238 APPLIED MECHANICS 
It was shown in the preceding Article that f= of, therefore 
f_w But . is the slope — BF of the tangent BE to the curve ABC 
v 
EF 
ds 
LN _ BF  s 
at B. If BL is the normal to the curve ABC at B, then —— BN EFT ) 
but BN=v, therefore LN=/; or the sub-normal of a velocity-space 
curve represents the acceleration. 
The scale with which to measure LN must 
now be determined. Let the velocity scale be = 3 
1 inch to m feet per second, the distance “8 \v 
scale 1 inch to m feet, and the acceleration S$ 7 ¥ a 
scale 1 inch to.x feet per second per second, AN n INE 
Let BF, EF, BN, and LN denote the lengths 4g Tistemes D 
of these lines in inches. Then, f=LN xa Fig. 452 : 
feet per second per second, and v=BN xm 
feet per second; BF represents a velocity BF x m feet per second, and EF 
represents a distance EF x m feet. Hence 2 = es <i = et: 
2 sm m2 3 
—s =, ANd aaa. 
n 
, therefore 
254. Conversion of Space-, Velocity-, and Acceleration-Time 
Diagrams.—It was shown in Art. 250 that the slope of the velocity-time 
curve represented the acceleration, and in Art. 251 that the slope of the 
space-time curve represented the velocity. These properties may be 
made use of in constructing any two of the three curves, space-time, 
velocity-time, or acceleration- time, from the third. 
The curve OABC (Fig 453) isa space-time curve plotted from the 
data in the following table :— 
0 2 4 6 8 10 12 14 16 18 
| Psat tag teh Sy 7 22 4] 64 90 122 | 160 | 197 | 228 
where ¢ is the time in seconds, and s the distance moved from rest in 
feet. Let A and-B be two points on the curve OA BC, the points being 
sufficiently near to one another to warrant the assumption that the part 
AB of the curve is straight. Drawing AD perpendicular to the ordinate 
through B, BD is the space covered during the interval of time AD, and 
the mean velocity during that interval is BD+AD,. In Fig. 453 AD 
is 2 seconds and BD is 26 feet, therefore the velocity at the time 
9 seconds, the middle of the interval AD, is 13 feet per second, and if 
the ordinate NP be made equal to 13 on the velocity scale, a point P on 
the velocity curve is determined. If equal intervals of time be taken, it 
is only necessary to take the distance BD in the dividers and step it out 
a fixed number of times on the mid ordinate to obtain a point on the 
velocity curve. 
The velocity curve in Fig. 453 has been found by taking intervals of 
one second, and making the mid ordinate ten times the increase in space 
for each second. 
