16 APPLIED MECHANICS 
line is the magnitude of the velocity or speed, and an arrow-head placed 
on the line shows the sense of the velocity. The sense may also be 
given by placing letters, say, A and B, one at each end of the line, and 
stating that the velocity is AB for one sense, and BA for the opposite sense. 
- Innear velocity is measured in units of distance per unit of time, as, 
feet per second, feet per minute, or miles per hour. <A knot is a linear 
velocity of one nautical mile (6080 feet) per hour. 
If v denote the linear velocity or speed of a point or body, and s the 
space or distance through which it moves in time ¢, then s=vt. The 
unit of distance used in measuring v must be the same as that used in 
measuring s, and the unit of time used in measuring v must be the same 
as that used in measuring ¢. For example, if v is in feet per second, 
s must be in feet, and ¢ in seconds. 
Angular velocity is measured in radians * per second, revolutions per 
second, or revolutions per minute. The Greek letter w is generally used 
to denote angular velocity in radians per second. If a point moves ina 
circle of radius 7 feet with a linear velocity of v feet per second, and if the 
point makes 2 revolutions per second or N revolutions per minute, then 
v 2rN 
18. Acceleration.—When a velocity is not uniform, its rate of change 
is called acceleration. Acceleration is positive or negative according 
as the velocity is increasing or decreasing. Negative acceleration is 
frequently called retardation. Linear acceleration is rate of change of 
linear velocity, and is generally measured in feet per second per second. 
Angular acceleration is rate of change of angular velocity, and is generally 
measured in radians per second per second. The symbols f and a will 
be used to denote linear acceleration and angular acceleration respec- 
tively. The linear acceleration due to gravity is denoted by the symbol 
g. The value of g will be taken as 322 feet per second per second. 
Acceleration, like velocity, isa vector quantity, and may be completely 
represented by a straight line. 
19. Kinematical Equations.—Let v, or w, denote the velocity of a 
point or body at a given instant, and let v or » denote the velocity after the 
. lapse of ¢ seconds, the acceleration being uniform and denoted by f or a. 
Then v=0,+/t, and w=, +at. 
During the interval of ¢ seconds the mean velocity is 
2( +v)=v,++4f/t for linear velocity, and 
4(o, +0) =0, + + lat for angular velocity. 
If s is the linear distance moved, or @ the angle described in the 
interval of ¢ seconds, then 
=1(1, + vt=v,t+t/¥, and 0=}(o, + o)t=o,t + pal®. 
Eliminating ¢, it follows that 
r=vi +2fs, and w? =; + 2a0. 
If V,=0 ‘and i then 
v=ft, s=2 ft, and = 2fs, 
also, o=at, 0= Taf2, and w? = 2a0, 
* A radian is the angle subtended at the ceritre of a circle by an are of that 
circle equal in length to the radius. Hence the number of radians in an angle 
or the circular measure of an angle subtended at the centre of a circle of radius 
v by an arc of length a@ is equal to a/r. 
