KINEMATICS. [242. 



or since v + v=v and // = rz> sinir = rz/ : 



242. Introducing polar co-ordinates in (71), the equation of 

 the orbit assumes the form 



or putting (7^ 2 p)//t 2 = C cos a, vjh = C sin , 



. (74) 



This equation might have been obtained directly by integrat- 

 ing (60), which in our case, with /(r)=/i/r 2 , reduces to 



the general integral of this differential equation is of the form 

 (74), C and a being the constants of integration. 



Equation (74) represents a conic section referred to the focus 

 as origin and a line making an angle with the focal axis as 

 polar axis. 



243. Exercises. 



(1) If 2 k be the chord intercepted by the osculating circle on the 

 radius vector drawn from the fixed centre, show that z?= k-f(r). 



(2) A point moves in a circle; if the acceleration be constant in 

 direction, what is its magnitude ? 



(3) A point moves in a circle ; if the acceleration be constantly 

 directed towards the centre, what is its magnitude ? 



(4) A point is subject to a central acceleration proportional to the: 

 distance from the centre and directed away from the centre ; find the 

 equation of the path. 



(5) A point P is subject to two accelerations, p?-OP directed 

 towards the fixed point O^ and p?-O 2 P directed away from the fixed 

 point <9 2 . Show that its path is parabolic. 



