Motion down inclined Planes. 207 



DE : DF : : GB : GJl 



that is, the length of the inclined plane is to its height, as the velocity 

 which gravity communicates to a free body, is to that with which it 

 urges the body along the inclined plane. 



329. Now as gravity gives to a free body, in a second of time, 

 a velocity by which a space of 32,2 feet are described uniformly 



in a second, it will be easy to determine the velocity acquired 273f ' 

 by a body in the first second of its descent down an inclined 

 plane. If, for example, the length of the plane is double the 

 height, the velocity acquired along the plane during the first 

 second, will be half of 32,2 feet ; that is, at the end of the first 

 second, if gravity ceased to act, the body would pass over 16,1 

 feet in a second. 



Having thus determined the velocity for the first second, we 

 shall have the velocity after any proposed number of seconds, 

 by multiplying this by the number of seconds ; also the space 

 is found by multiplying this first velocity by half the square of 26 ~- 

 the number of seconds. In short, it would be easy to determine 

 all the other circumstances of the motion in question, by articles 

 267, &c. We hence deduce the following propositions. 



330. If two heavy bodies, setting out at the same time from 



the point D, descend, one along the plane DE, and the other in p . 15g 

 the direction of the perpendicular DF, and we would know in 

 what part of the plane DE the first would be, when the second 

 had arrived at any given point A, we have only to let fall from 

 the point A upon DE the perpendicular AB ; and the point B 

 will be the place sought. Indeed if we represent by g the velo- 

 city that gravity communicates to a free body in one second, by 

 calling t the time employed in describing DA, we shall have 



L 



on the other hand the velocity acquired in a second by the body 

 that descends along the plane DE, is - ; accordingly by 



calling /' the time employed in descending from D to B, we shall 

 have 



DB - g * DF X ' t' 2 - 



UK _ ^ X t , 



