47 49] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 93 



49. Velocities. If a point change its position in space, its 

 motion may be described by giving the values of its coordinates 

 for every instant of time, by means of equations such as 



The functions / must be continuous, since the point cannot 

 jump from one position to another. 



We may describe the motion otherwise by giving two equations 

 F 1 (x, y, z) = 0, F 2 (x, y, z) 0, which denote the curve of intersection 

 of two surfaces along which the point moves. This curve is called 

 the path of the point. We must further give the distance s 

 measured along the curve, which the point has traversed, counting 

 from a fixed point on the curve. We must know s at all times t, 

 which is expressed by giving s as a continuous function of t, 

 s = <p (t). This, with the two equations of the path, gives as before 

 three equations to completely define the motion. 



The velocity of the point is defined as the limit of the ratio of 

 the distance As traversed in an interval of time A to the time 

 A when both decrease without limit, 



T . As ds 

 v = Lim = -=- . 

 A<=0 A2 dt 



A point travelling with a given numerical velocity may how- 

 ever be moving in any of an indefinite number of directions, 

 accordingly a velocity is completely specified only when we give its 

 direction and magnitude, or velocities are vector quantities. The 

 direction of the velocity is that of the tangent to its path. Its 

 direction cosines are accordingly 



dx n dy dz 



a = cTs' P^te' V = ds- 



Velocities are resolved and compounded like vectors in 

 genera], in particular the projections of v on the coordinate axes 

 are 



dx ds dx dx 



v x = va = v -j- = -j -j- = -j- , 

 ds dt ds dt 



dy ds dy dy 

 v v = vft = v -^ = -T: -r = TJ? > 

 ds dt ds dt 



dz ds dz dz 



V=V ry V - = .. - 



ds dt ds dt 



