28 I- KINEMATICS OF A POINT. LAWS OF MOTION. 



centimeter, written 1 cm 2 . In like manner the unit of volume is of 

 the dimensions [Z 3 ] and the unit is 1 cm 3 . The dimensions of 



velocity are pp or as we write for convenience, 

 Velocity = Length/ Time. 



Two quantities of different sorts do not have a ratio in the 

 ordinary arithmetical sense, but such equations as the above are of 

 great use in physics, and give rise to an extended meaning of the 

 terms ratio and product. 



The above equation is to be interpreted as follows. If any 

 velocity be specified in terms of units of length and time the 

 numerical factor is greater in proportion directly as the unit of 

 length is smaller, and as the unit of time is greater. For instance 

 we may write the equation expressing the fact that a velocity of 

 30 feet per second is the same as a velocity of 10 yards per second 

 or 1800 feet per minute. 



30 = 10-^ = 1800-5^. 



sec. sec. mm. 



We may operate on such equations precisely as if the units were 

 ordinary arithmetical quantities, for the ratio of two quantities of 

 the same kind is always a number. For instance 



30 yd. sec. 



10 ft. sec." 



The ratio T is the number 3, while - - = 1. Also 

 it. sec. 



1800 yd. min. Q CA 



10 " " ~ftT ITecT ~ 



Such an expression as '- is read feet per second. 



The unit of velocity is one centimeter -per -second, written, 



cm. 



- = cm. sec." 1 , 

 sec. 



Since acceleration is defined as a ratio of increment of velocity to 

 increment of time, we have 



, . -, [Velocity] [Length] r L 

 [Acceleration] = ^..^ = L * e2 j = [^ 



or the numeric of a certain acceleration varies inversely as the 

 magnitude of the unit of length, and directly as the square of the 

 unit of time. For instance, an acceleration in which a velocity of 

 10 feet per second is gained in 2 seconds is equal to one in which 

 a velocity of 9000 feet per minute is gained in a minute, 



10 ft. _ 10 ft. _ QQQQ ft. 



(2 sec.) 2 4 sec. 2 "" min. 2 



