258 MOTION UNDER A VARIABLE FORCE 



In the case in which V 2 > 2 ga, we obtain after integration of equation (e), 



(9) 



where C' is a new constant of integration. If the time is to be measured from 

 the instant at which the particle is projected, we must have t = when r a, 

 so that 



and on eliminating (7', we can again obtain the time required to describe any 

 portion of the path. The case in which V 2 < 2 ga can be treated in a similar 

 manner. This is left as an example for the student. 



EXAMPLES 



1. In the foregoing illustrative example, suppose that V 2 <2gra, and find 

 (a) the greatest height reached ; 



(&) the time of flight of the particle. 



2. A meteorite falls on to the earth. Assuming it to start from infinity with 

 zero velocity, and to fall directly on to the earth, find the velocity with which 

 it reaches the earth's surface, and the time taken to fall to the earth's surface 

 from a point distant r from the earth's center. 



3. A particle falls from distance a into a center of force which attracts 

 according to the law /A/r 2 . Show that the average velocity on the first half of 

 the path is to the average velocity on the second half in the ratio 



7T - 2 = 7T + 2. 



4. Find the time of falling to a center of force which attracts according 

 to the law pr~*. 



5. A particle moves in a straight line from a distance a to a center of attrac- 



tion towards which the force is Show that the time required to reach the 

 center is 



6. A particle begins to move from a distance a towards _a fixed center which 

 repels according to the law /wr. If its initial velocity is V/^a, show that it will 

 continually approach the fixed center, but will never reach it. 



