EXAMPLES. 73 



39. A heavy particle is projected from a point A so as to pass through 

 another point B ; show that the least velocity with which this is possible is 

 v/(2#Z) cos a, and that the highest point of the path is at a height I cos 4 a 

 above A, where AB=l and makes an angle a with the vertical. 



40. From a fort a buoy was observed at a depression i below the horizon, 

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

 water at a depression i . Show that to strike the buoy the elevation should 

 be 0, where 



cos 6 sin (6 + i) _ cos 2 / sin i 

 cos a sin (a -f i ) ~ cos 2 i sin i 



41. A particle is to be projected so as just to pass through three equal 

 rings, of diameter d, placed in parallel vertical planes at distances a apart, 

 with their highest points in a horizontal straight line at a height h above the 



point of projection. Prove that the elevation must be tan&quot; 1 - . 



42. A particle is projected from a point on a horizontal table so as to pass 

 through the four upper corners of a regular polygon of an even number of 

 sides which stands in a vertical plane with one side on the table. If R and r 

 are the radii of the circumscribed and inscribed circles of the polygon, prove 

 that the range on the plane is 2 V(^ 4 - 5R 2 r 2 + 8r*)/R and that the greatest 

 height of the particle above the polygon is %R 2 (R 2 - r 2 )/{r(2r 2 -R 2 )}. 



43. A man standing at a distance a from a net of height h wishes to 

 strike a ball over the net so that it may fall to the ground within a distance 

 b (&amp;lt; a) on the other side of the net. Prove that the square of the maximum 

 horizontal velocity which should be imparted to the ball increases in 

 harmonic progression as the height at which the ball is struck increases 

 in arithmetic progression, so long as the height does not exceed h(I+a/fy ; 

 and that for heights h and 2A these maximum horizontal velocities are in 

 the ratio J(a - 6) : Ja. 



44. A man travelling round a circle of radius a with speed v throws 

 a ball from his hand at a height h above the ground, with a relative velocity 

 T 7 , so that it alights at the centre of the circle. Show that the least possible 

 value of V is given by V 2 = v 2 + g y (a 2 + h 2 ) - h} . 



45. If A and B are two given points, and C a given point on the line 

 joining them, prove that, in the different trajectories possible under gravity 

 between A and B, the time of flight varies as *JCD t where D is the point 

 in which the trajectory meets the vertical through C. 



46. In any trajectory between two points A, B, the intercept on a vertical 

 line through a point C on AB between C and the trajectory is \gtfa, where ^ 

 is the time from A to the vertical through C, and Z 2 the time from that 

 vertical to B. 



47. A particle is projected with elevation a from a point on a plane of 

 inclination /3 in a vertical plane containing a line of greatest slope. Prove 

 that, if the elevation of the point of the path most distant from the inclined 

 plane is y, then tan a + tan = 2 tan y. 



