218 MOTION UNDER CONSTANT FORCES 



19. A howitzer gun has a muzzle velocity of 400 feet per second. What 

 distance immediately behind the top of a hill 200 feet high on a level plane 

 is safe, if the gun is distant 1000 yards from the point in the plane verti- 

 cally below the top of the hill ? 



20. The sights of a gun are inaccurately marked, the gun carrying 

 always 3 per cent farther than the distance indicated on the sights. A 

 marksman, not aware of the error, aims at a target distant 1000 yards. If 

 the velocity of the shot is 1200 feet per second, show that he will hit the 

 target about one yard too high. 



21. The sighting of a rifle is accurate, and to aim at a point on a ver- 

 tical target distant a feet the rifle ought to be elevated to an angle a. 

 Owing to the unsteadiness of the marksman's hand, the rifle is pointed in 

 directions which lie anywhere within a small angle 6 of the true direction. 

 Show that if a succession of shots is fired, the points at which they hit 

 the target will all lie within a small ellipse of semi-axes ad cos a and 

 a0(l tan 2 a) respectively. 



When a = ) the minor axis of this ellipse vanishes, so that the shots 

 ought all to lie in a straight line. Interpret this result. 



22. A particle slides down the outer surface of a smooth sphere, start- 

 ing from rest at the highest point. It leaves the sphere at a point P and 

 describes a parabola in space. Prove that the circle of curvature of the 

 parabola at P will touch the directrix. 



23. A particle is projected horizontally from the lowest point of the 

 interior of the surface of a smooth sphere. It leaves the surface of the 

 sphere at P, and after describing a parabola strikes the sphere again at Q. 

 Show that PQ and the tangent at P make equal angles with the vertical. 



24. Show that the whole area commanded by a gun planted on a hill- 

 side, supposed plane, is an ellipse, whose focus is at the gun, eccentricity 

 the sine of the inclination of the hill, and semi-latus rectum equal to twice 

 the height to which the muzzle velocity of the shot is due. 



25. Show that the area of a plane hillside of inclination a, which is 

 commanded by a gun placed on a fort of height H above the hill, is 



4 irh (h + H cos 2 a) sec 3 a, 

 where V2 gh is the muzzle velocity of the shot. 



26. A spherical shell of mass m explodes when moving with negligible 

 velocity at a height of h feet above the ground. The shell is divided into 

 very small particles, each of which moves, after the explosion, away from 

 the center of the shell with velocity v t and ultimately falls to the ground. 

 Find the total mass of the fragments which will be found per unit area a1 

 any specified distance from the point vertically underneath the shell. 



