58 



MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



The orbit in question 



is an ellipse if JPQ < SP i.e. if F 2 < J^ , 



2u 

 it is a parabola if \PQ = P i.e. if FS = ;j 



it is a hyperbola if \PQ > SP i.e. if F 2 > . 



56. Motion in a straight line with an acceleration to 



a point in the line varying 

 inversely as the square of 

 the distance. Let a point 

 N move in a straight line OA, 

 starting from A, so that, when 



Fig. 37. 



On OA as diameter de- 

 scribe a circle, and let C be 

 its centre, and a its radius; 

 draw NP at right angles to 

 OA, and consider the mo- 

 tion of the point P on the 

 circle. 



We shall show that, if P 

 describes the circle with an 

 acceleration towards 0, the 



point N will have the acceleration named. 

 By Example 4 of p. 53 we have 



Q 7 2 2 



acceleration of P= ---p 5 , where h is twice the rate at which OP 

 describes areas about 0. 



To resolve in direction AO multiply by ON /OP and observe 

 that ON:OP=OP:OA. Thus 



acceleration of N = 



ShWON ShWON 



P 6 (2a . ON) S a ON 2 ' 



Hence if we take the point N to start at a distance 2a from 

 and put h? = pa, then when ON = x, N will have an acceleration 



