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



65. Prove that if two heavy particles projected in the same vertical 

 plane at the same instant from two given points with the same velocity meet, 

 the sum of the inclinations of the directions of projection must be constant, 

 and that for a constant velocity of projection and different directions of 

 projection the locus of the point of meeting is a parabola. 



66. A man standing on the edge of a cliff throws a stone with given 

 velocity u at a given inclination to the horizon, in a plane perpendicular to 

 the edge of the cliff; after an interval r he throws another stone from the 

 same spot with given velocity v at an angle TT + # with the line of discharge 

 of the first stone and in the same plane. Find r so that the stones may strike 

 each other, and show that the maximum value of r for different values of 6 is 



) and occurs when sin 6=vlu, w being the vertical component of v. 



67. Two particles describe the same ellipse in the same time as a central 

 orbit about the centre. Prove that the point of intersection of their directions 

 of motion describes a concentric ellipse as a central orbit about the centre. 



68. Two particles are projected in parallel directions from two points in 

 a straight line passing through a point 0, with velocities proportional to their 

 distances from 0, and each particle has an acceleration to equal to 

 JJL (distance). Prove that all the tangents to the path of the inner cut off, 

 from that of the outer, arcs described in equal times. 



69. Two particles describe concentric and coaxial ellipses about the 

 common centre with accelerations which are equal at equal distances, the 

 sum of the axes of one ellipse being equal to the difference of the axes of the 

 other ; and the particles start in opposite directions from corresponding 

 extremities of the transverse axes. Prove that the line joining them is of 

 constant length, and turns with uniform angular velocity. 



70. From all points on the circumference of a circle, to the centre of 

 which tends a force varying as the distance, particles are projected towards a 

 point on the circumference with velocities varying as their distances from the 

 point. Prove that at any instant the particles lie on a circle. 



71. Particles are projected from points on a sphere of radius a with 

 velocity *J(gb) and move with an acceleration to the centre equal to gr/a 

 at distance r. Prove that the part of the surface on which they fall is 

 the smaller of the two segments into which the sphere is divided by a small 

 circle of radius b. 



72. A body is describing an ellipse of eccentricity ^ under a force to the 

 centre, and when it is at one end of the latus rectum the centre of force 

 is suddenly transferred to the foot of the corresponding directrix. Prove that 

 the times which elapse in the two possible cases before the body reaches the 

 major axis are to one another as 2 : 1. 



