212 



MOTION UNDER CONSTANT FORCES 



the envelope. Thus equation (64) is the equation of the envelope, 

 and is easily seen to give the same parabolic envelope as has already 

 been obtained. 



175. There is also a very simple geometrical way of deter- 

 mining the envelope of the system of parabolas. We notice first 

 that as the projectiles are all fired from the same point A with 

 the same velocity v, their paths must all have the same directrix 

 NM (fig. 123). 



Let any two parabolas of the system intersect in P, and let 



S, S f be the foci of these parabolas. Let AN y PM be the perpen- 



N M diculars from A and P to 



-\ the directrix. 



Then AS = AS', since 

 each is equal to AN, and 

 PS = PS', for each is 

 equal to PM. Thus S, S' 

 ^ are the two points of in- 

 tersection of two circles of 

 which the centers are A, P. 

 If the two parabolas are 



FIG. 123 



supposed to be adjacent, 



their foci S, S' are adjacent points, and therefore the two circles 

 touch, and ASP is, in the limit, a straight line. We now have 



AP = AS + SP 

 =AN+PM 



= the perpendicular from P on to a fixed horizontal 

 line at a distance AN above MN. 



The point P, then, satisfies the condition that its distance from 

 this fixed line is equal to its distance from the fixed point A. It 

 therefore is always on a certain parabola of focus A. But also it 

 always a point on the envelope, this being the locus of the point 

 of intersection of adjacent pairs of the parabolas of the systei 

 Thus the envelope is the parabola just obtained, of which the foci 

 is A, and this is the same parabola as was obtained before. 



