29-32] DEFINITIONS. 33 



velocity of the moving point at time t is -r , where s is the length of the arc 



of the path measured, in the sense of description of the path, from some par- 

 ticular point of it to the position of the moving point at time t. The 

 magnitude of the velocity of a point is often called its speed, and, when it is 

 independent of the time, the point is said to move with uniform speed whether 

 its path is straight or curved. 



31. Measurement of velocity. The measure of any par- 

 ticular velocity is a number expressing the ratio of the velocity to 

 the unit of velocity. 



The unit velocity is that with which a point describes one 

 unit of length uniformly in each unit of time. 



The number expressing a velocity is the ratio of a number ex- 

 pressing a length to a number expressing an interval of time. 

 It therefore varies inversely as the unit of length and directly as 

 the unit of time. 



Velocity is accordingly said to be a quantity of one dimension 

 in length and of minus one dimension in time, or its dimension 

 symbol is LT~ l , where L stands for length, and T for time. 



32. Acceleration. A point moving with a variable velocity, 

 relative to any frame, is said to have an acceleration relative to 

 that frame. 



When the point is moving in such a way that its velocity 

 increases by equal amounts in equal intervals of time, however 

 short the intervals may be, it is said to have a uniform accelera- 

 tion, provided the velocity acquired in every interval has the same 

 direction and sense. 



Uniform acceleration is determined as regards magnitude, 

 direction, and sense by the velocity added in a unit of time. 



When the acceleration is not uniform, the moving point is 

 said to have a variable acceleration. 



The acceleration of a point moving in a straight line is the 

 rate of increase of its velocity per unit of time. This is a short 

 way of expressing the following definition : 



Let v be the velocity of the point at time t, and v its velocity 



i) f ~~ v 

 at time if, then its acceleration is the limit of the fraction , 



when the interval tf tis indefinitely diminished, or in words it is 

 L. 3 



