A FAMILY OF PROJECTILE PARABOLAS. 



BY HAROLD C. BARKER, PH. D. 



In the following discussion of an interesting group of 

 mathematically and physically related curves, my aim is 

 simply to present some more or less well known facts in a 

 brief and connected manner. 

 Let us consider : — 



I. The equation of the family of parabolas that 

 are the paths, in a medium offering no resist- 

 ance, of projectiles with a given speed of pro- 

 jection in a given vertical plane. 

 II. Their common directrix. 

 III. The locus of their vertices. 

 IV. The locus of their foci. 

 V. The loci of their latus rectum extremities. 

 VI. The envelope of the family. 



I. The equation of the path of a projectile in an unresist- 

 ing medium can be found as follows : Taking horizontal and 

 vertical coordinate axes in the plane of motion, and the origin 

 at the point of projection, the coordinates of the projectile at 

 any time t after projection are given by 



X = V cosa t y = V sina t — 



2 



where V is the speed of projection, a the angle of projection 

 measured from the x-axis, and g represents the magnitude of 

 gravitational acceleration. 



Eliminating t between these equations, we have 



&^' f I ) 



y = X tana ;- y^-' 



2 V'' cos^a 



which is the equation of the path, recognizable as a parabola 

 with its axis vertical. 



