284 APPLIED MECHANICS 
will be a maximum when A is at M, where P is equal to R. Beyond 
M the speed will diminish, and if the body comes to rest at J, then 
We 4 work represented by area OBDEJ=work represented by area 
2 
OFDHJ, or tt + work represented by area BDF = work represented by 
area EDH, and if y,=0, area BDF =area EDH. 
It is sometimes convenient to plot the unbalanced effort pr on a 
straight space base, as shown in Fig. 
446. The area of this diagram be- b 
tween the ordinates through O and 
L then represents the increase in the 
kinetic energy of the body when it 9 
has moved through the distance OL, 
areas above OL being reckoned as posi- 
tive, and areas below OL as negative. 
The same result is represented by the 
portion FBDEHDF of the original diagram (Fig. 445), areas above 
FDH being reckoned as positive, and areas below FDH as negative. 
247. Effort - Time Diagram.—Referring to Fig. 445, if OX is a 
time base, that is, if ON represents the time during which the body. A 
has been moving while the effort changes from OB to Ng, then the area 
of the figure OBpN represents the momentum added to A by the effort P 
during the time ON. If the resistance R be also plotted on the time 
base OX, the result being the curve FrDH, then the area of the figure 
OFrN represents the decrease in the momentum of A during the time 
ON, and the area of the figure FBpr represents the net increase in the 
momentum of A, due to the simultaneous action of P and R during the 
time ON. 
If the unbalanced effort be plotted on a straight time base OX 
(Fig. 446), then the area of the diagram between ordinates through O and 
L represents the net increase in the momentum of A during the time OL. 
248. Space Average and Time Average of a Force.—When the 
magnitude of a force, acting on a body in the direction of its motion, is 
plotted on a straight base which represents the space or distance through 
which the force acts, the average value of the magnitude of the force, or 
the mean height of the diagram, is the space average of the force. Again, 
when the magnitude of the force is plotted on a straight base which 
represents the time of the motion, the average value of the magnitude of 
the force, or the mean height of the diagram, is the time average of the 
Sorce. 
When the space average of a force is used, it is a question of work ; 
and when the time average of the force is used, it is a question of 
momentum. 
The space and time averages of a force are obviously equal when the 
magnitude of the force is constant; they are also equal when the body 
upon which the force acts moves with a uniform velocity, however the 
force may vary in magnitude; but if the velocity of the body is not 
uniform, and the magnitude of the force varies, the space and time averages 
of the force may be very different. 
Fiag. 446. 
