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 15 r 2 , and the velocities in these directions are u lt u 2 ; prove 

 that the accelerations in the same directions are 



* 1 + I ^(V- Y^ 2 ), and u, + \ 



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

 are r 15 r 2 , r 3 , and the velocities in these directions are u^ u 2 , u 3 , prove that 

 the accelerations in these directions are 



i + u i (7 + 7 s ) ~ 7 ( cos 12 + 3 cos 18 ), 



\ 2 r 3/ r l 



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 l5 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 BC, v along CA, w along AB. Prove that 



BC CA AB =Q 



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 g cot a. 



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 2 ; show that 

 he must estimate his position within a length 



.4 



h 



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, 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 

 /? tanusin/3 on one side, and .ft 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 7tan/3, where 2/3 is the angle which AB makes with the vertical. 



