257, 258] 293 



PRINCIPLES OF ENERGY AND MOMENTUM. 



258. We have remarked that there are numerous cases in 

 which the principles of energy and momentum supply all the first 

 integrals of the equations of motion of a system, and thus suffice 

 to determine the velocities of the parts of the system in any 

 position. Applications of this method have already been made in 

 parabolic motion, elliptic motion, the problem of two bodies, and 

 other cases. 



To illustrate further take the following problem : 



Two particles A and B placed on a smooth horizontal table are connected 

 by an elastic string of negligible mass. When the string is straight, and of 

 its natural length, one of the particles is struck by a blow in the line of the 

 string and away from the other particle,; determine the subsequent motion. 



Let m be the mass of the particle struck, m that of the other, V the 

 velocity with which m begins to move. There is no tension in the string 

 until it is extended, and thus at first m has no velocity. 



The centre of inertia moves on the table with uniform velocity u, 

 =mV/(m + m \ in the line of the string. Let x be the increase in the 

 length of the string at time t, then the velocities of the particles are 



m x mx 



u + - , , u --- , . 

 m+m m+m 



Hence the kinetic energy is - (m+m )u 2 +~ 



t 



2 m+m 



the potential energy is - - # 2 so long as x is positive, a being the natural 

 length of the string, and X the modulus of elasticity. 

 Thus the energy equation is 



1 m 2 T79 1 mm . 1 X 1 

 ~ x = 



o / 



2 m+m 2 m + m 2 a 2 



showing that the motion in x is simple harmonic motion of period 

 2n- &amp;lt;J{mm a/(m + m ) X}, 



so long as x remains positive. Whenever the string is unstretched we have 

 x= + V. When x vanishes the string has its greatest length 

 a + VJ{mm a/(m + m ) X}. 



We can thus describe the whole motion : m moves off with velocity V 

 which gradually diminishes, and m moves in the same direction from rest 

 with gradually increasing velocity ; the string begins to extend, and continues 

 to do so until it attains its greatest length; this happens at the end of a 

 quarter of the period of the simple harmonic motion, and at this instant the 

 particles have equal velocities u. The velocity of m continues to diminish 



