EXAMPLES. 75 



they are all equally elastic and impinge on a vertical wall the vertices of their 

 subsequent paths also lie on an ellipse. 



57. A shot is fired with velocity \[(Zgk} from the top of a mountain 

 which is in the form of a hemisphere of radius r. Show that the furthest 

 points of the mountain which can be reached by the shot are at a distance 

 (measured in a straight line) r J(r 2 4rh} from the point of projection. 



58. A gun is placed on a fort situated on a hill side of inclination a to 

 the horizon. Show that the area commanded by it is 4?r h (h + d cos a) sec 3 a, 

 where /^(^gk) is the muzzle-velocity of the shot, and d the perpendicular 

 distance of the gun from the hill side. 



59. A gun is mounted at a given spot so as to command the horizontal 

 plane on which it stands. Its mounting is such that the direction, in which 

 it is pointed, must lie in a given plane inclined to the horizontal at an angle a. 

 Prove that the part of the plane commanded is an ellipse of eccentricity sin a, 

 the muzzle-velocity being constant. 



60. At a horizontal distance a from a gun there is a wall of height 

 h(&amp;gt;a ga 2 /v 2 ), and a shot is fired with velocity v in the vertical plane at 

 right angles to that of the wall. Prove that the distance commanded by the 

 gun on the other side of the wall is 



provided this expression is real. 



61. It is required to throw a ball from a given point with a given velocity 

 V so as to strike a vertical wall above a horizontal line on the wall. It is 

 found that when projected in the vertical plane at right angles to the wall the 

 elevation must lie between 6 l and $ 2 . Prove that the points on the wall 

 towards which the ball may be directly projected lie within a circle of radius 



62. Water issues from a fountain jet in such a manner that the velocity 

 of emission in a direction making an angle 6 with the vertical is *J(ga cosec 6} y 

 the jet being at a height h above the centre of a circular basin. Prove that, 

 if all the water is to fall into the basin, its radius must not be less than 



63. Prove that if the sole effect of a wind on the motion of a projectile is 

 to produce an acceleration / in a horizontal direction, the locus of points in a 

 horizontal plane which can just be reached with a given velocity v of projec 

 tion is an ellipse of eccentricity //V(/ 2 +g 2 } and area 7rv i ^(f 2 +g 2 )/g 3 . 



64. A particle is projected so as to enter in the direction of its length a 

 smooth straight tube of small bore fixed at an angle of 45 to the horizon, and 

 to pass out again at the other end of the tube. Show that the latera recta of 

 its paths before entering and after leaving the tube differ by */ 2 times the 

 length of the tube. 



