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41 method conceptually similar to momentum is derived from the conc ts in phjsics of velocity and acceleration, both elements in the science of motion. Velocity, as defined in mechanics, is the rate of change of position with respect to time (also called speed). There are two tjpes of velocity, average and instantaneous. The average velocity is simply calculated as the mean velocity over a fbsed distance and for a fixed time interval. In working with commodity prices, the time interval used will be dajs and the distance is measured in points: so that if silver moved 40 points in 6 dajs, its average velocity is 40 2 V = zr = b  points per day o 3 In general, the average velocity is expressed D where D is the total elapsed distance over the time interval T For a geometric interpretation of momentum, D (ihe change in price) can be related to 7" (the length of the momentum span) to get exactly the same results for average velocity as for slope (see Figure 613) The instantaneous velocity /, which is the velocity calculated at a specific point in time, will be different. To determine the itistantaneous velocity, a mathematical technique called differentiation is used. It effectively looks at smaller and smaller time intervals and consequently smaller distances on the price curve until the slope calculation is reduced to a single point. The results of the process of differentiation is called the derivative and is expressed dD f, = lim  = 7 This shows that the velocity taken at any point is the result of the time interval (t) becoming progressively smaller without reaching zero. The rules for differentiation can be found in any advanced mathematics book. Only the results will be presented. The velocity v, represents the speed or momentum of the price at the point in time t. If r gees larger for /, Ci, tt, . . , then the velocity is increasing; If v gets smaller, the velocity is decreasing. Because the velocity also denotes direction, it can be both positive and negative in value and appear similar to a momentum indicator Systems applied to momentum may equally be applied to velocity. Of course, some of the basic equations have constant velocity and cannot be used for a velocity trading plan, because the values never chaiige. The strght line, simple FIGURE 613 (a) Average velocity, (b) Instantaneous velocity
and weighted moving averages, and exponential smoothing all have constant velocities. Only those equations with secondorder smoothing will work. When the process of differentiation is reapplied to the equation for velocity, it results in the rate of change the apeed with respect to time, or acceleration. This tjpe of acceleration tells whether the velocity is increasing or decreasing at any point in time. The acceleration, also called the second derivative, adds another dimension to momentum and may improve the timing of trades. Lets assume that the velocity and acceleration have been calculated (Table 62). The following are the possible combinations that can occur:   Price Movenient    Price is moving up at an increasing rate    Price is moving up at a constant rate    Price IS moving up at a decreasirig rate    Price is stadc    Price is moving down at a decreasing rate    Price is moving down at a constant rate    Price is moving down at an increasing rate 
TABLE 62 Equations for Velocity and Acceleration
Straight line Curvinear Logandimic (natural to Eiponeiwl Moving aversse y, = In X, Weighted moving average Exponential smoodiwi v, = b v, = b + 2o, X.* Aecetemwi *,* v,i;(x,ino) o,=i/(i;ino) c, = 0 * Because velocily and are time iJerivaoves.all c< ni Hiylicldx indude Ihe hoar as parr of Ihe rifhi nwnber Using the acceleration feature, a change of velocity (or momentum) can be detected, or the strength of a current price move can be confirmed. Quick Calculation of Velocity and Acceleration A less precise but very convenient way of determining velocity and acceleration is the calculation of first and second differences. The pu ose of these values is to find more sensitive indicators of price change, and most traders will find this quick calculation satisfactory. The results can be used in exactly the same way as the formal mathematical results. Consider the following examples: 1. A price series 10, 20, 30,40.... is moving higher by a constant value each day. The first differences are 10,10, 10,..., showing a consistent velocity of 10. The second differences, formed by subtracting sequential values in the firstdifference series, are 0, 0, 0, ..., showing that there is no change in speed; therefore, the acceleration is 2. Another price series is shown with its first and second differences as where S is the price series V is the velocity (first differences; A is the acceleration (second differences) The original series S has two tums in trend clearly shown by the velocity and acceleration. The velocity V continues to be positive through the sixth value as the underljing price moves from 10 to 50. Whenever prices change direction, the velocity changes sign. The basic upward trend can be associated with a positive velocitj and a downward trend with a negative one. The acceleration, or second difference, shows the change in speed. At the sixth item, the acceleration becomes negative, even though the velocity was positive, because prices moved higher at a slower rate. They had been gaining by 5, 10, and 15 points each day, but on day 6, the gain was only 5 points. This reversal in acceleration was a leading indicator of a trend change. A similar situation occurred on day 8, when the acceleration slowed and reversed on day 10, one day shead of the actual price reversal. The relationship between velocity and acceleration using first and second differences can be interpreted using the
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