208 
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 poipt 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 
dd 
ae” 
By taking this triangle very small the average velocity may be made 
space passed over designated by the small triangle will be y= 
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- 
( 
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 anda 
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 
: : ; . : dy 
instantaneous values of acceleration. We may therefore write a =a 
( 
