34 



II. PARTICULAR MOTIONS OF A POINT. 



where # , y Q , # are the coordinates of the point at the time t = 0. 

 Eliminating t between the first two of equations 3), we obtain 



which shows that the motion is in a vertical plane. (The twisted 

 curves sometimes described by a base-ball, golf or tennis-ball or 

 rifled shot are the results of actions due to the air and the rotation 

 of the ball and not here contemplated.) If we choose this vertical 

 plane for the plane of XZ, we shall have y = 0, V y = 0, and the 

 equation of the path is found by eliminating t between the first and 

 third of equations 3) giving 



the equation of a parabola with axis vertical. If V z is positive, the 

 projectile will rise until ^- = 0, or 

 The height reached at this point is 



projectile will rise until - = 0, or - = 0, that is x # = 



6) ,^^-V.i 



2<jT 



It will be observed that this is independent of the horizontal component 

 of the velocity, V x , and is therefore the height that would be reached 

 by a projectile thrown vertically upward, or in other words 



7) 



is the velocity that would be attained 

 by a body falling from rest vertically 

 through the height k. 



If a be the angle of elevation 

 of the path at the start, V the 

 velocity of projection, we have, 



V = Fcos a 



Fig. 



and the range or horizontal distance 

 traversed by the projectile until it 

 has fallen to the original level is 



twice the value of (x X ) calculated for the highest point, or 



F 2 sin 2 a 



As we vary the elevation the range is accordingly greatest 

 when cc = 45. 



