PROJECTILES 



209 



To obtain the range on a horizontal plane, we have to find the 

 point in which the parabola intersects the line y = 0. Putting 

 y = in equation (58), we obtain at once 



2 v 2 cos 2 a v* sin 2 a 



x = tan a = > 



9 9 



agreeing with the value obtained in 168. 



Range on an Inclined Plane 



171. Suppose, next, that the projectile is fired so as to strike 

 an inclined plane through 0, the point of projection. Let ft be 

 the inclination of this plane to the 

 horizon, and let r be the range of the y 

 projectile on this plane. Then the co- 

 ordinate of the point at which the 

 projectile meets the plane must be 



x = r cos ft, y = rsinft. 



This point is a point on the parab- 

 ola, so that its coordinates must sat- 

 isfy equation (58). Substituting these 

 coordinates, we obtain 



n or 2 cos 2 ft 



r sin p = r tan a cos p - > 



2 v 2 cos 2 a 



giving as the value of the range r, 



_ 2v 2 cosasm(a ft) 

 g ' cos 2 /3 



Since 2 cos a sin (a ft) = sin (2 a ft) sin/3, (61) 



it is clear that if a alone is allowed to vary, the range r will be a 

 maximum when sin (2 a ft) is a maximum, i.e. when it is equal 

 to unity. To obtain this value, we make 



FIG. 120 



