208 MOTION UNDER CONSTANT FORCES 



v 2 

 v will be j and to obtain this range we make sin 2 a: = 1, or 



9 



a = 45 degrees. Thus, to send a projectile as far as possible on 

 a horizontal plane we project it at an angle of 45 degrees. 



170. These results can also be obtained analytically. Let us 

 take the point of projection for origin, and the plane in which 

 the flight takes place as plane of xy, the axes of 

 x and y being respectively horizontal and vertical. 

 The ^-coordinate of the point reached by the 

 particle after time t is equal to the horizontal 

 distance described in time t with uniform hori- 

 ~x zontal velocity v cos a. Thus 



FIG. 119 



x = vcosa-t. (56) 



Similarly the ^-coordinate of this point is the distance described 

 in time t, starting with initial velocity v sin a, and with retarda- 

 tion a. Thus . , , 2 /cr?x 

 y = v sin a; t \g&. (57) 



If we eliminate t between equations (56) and (57), we obtain 

 the equation of the path. It is found to be 



/~r, (58) 



2 v 2 cos 2 a 

 This can be expressed in the form 



1 v 2 sin 2 a g I v 2 sin a cos a\ 



9 



r 



which is clearly the equation of a parabola, of which the vertex is 



at the point . 



v z sin a cos a: 1 tr sin 2 a /Km 



(59) 



9 2 9 



and of which the latus rectum is of length 



2 v 2 cos 2 a 



