THE MOTION OF A SPHERE 191 



Putting v = -j- and integrating again with condition y = o when t = o we get 



y = V\t- l -(l-e-)\ -.. . . . . (6) 



2. Horizontal motion with initial velocity H. 



If u is horizontal velocity at time t, the equation of motion is now simply 



du 



dt=- cu ' 

 or 



du 



U -y- = - CM. 



ax 

 Therefore 



d = - cote, 

 and hence 



H-M = CX . (7) 



But a; = X for u = o, therefore H = cX. 



From the last expression and (3) we obtain 



H=f (8) 



Proceeding with the integration, from (7) we have 



dx 



~dt =u 

 = H-cx 



= e(X-z). 



Integration with initial condition x = o when t = o leads to 



*=X(l-e-<*) (9) 



3. The equation of the path of a sphere projected horizontally under gravity 

 is obtained at once by the elimination of t from the two equations (6) and (9) ; 

 and replacing c by its value ^ we have finally 



"7 1 X'-xH! (i) 



