286 APPLIED MECHANICS 
a time base corresponding to the positions 1’, 3’, 5’, etc., respectively of - q 
the-crank pin. The following table may now ‘be constructed :— ra 
Position of , , , , , ’ t ’ , ; 
crank pint} 2 48 | BE fe Pe) a) 16 eae 
a(time). .|1]3 | 5 7 9 it’ 1°13 |.) 0 ee 
Or aa | Bf BTL ABT | GB? 81° 99° | 117° | 135° | 153° | 179° 
Pinoys) ceyl ey bow = 150 | 109°9 | 71°11 | 51°88 | 41°27 | 35°15 | 31°73 | 30°19 | | 
from which the mean value of P is 82°12, and this is the time average 
of P. 
The time average of the pressure on the piston is therefore, in this 
case, 4°12 lbs. per square inch, or 5°3 per cent. greater than the space 
average. These results are not quite accurate, but they are sufficiently 
approximate for practical purposes. More exact results would of course 
be obtained by making the space and time intervals shorter, and corre- 
spondingly more numerous. 
If the effect of the obliquity of the connecting-rod be considered, and 
the length of the connecting-rod is five times the length of the crank, it 
will be found that in the foregoing example the time average of the 
pressure is only 1°56 per cent. greater than the space average. 
249. Acceleration-Time Diagram.—In Fig. 448, OD is a time 
base, and the curve ABC is such that any ordinate FN represents the 
acceleration f of the motion of a body after the 
lapse of time ¢, represented by the abscissa ON. - 
If f, is mean acceleration during the interval of & 
time ¢, or the mean height of the curve AF above |§ 
ON, then the area of the diagram OAFN repre- 8h t f 
sents f,,t, or v the added velocity. Ifv, and vy, wid 
are the velocities at the beginning and end of the Time D 
interval of time ON =¢, then v=v, —0,=fnt. Wel ad : 
Since acceleration is proportional to the 
force producing it, it is evident that a curve of unbalanced effort will 
also be a curve of acceleration, but to a different scale. 
250. Velocity-Time Diagram.—OABCD (Fig. 449) is a velocity- 
time diagram for the motion of a body. An ordinate BN of the velocity 
curve ABC represents the velocity v after the 
lapse of time ¢, represented by ON. oe cae 
The area of the diagram between the ordi- 3 i 
nates AO and BN represents the distance ‘8 v 
‘travelled by the body in the time 7, for if v,, S t Fret 
is the mean velocity between O and N, the dis “4a “ wily 4 
tance travelled in the time ¢ is v,¢; but vy, is 9 Tage 
the mean height of the curve AB above ON, and fra hld 
the area of OABN is therefore v,, x ON = Of. 
The slope of the curve ABC at any point B is equal to the acceleration 
at the time ON, for if a point b be taken on the curve ABC near to B, 
and if the ordinate bn =v+6v, and the abscissa On =¢+ St, the slope of 
Bod is “f and in the limit when b coincides with B, the slope of Bd 
