CHAPTEE VIII 

 MOTION OF A PARTICLE UNDER CONSTANT FORCES 



154. The simplest case of motion of a single particle occurs 

 when the particle is acted upon only by constant forces and moves 

 in a straight line. 



If P is the component force in the direction of the motion of 

 the particle, there will, by the second law of motion, be an 



acceleration / given by 



P = mf, 



where m is the mass of the particle. Since the forces are, by 

 hypothesis, constant, the acceleration / is also constant. 



Let the particle start with a velocity u, and move with a con- 

 stant acceleration /. In time t the increase in velocity is ft, so 

 that, after any time t, the whole velocity is u + ft. Denoting this 



velocity by v, we have 



v = u+ft. (44) 



By definition, v is equal to > where s is the space described 



Cbv 



from the beginning of the motion. We accordingly have 



ds 



an equation giving the rate of increase of s at any instant. Integrat- 



ing, we obtain 



s = ut + \ft\ (45) 



no constant of integration being needed, because the distance 

 described at time t = has to be 0, from the definition of s, 

 By equation (44), u = v ft, so that equation (45) can be written 



. = vt - \ft\ (46) 



188 



