becomes ot the slope of the tangent to the curve at B, namely, BF 
EFFORT, ACCELERATION, AND VELOCITY DIAGRAMS 287 
EF’ But 
dv 
Pa is the rate of increase of the velocity, and is therefore equal to the 
acceleration 7 In measuring the slope of BE, the height BF must be 
measured with the velocity scale, and the base EF with the time scale. 
251. Space-Time Diagram.—OBCD (Fig. 450) is a space-time 
diagram for the motion of a body. An ordinate BN of the space or 
distance curve OBC represents the distance s 
travelled after the lapse of time ¢, represented 
by the abscissa ON. 
The slope of the curve at any point B 
is equal to the velocity at the time ON, 
for if a point } be taken on the curve near 
to B, and if the ordinate n=s+6s, and the 
abscissa On =¢+ dt, the slope of Bd is = and 
dt 
in the limit when 2 coincides with B, the slope of Bb becomes a the 
slope of the tangent to the curve at B, namely, ae But . is the rate 
t 
of change of position, and is therefore equal to the: velocity v. In 
measuring the slope of BE, the height BN must be measured with the 
distance scale, and the base EN with the time scale. 
252. Acceleration-Space Diagram.—Fig. 451 shows an acceleration- 
space diagram, any ordinate BN of. the curve 
ABC representing the acceleration when the 
distance moved by the body is represented by 
the abscissa ON. Consider an indefinitely 
narrow vertical strip of the diagram. Let ds 
be the width of this strip and f its height, 
dv_ds dv_ dv 
oAcceleration. > 
then its area is fds. But Dog eet as silat: 
therefore fds =vdv. Fig. 451. 
Let v be the velocity of the body when at O, and », its velocity when at 
D, then the area OABCD = fie= do 1% If 7,, is the mean 
a 9 
acceleration between O and D, and OD =s,, then /,,3, = vi > v, or twice the 
area of the diagram represents the difference between the squares of the 
velocities of the body at the ends of the space base. If v,=0, or the 
body is at rest when at O, then /,,s, = a or twice the area of the diagram 
represents the square of the velocity of the body at the other end of the 
space base. 
253. Velocity-Space Diagram.—ABC (Fig. 452) is a curve such 
that any ordinate BN represents the velocity v of a body when it has 
moved a distance s, represented by the abscissa ON. 
