a Sphere in a Viscous Fluid. 525 



9. Accelerated Motion of the Falling Sphere. 



We may conveniently consider the motion of a sphere 

 falling in a viscous fluid to be divided into two stages. In 

 the first stage the sphere is moving with continually diminish- 

 ing acceleration ; in the second it is moving with constant 

 " terminal " velocity. Theoretically the second stage is 

 reached only after the lapse of an infinite time, during 

 which the limiting value is approached asymptotically; 

 practically this stage is reached after a very short time. 



Measurement of the distance between two images on a 

 plate during the accelerated motion gives the average 

 velocity of the sphere in moving from one position to the 

 other. This average velocity is the velocity at the middle 

 of the time between the two positions, and therefore, except 

 at the very beginning of the motion, is nearly the same as 

 the velocity at the point of space halfway between the two 

 positions. 



If a series of photographs of the same sphere at different 

 depths is obtained, we have the means of plotting a curve 

 showing the velocity attained after falling through any 

 height. Such a curve is drawn for the smallest ball, radius 

 •1590 centim., falling in a vessel 11'5 centim. long and 3 

 centim. wide, in fig. 3. The result would be more regular if 

 greater care had been taken in measuring the height of fall; 

 when these photographs were taken this was only done 

 roughly to serve as a check in determining whether the 

 terminal velocity had been attained. 



In the same diagram is shown, as far as limits of space 

 allow, a portion of the corresponding curve for the fall of 

 a body in vacuo. Comparison of these two curves shows how 

 effective is the resistance of the fluid in destroying the 

 acceleration of the sphere. The velocity of the sphere has 

 become practically constant after a fall of 20 centim. 



Fluid Resistance in Accelerated Motion. 



In the case of the smallest ball used, the determinations of 

 velocity at different depths were sufficiently numerous to 

 enable a curve to be drawn showing the relation between 

 the velocity and the fall. The slope of this curve at any 



dY 

 point gives the value of -j- at the corresponding depth, and 



dV 

 so the value of the acceleration V-^— may be found. 



The ratio of this acceleration to the acceleration of a body 

 Phil. Mag. S. 5. Vol. 50. No. 306. J\ T ov. 1900, 2 O 



