I59-] 



ACCELERATION. 



8l 



both in magnitude and direction ; i.e. the velocity of the hodo- 

 graph is the acceleration of the original motion. 



It is easy to see how these considerations might be extended. 

 We might construct the hodograph of the hodograph ; its 

 velocity might be called^ the acceleration of the second order for 

 the motion of P ; and so on. 



It is sometimes convenient to draw the radii vectores of the 

 hodograph not parallel to the velocities of the point P, but so as 

 to make some constant angle (usually a right angle) with these 

 velocities. 



157. Exercises. 



(1) Discuss rectilinear motion as a special case of plane motion. 



(2) Show that the hodograph of rectilinear motion is a straight line. 



(3) Show that the velocity of a moving point is increasing, constant, 

 or diminishing, according to the value of the angle if/ between v and j 

 (Fig. 37)- 



158. Acceleration having been defined as a vector, the rules 

 for vector composition and resolution may be applied to accelera- 

 tion just as they were before applied to displacements and to 

 velocities. Thus, a point subjected to two or more simultaneous 

 accelerations will have a resulting acceleration found by geo- 

 metrically adding the component accelerations ; and conversely, 

 the acceleration of a point may be resolved in various ways. 



159.- Let the vector / which represents the acceleration of 

 the point P at the time /, make an angle -^ with the vector 

 representing the velocity v at the same time (see Fig. 37). 

 Resolving the vector j along the tangent and normal at P, we 

 obtain the tangential acceleration j t =jcos-^ and the normal 

 acceleration /=/ sin i/r. 



To find expressions for these components, let us regard PP ? 

 in Fig. 37 as the element ds of the path described by P ; then 

 the length of P'T', or of OV\ is v' = v + dv, and the angle 

 VOV 1 , being equal to the angle between two consecutive 



PART / 6 



