74 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



48. A particle is projected with velocity V at any elevation, a, greater 

 than the least positive value of cos" 1 ^; show that its path will cut two 

 planes through the point of projection at right angles ; that, if their inclina- 

 tions to the horizontal are /3 and y, then /3+y = a; and that the time of 

 passing from one to the other is 



sin (/3~y) V/ff. 



49. A heavy particle starts, with a velocity u at an inclination y to the 

 horizontal, from a point in a plane of inclination a, such that 2^2 tan a = ^/Stan-y. 

 Show that, for different positions of the vertical plane of motion, the greatest 

 projection of the range on a horizontal line perpendicular to the line of 

 greatest slope is 



50. Two inclined planes intersect in a horizontal line and are inclined to 

 the horizontal at angles a and 0. A particle is projected from a point in the 

 former, distant a from the intersection, so as to strike the latter at right 

 angles ; show that the velocity of projection is 



A/(2</a) sin j3/<J{sin a sin cos (a -t-0)} 



51. If the velocity v at any point of the path of a projectile under gravity 

 is suddenly diminished by one-half, prove that the focus of the new trajectory 

 is nearer to the projectile by the distance f v 2 /g, and that the curvature of the 

 path is quadrupled. 



52. Two heavy particles are projected from a point with equal velocities, 

 their directions of projection being in the same vertical plane; t, t' are 

 the times taken by the particles to reach the other point where their paths 

 intersect, and T, T' are the times taken to reach the highest points of the 

 paths : show that tT+t'T' is independent of the directions of projection. 



53. Three particles are projected from the same point in the same 

 vertical plane with velocities v lt v 2 , v 3 at elevations 1} 2 , 3 . Prove that 

 the foci of their paths lie in a straight line if 



sin 2 (0 2 -0s) sin 2(03 0,) sin2(0 1 -0 2 )_ 



V V V 



54. Three particles are projected from a given point in given directions. 

 Prove that after an interval of time t they form a triangle of area proportional 

 to t 2 . If the directions of projection of two of them are in the same vertical 

 plane, show that the plane of the triangle will pass through the point of pro- 

 jection after a time ^- SL where u, v are the initial velocities and 



g u cos a v cos ' 



a, the initial elevations of these two particles. 



55. A number of particles are projected simultaneously from a point, and 

 move under gravity ; prove that, if tangents are drawn to their paths from 

 any point in the vertical line through the point of projection, the points 

 of contact will be simultaneous positions of the particles. 



56. Particles are projected from the same point with equal velocities 

 under gravity ; prove that the vertices of their paths are on an ellipse. If 



