334 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



that they will come to rest at the same time and that their distance apart 

 will then be (m~m ) /{&amp;lt;*?/ X(m + )}, where u and v are their initial 

 velocities. 



147. Find the charge of powder required with an elevation of 15 to 

 send a 32 Ib. shot over a range of 1600 yards, being given that the initial 

 velocity is 1600 feet per second when the charge is half the weight of 

 the shot. 



Prove that, if the gun is moveable on a smooth horizontal plane, and if 

 the weight of the gun is n times that of the shot, while the charge is that 

 just found, then the range is 



6400ft/(4?i + 2- v /3) yards. 



148. A gun is suspended freely at an inclination a to the horizontal by 

 two equal parallel cords in a vertical plane containing the gun, and a shot 

 whose mass is Ijn of that of the gun is fired from it. Prove that the range 

 on a horizontal plane through the muzzle is 4n(l + ri) /itana, where h is the 

 height through which the gun rises in the recoil. 



149. A regular polygon of n sides is placed in a vertical plane with its 

 two highest sides equally inclined to the vertical. A particle slides down 

 from the highest point and V K is its velocity on arriving at the *th angular 

 point from the top, there being no restitution in the impacts. Prove that 

 the particle will or will not pass through the (fc-fl)th angular point ac 

 cording as 



^i 2 *- (2 + l)w (2-l)ir 



or &amp;gt;\aq cosec cos 2 - -- sec - - . 



n 



150. A wedge of mass M and angle a rests on a smooth horizontal table, 

 and a particle of mass m, moving on the table in a vertical plane through 

 the centre of inertia of the wedge and a line of greatest slope on its inclined 

 face, comes to the edge of the wedge. Prove that, if there is no restitution 

 between the wedge and the particle, and if the wedge is high enough, the 

 particle will ascend through a vertical height 



hM 2 cos 2 al{(M+ m) (M+ m sin 2 a)}, 



where h is the height to which the velocity of the particle before reaching the 

 wedge is due. 



151. A wedge of angle a and mass M is free to move on a fixed hori 

 zontal plane. Another wedge of angle a and of mass M is laid upon it so 

 that its upper surface, on which there is a particle of mass m, is horizontal. 

 The surfaces are all smooth and the motion takes place in a vertical plane. 

 Prove that the pressure of the particle m on the plane with which it is in 

 contact is 



MM mgJ{MM + (M+ M } (m + M } tan 2 a}. 



152. Prove that, in the system of Example 151, the total weight exceeds 

 the pressure on the fixed horizontal plane by 



{(M+ M } (M + m} 2 g sin 2 a}/{( M+ M } (M + m) sin 2 a + MM cos 2 a}. 



