.7 1- M AT II KM ATICAL PROBLEMS. 



1313. An imperfectly elastic particle is projected from a 

 piv.-u point with given velocity, so as, after one rebound at m 

 inclined plane passing through the point, to return to the point of 

 projection ; prove that the locus of the point where the particle 

 strikes the inclined plane is the ellipse 



or, y being horizontal and vertical through the given point, e the 

 elasticity, and h the space due to the given velocity. 



1314. A particle is projected from a given point so as just 

 to pass over a vertical wall whose height is 6, and distance from 

 the point of projection a ; prove that when the area of the para- 

 bolic path described, before meeting the horizontal plane through 



the point of projection, is greatest, the range is -^ , and the height 



.96 



of the vertex is - . 

 o 



1315. A heavy particle is projected from a point in a plane 

 whose inclination to the horizon is 30 with given velocity, in a 

 vertical plane perpendicular to the inclined plane ; prove that, if 

 all directions of projection in that vertical plane are equally pro- 

 bable, the chance of the range on the inclined plane being at least 

 one third of the greatest possible range is J. 



III. Motion on a Smooth Curve under tlie, action of Gravity. 



1316. A heavy particle is projected from the vertex of a 

 smooth parabolic arc, whose axis is vertical and vertex downwards, 

 with a velocity due to a height h, and after passing the extremity 

 of the arc proceeds to describe an equal parabola freely ; prove 

 that, if c be the vertical height of the extremity of the arc, the latus 

 rectum is 4 (h - 2c). 



