EXAMPLES 219 



27. A shell bursts while in the air, all the fragments receiving equal 

 velocities from the explosion. Show that the fragments at any instant lie 

 on a sphere, that the foci of the paths they describe also lie on a sphere, 

 and that the vertices lie on a spheroid. 



28. A particle slides down a rough inclined plane AB, starting at rest 

 from A and describing a parabola freely after leaving the plane at B. If 



F is the focus of the parabola described, show that the angle AFB = - -f e, 

 where c is the angle of friction. 



29. From a fort a buoy was observed at a depression i below the horizon ; 

 a gun was fired at it at an elevation a, but the shot was observed to strike 

 the water at a point whose depression was i'. Show that, in order to strike 

 the buoy, the gun must be fired at an elevation B, where 



cos 6 sin (0 -f f) _ cos 2 i sin i' 

 cos a sin (a + i') cos 2 i' sin i 



30. Show that the least energy which will project a particle over a wall 

 which is at distance a from the point of projection is 



1 + tan a 



\ mqa , 



1 - tan i a 



where a is the elevation of the top of the wall at the point of projection. 



31. A mill wheel of radius a revolves so that its rim has a velocity F, 

 and drops of water are thrown off from the rim of the wheel. Show that 

 the envelope of their paths is a parabola whose axis is vertical and whose 



focus is at a distance -j- vertically above the center of the wheel. 



