210 



MOTION UNDER CONSTANT FORCES 



Thus, to get the maximum range, we project in the direction which 

 bisects the angle between the inclined plane and the vertical. 



When projection takes place in this direction, the maximum 

 range R is given by putting sin (2 a ft) = 1 in the value for r 

 given by equation (60). Thus we have 



_ v 2 2 cos a sin (a ft) 

 ~~^~ cos 2 ft 

 _ v 2 sin (2 a - ft) - sin ft 

 g cos 2 ft 



v*l- sin ft 



= 

 g 



cos ft 





172. This equation enables us to find the greatest distance 

 which can be reached in any direction by a projectile fired with 



velocity v. Let us replace ft by 6, so that 6 is the angle 



which the direction makes with the vertical. Then the relation 

 between R and 6 is v 2 



r*r ( 63 ) 



Regarded as an equation in polar coordinates R, 0, this is clearly 

 the equation of a curve such that we can hit any point inside it 



with a projectile fired with 

 velocity v, but cannot reach 

 any point outside it. The 

 polar equation of a parabola 

 of latus rectum I, referred 

 ^ ** \ to its focus and axis, is 



known to be 



FIG. 121 1 + cos 



Comparing this with equation (63), we see that this equation 

 represents a parabola, of which the point of projection is the 

 focus, the axis is vertical, and the semi-latus rectum is v*/g. 



