EXAMPLES. 335 



153. A smooth bore gun and carriage, together of mass M tons, are 

 placed on a railway truck of mass M tons which runs on a smooth level 

 railway. A projectile of mass m tons is fired from the gun parallel to the 

 rails. Show that, if the gun is fixed to the carriage, if the powder gas exerts 

 a uniform thrust equal to the weight of Q tons on the shot and gun, lasting 

 till the shot has traversed the bore, a length I feet, and if the resistance to 

 sliding between the gun carriage and the truck is constant and equal to the 

 weight of R tons, then the velocity imparted to the shot is 



QJ[2Mgll{m(m+M] Q-m 2 R}] feet per second, 

 and the total length of recoil of the gun carriage on the truck is 



IQm (M (Q-R)- MR)I[R (M+ M } {(M+ m} Q - mR}] feet. 



154. In a truck of mass M is fixed a fine vertical tube inside which is 

 fastened a particle of mass m. The truck is made to slide on a smooth 

 horizontal plane by a massless horizontal chain which passes over a fixed 

 smooth pulley and supports a body of mass M . Motion ensues for time t 

 after which the particle is allowed to fall down the tube. Prove that the path 

 of the particle is a parabola of latus rectum 



2 M 2 (M+ M + m} gt 2 l{(M+ M + m) 2 + M 2 }l 



155. A railway carriage of mass M moving with velocity v impinges on a 

 carriage of mass M at rest. The force necessary to compress a buffer 

 through the full extent I is equal to the weight of a mass m. Assuming that 

 the compression is proportional to the force, prove that the buffers will not be 

 completely compressed if 



Prove also that if v exceeds this limit, and if the backing against which 

 the buffers are driven is inelastic, the ratio of the final velocities of the 

 carriages is 



156. Two particles of masses m and m , joined by an elastic thread of 

 natural length I and modulus X, are placed on a smooth table with m at the 

 edge and m at a distance I in a line perpendicular to the edge. The particle 

 m is then just pushed over the edge. Prove that, if the length of the thread 

 at any time is l-\-s, then 



s 2 = 2gs - \s 2 (m + m )/mm L 



Also, if at time t, m has fallen through z and m is at a distance x from 

 the edge, prove that 



m (l x)+ mz = \ mgt 2 . 



157. An elastic circular ring of radius c sin a is placed unstretched in a 

 horizontal plane over a smooth sphere of radius c. Prove that if it just 

 slips over the sphere the position of equilibrium is defined by the equation 



4 (sin 6 - sin a) 2 (1 + sin a) = tan 2 6 (1 - sin a) 3 . 



