the vertical direction can be ignored (see chapter 8). Without fluid forces in the vertical direction, projectile motion is uniformly accelerated by one force, the force of gravity. There are exceptions, of course (e.g., skydiver, badminton shuttle), but for the majority of human projectiles we can take advantage of the special conditions of vertical motion to simplify kinematic description of the motion. The Italian Galileo Galilei is often credited with discovering the nearly constant nature of gravitational acceleration using some of the first accurate of measurements of objects falling and rolling down inclines. This section will briefly summarize these mathematical descriptions, but will emphasize several important facts about this kind of motion, and how this can help determine optimal angles of projection in sports.

When an object is thrown or kicked without significant air resistance in the ver

Figure 5.6. Softballs (left) and soccer balls (right) are projectiles because they are not self-propelled when thrown or kicked.

tical direction, the path or trajectory will be some form of a parabola. The uniform nature of the vertical force of gravity creates a linear change in vertical velocity and a second-order change in vertical displacement. The constant force of gravity also assures that the time it takes to reach peak vertical displacement (where vertical velocity is equal to zero) will be equal to the time it takes for the object to fall to the same height that it was released from. The magnitude of the vertical velocity when the object falls back to the same position of release will be the same as the velocity of release. A golf ball tossed vertically at shoulder height at 10 m/s (to kill time while waiting to play through) will be caught at the same shoulder level at a vertical velocity of -10 m/s. The velocity is negative because the motion is opposite of the toss, but is the same magnitude as the velocity of release. Think about the 1 g of acceleration acting on this golf ball and these facts about uniformly accelerated motion to estimate how many seconds the ball will be in flight.

This uniformly changing vertical motion of a projectile can be determined at any given instant in time using three formulas and the kinematic variables of displacement, velocity, acceleration, and time. My physics classmates and I memorized these by calling them VAT, SAT, and VAS. The various kinematic variables are obvious, except for "S," which is another common symbol for displacement. The final vertical velocity of a projectile can be uniquely determined if you know the initial velocity (Vj) and the time of flight of interest (VAT: Vf2 = Vi + at). Vertical displacement is also uniquely determined by initial velocity and time of flight (SAT: d = Vit + 0.5at2). Finally, final velocity can be determined from initial velocity and a known displacement (VAS: Vf2 = Vi2 + 2ad).

Let's consider a quick example of using these facts before we examine the implications for the best angles of projecting objects. Great jumpers in the National Basketball Association like Michael Jordan or David Thompson are credited with standing vertical jumps about twice as high (1.02 m or 40 inches) as typical college males. This outstanding jumping ability is not an exaggeration (Krug & LeVeau, 1999). Given that the vertical velocity is zero at the peak of the jump and the jump height, we can calculate the takeoff velocity of our elite jumper by applying VAS. Solving for Vj in the equation:

We select the velocity to be positive when taking the square root because the initial velocity is opposite to gravity, which acts in the negative direction. If we wanted to calculate his hang time, we could calculate the time of the fall with SAT and double it because the time up and time down are equal:

d = Vjt + 0.5at2 -1.02 = 0 + 0.5(-9.81)t2 t = 0.456 s

So the total fight time is 0.912 seconds. If you know what your vertical jump is, you can repeat this process and compare your takeoff velocity and hang time to that of elite jumpers. The power of these empirical relationships is that you can use the mathematics as models for simulations of projectiles. If you substitute in reasonable values for two variables, you get good predictions of kinematics for any instant in time. If you wanted to know when a partic ular height was reached, what two equations could you use? Could you calculate how much higher you could jump if you increased your takeoff velocity by 10%?

So we can see that uniformly accelerated motion equations can be quite useful in modeling the vertical kinematics of projectiles. The final important point about uniformly accelerated motion, which reinforces the directional nature of vectors, is that, once the object is released, the vertical component of a projectile's velocity is independent of its horizontal velocity. The extreme example given in many physics books is that a bullet dropped the same instant another is fired horizontally would strike level ground at the same time. Given constant gravitational conditions, the height of release and initial vertical velocity uniquely determine the time of flight of the projectile. The range or horizontal distance the object will travel depends on this time of flight and the horizontal velocity. Athletes may increase the distance they can throw by increasing the height of release (buying time against gravity), increasing vertical velocity, and horizontal velocity. The optimal combination of these depends on the biomechanics of the movement, not just the kinematics or trajectory of uniformly accelerated motion. The next section will summarize a few general rules that come from the integration of biomechanical models and kinematic studies of projectile activities. These rules are the basis for the Optimal Projection Principle of biomechanics.

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