70 KINEMATICS. [137. 



III. Plane Kinematics. 



I. VELOCITY J COMPOSITION OF VELOCITIES ; RELATIVE 

 VELOCITY. 



137. The motion of a point in a curved path would not be 

 completely characterized by its velocity and acceleration as 

 defined in the preceding section ; the varying direction of the 

 motion, and the rate of change of direction, must be taken into 

 account. It is convenient to incorporate these ideas in the 

 definitions of velocity and acceleration. By this generalization 

 of their original meaning, velocity and acceleration become 

 vectors, i.e. magnitudes having both length and direction. 



138. The generalized idea of velocity as a vector may be 

 derived as follows : 



Consider a point P moving in a curve (Fig. 32). Let P be 



its position at the time t, P' 

 its position at the time /+ A/, 

 and let P Q P = s ) PP' = bs. 

 The space s described in any 

 time / may be regarded as 

 some function of the time /, 

 say *=/(/). 



The mean velocity 



Fi 32> for the time A/ during which 



the point passes from P to 



P'may be represented by a length PS laid off on the chord 

 PP' from P. As A/ diminishes, P' approaches P, and in the 

 limit when Aj/A/ becomes the derived function ds/dt=f'(t), 

 the chord merges into the tangent at P. This leads us to rep- 

 resent the velocity at the time t, or at the place P, by a length 

 PT proportional to ds/dt laid off on the tangent at P from this 

 point in the sense of the motion. The vector PT will then 

 completely represent the velocity at the time /. 



