44-46] PARABOLIC MOTION. 45 



Thus, after au interval measured by V sin a/g, y vanishes, and 

 the particle has no velocity parallel to the axis y, it is therefore 

 moving parallel to the axis x. Previously to this it had a velocity 

 in the positive direction of the axis y, and after this it has a 

 velocity in the negative direction of the same axis. Its path 

 therefore has a vertex, which is reached after an interval 



Vsma/g, 1 say, 

 from the beginning of the motion. 



If we refer the motion to parallel axes of #', y' (y' being 

 positive in the opposite sense to y} through the vertex A, and 

 take t' to measure the time of moving from the vertex A to any 

 point P we shall have 



d?x dx' 



,-- /2 = 0, with t V cos a, and x = 0, at time t' 0, 



dV dv' 



and -=pj = g, with j,, = 0, and y' = 0, at time t' = 0. 



Hence x' = V cos at', y' = \g^. Eliminating ', we have 

 /a 2 F 2 cos 2 a , 



+-1, 



so that the path of the particle is a parabola with vertex at A. 



We might have deduced this result analytically from the equations #=0, 

 y -g. Integrating and determining the constants so that when =0, x=x^ 

 x Fcosa, and y=y > y ^sina, we find 



Eliminating t we have 



the equation of a parabola whose axis is parallel to the axis y, and whose 

 vertex is at the point 



F 2 sin a cos a F 2 sin 2 a 



The theorem of this Article was discovered by Galilei. 



46. Examples. [In these examples the axis y is supposed to be the 

 vertical at a place.] 



1. Write down the length of the latus rectum of the above parabola. 



2. Show that the height of the directrix above the starting point is F 2 /2<7. 



3. If v is the velocity at any point of the path show that the point is at a 

 distance v 2 /2# below the directrix. 





