122.] 



RECTILINEAR MOTION. 



6l 



(4) A particle is projected vertically upwards from the earth's surface 

 with an initial velocity V Q . How far will it rise ? 



(5) If, in (4), the initial velocity be v = \gft, how high and how 

 long will the particle rise ? How long will it take the particle to rise 

 land fall back to the earth's surface ? 



(6) A body is projected vertically upwards. Find the least initial 

 ^elocity that would prevent it from returning to the earth, taking 

 r = 32, R = 4000 miles. 



121. Acceleration directly proportional to the distance, i.e.j = /cs, 

 where A: is a constant and s is the distance of the moving point 

 from a fixed point in the line of motion. 



The equation of motion 



%-" .' : (I8) 



can be integrated by the method used in Art. 117. The result 

 of the second integration will again be different according to 

 the sign of K. We shall here study only a special case, reserv- 

 ing the general discussion of this law of acceleration for later 



(see Arts. 177 sq.). 



122. It is shown in the theory of attraction that the attrac- 

 tion of a spherical mass such as the earth on any point within 

 the mass produces an acceleration directed to the centre of the 

 sphere and proportional to the distance 



from this centre. Thus, if we imagine 

 a particle moving along a diameter of 

 the earth, say in a straight narrow tube 

 passing through the centre, we should 

 have a case of the motion represented 

 by equation (18). 



To determine the value of K for our 

 problem we notice that at the earth's 

 surface, that is, at the distance OP 1 = R 

 from the centre O (Fig. 31), the accel- 



Fig. 31. 



eration must be =g. If, therefore,/ denote the numericalvalue 



