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



30. The radii vectores from two fixed points distant c apart to the position 

 of a particle are r lt ?* 2 , and the velocities in these directions are it l5 u 2 ', prove 

 that the accelerations in the same directions are 



31. The radii vectores from three fixed points to the position of a particle 

 are r lf r 2 , 7* 3 , and the velocities in these directions are u^ u 2 , u 3 ; prove that 

 the accelerations in these directions are 



? ( W 2 cos 12 + u a cos 13 ), 



7 1 



and the two similar expressions, in which 23 , 31 , 12 are the angles contained 

 by the directions of (r 2 , r 3 ), (r 3 , r x ) and (r^ ?' 2 ). 



32. Three tangents to the path of a particle whose acceleration is constant 

 and always in the same direction form a triangle ABC; the velocities are u 

 along BCj v along CA, w along AB. Prove that 



*0 CA AE^ 



U V W 



33. Prove that the angular velocity of a projectile about the focus of its 

 path varies inversely as its distance from the focus. 



34. Prove that when a shot is projected from a gun at any angle of 

 elevation, the shot as seen from the point of projection will appear to descend 

 past a vertical target with uniform velocity. 



35. A particle is projected from a platform with velocity V and elevation 

 /3. On the platform is a telescope fixed at elevation a. The platform moves 

 horizontally in the plane of the particle's motion, so as to keep the particle 

 always in the centre of the field of view of the telescope. Show that the 

 original velocity of the telescope must be Fsin (a /3) cosec a, and its accelera- 

 tion ffcota. 



36. A cricketer in the long field has to judge a catch which he can secure 

 with equal ease at any height from the ground between k and k z ; show that 

 he must estimate his position within a length 



where R is the range on the horizontal and h the greatest height the ball 

 attains. 



37. If a is the requisite elevation of a cannon for a mark on a target at a 

 horizontal range R y and if the axis of the trunnions of the cannon is inclined 

 to the horizontal at an angle /3, the shot will strike the target at a distance 

 R tan a sin on one side, and R tan a (1 - cos /3) below the mark aimed at. 



38. A heavy particle is projected from a point A with the least velocity 

 of projection V so as to pass through a point B ; show that the velocity at B 

 is Ftan/3, where 2/3 is the angle which AB makes with the vertical. 



