72 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



rest, which afterwards proceeds to execute an observed swing, 

 measures the time -integral of an impulsive force. 1 ) 



In order to find the work done by a given impulse, let us make 

 use of the equation of work and energy, 22), which says that the 

 work done is equal to the increase of the kinetic energy. The latter 

 may be written, bearing in mind the definition of momentum, 



T = I m (vl + vl + O = ~ (M x v x + M y v y + M,vJ. 



Suppose now the particle set in motion by an impulsive force, from 

 rest. The kinetic energy acquired, and accordingly the work done, 

 is then one -half the geometric product of the impulse and the 

 velocity generated, or in other words, the geometric product of the 

 impulse and the average value of the velocity at the beginning and 

 the end of the impulsive action. This may be otherwise shown, 

 whether the particle start from rest or not, by the following 

 considerations. 2 ) Since the interval of time and the distance moved 

 are infinitely small, we may consider the motion as rectilinear. 

 Suppose the initial velocity to be v Q9 and the final value i\, and 

 let s be a parameter which during the interval runs rapidly through 

 all values from to 1, so that at any part of the interval 



But as the momentum always increases at a rate proportional to the 

 increase of velocity, we have also 



M = M, + s(M l - M.] = M + tl, 



1) Suppose that a body which swings according to the law of the pen- 

 dulum, or equation 8), 19, receives, when in its position of equilibrium, an 

 impulse I. It swings out according to the equation 



dx 



x = asmnt, -j- = ancosnt 



Ct v 



during a time t = ?t/%n to a maximum excursion a, at which its velocity 

 vanishes, and it turns back. If its mass is m, the momentum communicated 

 to it while at rest was 



T / dx\ 



I = l m \ = man 

 \ dt/t=o 



so that if we know m , a , and n = 2 jt/period , we can measure the impulse of 

 the impulsive force. This is the mode of use of the ballistic galvanometer and 

 electrometer, as well as of the ballistic pendulum formerly used in gunnery. 

 The same formula applies (see Chapter X) , to the heeling of a ship when a shot 

 is fired from a cannon. 



2) Thomson and Tait, 308. 



