72 KINEMATICS. [142. 



have dx=ds cos a, dy=ds sin a. Divid| I by dt and comparing 

 with the preceding equations, we find 



Conversely, knowing the velocities of the moving point paral- 

 lel to the axes, we find its resulting velocity from the relation 



7i = J(^V#Y 

 ^\dti + \dt)- 



(2) 



142. If the equation of the path be given in polar co-ordinates, 

 it may be convenient to resolve the velocity v along the radius 

 vector OP and at right angles to it (Fig. 34). 



Fig. 34. 



Let r, 6 be the polar co-ordinates, a the angle between v and r', 

 then v r =v cos a,v e = v sin a. The element ds of the curve has in 

 the same directions the components dr=dscosa, rd0 = 

 Hence, dividing by dt, we find 



and 



v = 



143. In the case of relative motion we have to distinguish 

 between the absolute velocity v of a point, its relative velocity v l9 

 and the velocity of the body of reference v 2 . 



