96.] VELOCITY. 47 



p and we have of course the proportion : 30 miles an hour is to 

 I mile an hour as 44 ft. per second is to I ft. per second, etc. 



94. The full meaning of the equation of dimensions V = LT~ 1 

 is obtained if we substitute V/V^ for V, L/L l for I, T/T^ for'T, 

 where V, L, T are the concrete quantities and V lt L lt 7\ their 

 units. We find 



Z -L. !i 



V,~ L,' T^ 



and this equation shows two things which are of frequent appli- 

 cation in reductions between different systems of units : 



(a) The numerical value V/V^Q/l a velocity varies directly as 

 the unit of time and inversely as the unit of length ; 



(b) the unit of velocity V^ varies directly as the unit of 

 length and inversely as the unit of time.* 



95. In speaking of velocities, the time unit (usually the 

 second) is frequently understood without being mentioned. 

 This has led to considering velocity as a length (viz. the length 

 passed over in unit time) ; it can then be represented graphi- 

 cally by a segment of a straight line, and if in addition we, com- 

 bine with the idea of velocity that of the direction and sense of 

 the motion, its geometrical representative will be a vector (see 

 Art. 45). We shall see later that this view is of particular 

 advantage in studying the velocity of curvilinear motion. 



Some recent writers on mechanics use the term velocity 

 exclusively in this meaning, i.e. as denoting a vector, and apply 

 the term speed to denote the numerical magnitude of this 

 vector. In linear kinematics the direction is given, and the 

 "speed" alone is the subject of investigation. The + or 

 sign of the "speed " expresses the sense of the motion.f 



96. Exercises. 



(i) A train leaves the station A at 9 h. 5 m., passes (without stop- 



*See J. D. EVERETT, C. G. S. system of units, 1891, p. 3. 



f See Syllabus of elementary dynamics, Part I., prepared by the Association for 

 the Improvement of Geometrical Teaching; London, Macmillan, 1890, p. 8. 



