294 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



until it is reduced to F(w-m')/(m + m'), and the velocity of m' continues to 

 increase until it has the value 2mF/(ra+m'), these values are attained at the 

 same instant ; in the meantime the string contracts to its natural length a, 

 which it attains at the instant in question, and this happens at the end of 

 half a period from the beginning of the motion. The particles then move 

 with the velocities they have attained until m! overtakes m, when a collision 

 takes place. The subsequent motion depends on the coefficient of restitution 

 between the two particles ; if this is unity, the relative motion is reversed. 

 In any case the description of the subsequent motion involves nothing new. 



259. Examples. 



1. A shot of mass m is fired from a gun of mass M placed on a smooth 

 horizontal plane and elevated at an angle a. Prove that, if the muzzle velocity 

 of the shot is F, the range is 



2Fj> (1+Mt/lQtana 

 g 



2. A smooth wedge of mass M whose base angles are a and /3 is placed 

 on a smooth table, and two particles of masses m and m! move on the faces 

 being connected by an inextensible thread which passes over a smooth pulley 

 at the summit. Prove that the wedge moves with acceleration 



(m sin a ~ m' sin /3) (m cos a + m' cos /3) 

 ^ (m + m') (M+ m+m')- (m cos a + m' cos /3) 2 ' 



3. Two bodies of masses m 1} m 2 are connected by a spring of such 

 strength that when m l is held fixed m z makes n complete vibrations per 

 second. Prove that, if w 2 is held, m t will make n^/(m 2 /m 1 \ an( ^ that, if both 

 are free, they will make n^/^m^+m^/m^ vibrations per second, the vibrations 

 in all cases being in the line of the spring. 



4. Three equal particles are attached at equal intervals to an inextensible 

 thread, and when the thread is straight, the two end ones are projected with 

 equal velocities in the same sense at right angles to the thread. Prove that, 

 if there are no external forces, the velocity of each of the end particles at the 

 instant when they impinge is ^*J3 of their initial velocity. 



5. A particle is attached by an elastic thread of natural length a to a 

 point of a smooth plank which is free to slide on a horizontal table, and the 

 thread is stretched to a length a + c in a horizontal line passing over the 

 centre of inertia of the plank and the system is let go from rest. Prove that, 

 if the plank and particle have equal masses, and the modulus of elasticity of 

 the thread is equal to the weight of the particle, the velocity of the particle 

 relative to the plank when the thread has its natural length is that due to 

 falling through a height c 2 /a. 



6. A spherical shell of radius a and mass m contains a particle of the 

 same mass which is attached to the highest point by an elastic thread of 

 natural length a stretched to length a+c and also attached to the lowest 



