PROFESSOR TAIT ON IMPACT. 233 



The actual value, as given by the equations for the two parts (fi v fi 2 ) of the first 



rebound, is 



J(125-73 + 120-81) = 123-27 , 



the difference being less than 0'05 per cent. In this case the acceleration due to friction 



bears to that of gravity the ratio 



246 : 12327 ; 

 almost exactly 2 per cent. 



From the data (y lt y 2 ) for the second rebound we find the actual value of B to be 



i(131 -31 + 12146) = 126-33; 

 and the percentage of acceleration due to friction rather less than 4. As the whole rise 

 in this second rebound was considerably less than an inch, these results are highly satis- 

 factory. 



It is a fairer mode of proceeding, however, to calculate the value of N from that of 

 B , by means of the above relation. The values, thus calculated, are inserted in the 

 tables below, in the same column as the measured value of N, with the prefixed letters 

 )8, y, &c, to show from which rebound, the first, second, &c, they have been calculated. 

 These agree in a very satisfactory manner with the value of N given by the record of the 

 tuning-fork. 



From the facts, that the time of impact is nearly the same for all small distortions, 

 and that it diminishes rapidly as the distortion is greater, it follows that the equation of 



motion must be of the form 



Mx = - Cx - X 



during the first stage of the impact ; and of approximately the same form, but with the 

 square of the coefficient of restitution as a factor of the right, during the second stage. In 

 this equation x (which is confined to positive values) is measured from the datum line, so 

 that no term in g comes in explicitly. X is a function of x, which is small for small 

 values of x, but increases faster than does the first power of x for larger values. Hence, 

 for small relative speeds, the time of compression is 



7T /M 

 2V c 



and that of rebound 1/e times as much. The utmost distortion is 



V c + c 



V 



where V is the speed at the datum line. The first term is due to the fall; the 

 second, which is due to the weight of the block, does not appear in our Tables, as the 

 measures are made from the datum line. Its value, however, is usually only a small 

 fraction of that of the first term. 



To compare the distortion with the duration of impact in experiments made with the 

 same mass, falling from different heights, the following equation was tried : — 



2 Sx 2 

 x = — n z x — — , 

 2a 



VOL. XXXVI. PART I. (NO. 8). 2 N 



