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end of a certain time, however, equal increments of time show equal in- 

 crements of distance. The curve then becomes straight because the rate 

 of motion has become constant. 



Velocity or the average rate of motion is defined as the space passed 

 over divided by the time required for passage. The average velocity 

 through any point then may be found by dividing small increments of 

 distance by the corresponding increments of time. By taking these in- 

 crements sufficiently small we may make the average velocity approach 

 the true instantaneous velocity through any given point, as closely as we 

 please. At the limit or when the increments become zero these velocities 

 are equal. 



Near the point "P" on the distance curves shown in Figs. 1 and 2 are 

 drawn small triangles having for their vertical components small distances 

 •'dd - ' and for their horizontal components the corresponding increments 

 of time "dt." From the above definition the average velocity for the 



space passed over designated bv the small triangle will be v = -^— . 



ilt 



By taking this triangle very small the average velocity may be made 

 to very closely approximate the instantaneous velocity at the point "P." 



It is also to be noted that the ratio -;- is the expression for the tan- 



dt 



gent of the angle included between the line "dt" and that portion of the 

 curve which completes the triangle. Values proportional to "v" may 

 therefore be found at any point on the distance curve by drawing a tan- 

 gent line at that point and finding the tangent of the angle between this 

 line and the horizontal. Plotting these values multiplied by a constant 

 gives the velocity curves "V" (See Figs. 1 and 2). From this curve we are 

 able to determine the velocity of the car at any time "t." 



By scanning curve "V" we note that the velocities for different 

 time values until that time is reached where the distance curve became 

 a straight line. At this point the tangent values become constant and 

 the velocity curve becomes horizontal. 



Just as velocity may be determined by dividing space passed over by 



the time required, so may the acceleration be determined by dividing the 



velocity change by the time required to make the change. The statements 



relative to average and instantaneous velocity also hold for average and 



dv 

 instantaneous values of acceleration. "We mav therefore write a=-^- 



dt 



