62 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



3. Apply the results of Example 2 to obtain a formal proof of the 

 statement that, when the acceleration of a moving point is always directed 

 towards or from a point, fixed relatively to a frame, the path of the point, 

 relative to that frame, is a plane curve. 



4. The motion of each of two points relative to a certain frame is 

 uniform rectilinear motion, and the straight paths intersect. Prove that the 

 acceleration with which the distance between the points increases is inversely 

 proportional to the cube of that distance, and find the path of either point 

 relative to the other. 



5. Relatively to a certain frame a point describes a straight line 

 uniformly with velocity F, and a second point P describes a curve in such 

 a way that the line OP describes areas uniformly ; prove that the resolved 

 part perpendicular to OP of the acceleration of P is 2 Vv sin (f>/OP, where v is 

 the velocity of P, and < the angle the tangent to its path makes with OP. 



6. Relatively to a certain frame, a point A describes a circle (centre 0} 

 uniformly, and a point B moves with an acceleration always directed to A. 

 If the area covered by the line AB is described uniformly, prove that the 

 resolved part parallel to OA of the velocity of B is proportional to the 

 perpendicular from B on OA produced. 



60. Differential equation of central orbit. The equations 

 that hold for a point describing a central orbit about the origin 

 with acceleration /to wards the origin are 



r-r6*=-f, 



r n -0 = h, 

 where h is a constant determined by the initial conditions. 



By using the second of these equations we can change the 

 independent variable in the first from t to 6. We write u for r" 1 , 

 then we have 



so that the first equation becomes 



-* f> 



-, . -dr du , . . 

 and, since u 2 ^ = - -^ , and h is constant, this is 



d?u _f 



This is the differential equation of the path. The integral of 

 the equation will be a relation between u and containing two 

 arbitrary constants, and this is the polar equation of the path. 



