CHAPTER IV. 



DESCRIPTION OF MOTION IN TERMS OF ACCELERATION. 



41. In this chapter we propose to discuss with some detail a 

 number of important particular cases in which the motion of a 

 point is deduced from the value of its acceleration relative to a 

 frame : in all these cases the path is a straight line or a plane 

 curve. 



The moving point will be considered as defining the position 

 from time to time of a very small part of a body, and will fre- 

 quently be described as a particle. 



42. Rectilinear motion with uniform acceleration. 



Let the point move in a straight line, say the axis of x with 

 uniform acceleration f in the positive direction of the axis ; let 

 # be the coordinate of its position at the instant from which t is 

 measured, and let u be the velocity of the moving point in the 

 positive direction of the axis when t = 0. 



Then we are given x =/, 



with the conditions X = X Q when t 0, and x = u when t = 0. 



Writing v for x, so that v is the velocity at time t, we are given 



*=/. 



with the condition v = u when t = 0. 



Now one function of t having the constant /for its differential 

 coefficient is the function ft, and the most general expression for 

 a function having this differential coefficient is ft + C, where C is 

 an arbitrary constant. Hence v must be of the form ft + C. 



Putting = 0, we find u = C, so that the constant is determined. 



Hence v = u+ft, or x = u+ft. 



