so that 



VELOCITY 



= Vdt : v z dt 



and hence 



Ar : Ap = 



Thus Ar represents the magnitude of the velocity V on the 

 same scale as that on which Ap represents the velocity v 2 . Also 

 since Ar is in the direction of AF, the resultant motion, we see 

 that Ar represents the velocity Fboth in magnitude and direction. 

 We have accordingly proved the following theorem : 



THEOREM. If two velocities are represented in magnitude and 

 direction l>y the two sides of a parallelogram which start from any 

 point A, then their resultant is represented in magnitude and direc- 

 tion on the same scale "by the diagonal of the parallelogram which 

 starts from A. 



This theorem is known as the parallelogram of velocities. We 

 may illustrate its meaning by two simple examples. 



1. Suppose that a carriage is moving on a level road with velocity F. 

 As a first frame of reference let us take the body of the carriage; as 

 a second frame take the road itself. The velocity of frame 1 relative to 

 frame 2 is then F. Relatively to frame 1 , the center of any wheel P is 

 fixed, so that any point 

 on the rim describes 

 a circle about P. Rela- 

 tively to frame 1 the 

 road is moving backward 

 with velocity F, so that 

 if there is to be no slip- 

 ping between the rim and 

 the road, the velocity of 

 any point on the rim, rel- 

 ative to the first frame 

 (the carriage), must be F. 

 Thus the velocity of any 

 point Q on the rim rela- 

 tive to frame 1 will Jbe a velocity F along the tangent QT. Representing 

 this by the line QT, the velocity of the carriage relative to the road is 

 represented by an equal line QH parallel to the road. Thus the resultant 

 velocity of the point Q is represented by the diagonal QS of the parallelo- 

 gram QHST. Clearly its direction bisects the angle HQT. Let L be the 



H" 



FIG. 4 



