66 KINEMATICS. 



(4) Find the whole time of ascent in (3) and the height to which 

 the particle rises. 



(5) Show that owing to the resistance of the air a particle thrown 

 vertically upwards returns to the starting point with a velocity less than 

 the initial velocity of projection. 



(6) A particle begins moving with an initial velocity VQ in a medium 

 of constant density whose resistance is proportional to the velocity. 

 Express s and v in terms of /, and v in terms of s. 



(7) A body falls from rest in a medium whose resistance is propor- 

 tional to the velocity. Investigate its motion. 



4. ROTATION ; ANGULAR VELOCITY ; ANGULAR ACCELERATION. 



128. A motion of rotation about a fixed axis can be treated 

 in precisely the same way in which we have treated rectilinear 

 motion in the preceding sections. It is only to be remembered 

 that rotations are measured by angles (see Arts. 11-15), while 

 translations are measured by lengths. 



129. The rotation of a rigid body (see Art. 8) about a fixed 

 axis is said to be uniform if the circular arcs described by the 

 same point in equal times are equal throughout the whole 

 motion; in other words, if the angle of rotation is proportional 

 to the time in which it is described. In this case of uniform 

 rotation, the quotient obtained by dividing the angle of rotation, 

 6, by the corresponding time, /, is called the angular velocity. 

 Denoting it by w we have w = Q/t ; and the equation of motion is 



Thus, expressing the time in seconds and the angle in radians 

 (Art. 15), the angular velocity is equal to the number of radians 

 described per second. (Compare Arts. 88-90.) 



130. If the time of a whole revolution be denoted by T, we 

 have, from (i), 2ir=o)T' y hence 



' - """ (2} 



