I44-] VELOCITY. 73 



To fix the ideas, imagine a man walking on deck of a steam- 

 boat. His velocity of walking is his relative velocity z/j; the 

 velocity of the boat (say with respect to the water or shore 

 regarded as fixed), or more exactly speaking, the velocity of that 

 point of the boat at which the man happens to be at the time, 

 is the velocity z/ 2 of the body of reference ; and the velocity with 

 which the man is moving with respect to the water or shore, is 

 his absolute velocity. 



Representing these three velocities by means of their vectors, 

 we evidently find the absolute velocity v as the geometric sum of 

 the relative velocity Vj and the velocity v 2 of the body of reference, 

 just as in the case of displacements of translation (Art. 53). 

 And conversely, the relative velocity is found by geometrically 

 subtracting from the absolute velocity the velocity of the body of 

 reference. 



It is often convenient to state the last proposition in a some- 

 what different form. Imagine that we give the velocity z/ 2 . 

 both to the man and to the boat ; then the boat is brought to 

 rest, and the resulting velocity of the man is what was before 

 his relative velocity. Hence the relative velocity is found as the 

 resultant of the absolute velocity, and the velocity of the body of 

 reference reversed. 



144. Exercises. 



(1) A straight line in a plane turns with constant angular velocity o> 

 about one of its points O, while a point P, starting from O, moves along 

 the line with a constant velocity V Q . Determine the absolute path of 

 P and its absolute velocity v. 



(2) Show how to construct the tangent and normal to the spiral of 

 Archimedes r = aO, where 9 = o>/. 



(3) A wheel of radius a rolls on a straight track with constant velocity 

 (of its centre) z/ . Find the velocity v of a point /'on the rim. 



(4) Show that the tangent to the cycloid described by P, Ex. (3), 

 passes through the highest point of the wheel. 



(5) Show that the tangent to the ellipse bisects the angle between 

 the radii vectores r, r< drawn from any point P on the ellipse to the 

 foci S, 8. 



