Problem 1C-02*: Velocity and acceleration vectors of a straight line motion

The motion of a particle in a flat plane is described as

                x(t)=At+B,
and                                               with A ≠ 0 and C ≠ 0                                             (1)
                y(t)=Ct+D.      

Use the concepts of velocity vector and acceleration vector to describe the motion.






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Solution:

Let the velocity vector be denoted in the rectangular coordinate representation as

                 v(t) = (vx(t), vy(t))

and the acceleration vector as

                 a(t) = (ax(t), ay(t)).

Since from Eqs.(1) we have


the velocity vector v = (A, C) is independent of time t, and has a slope of C/A with regard to the x-axis. From part (c) of Problem 1C-01 we know already that the trajectory as represented by Eqs.(1) is a straight line of slope C/A. Thus the velocity vector v is aligned with the trajectory. The speed of the particle equals the magnitude of the velocity vector, that is,
        
Thus the particle starts from the point (B, D) at t = 0, and moves along the trajectory with a constant speed |v| as given in Eq.(2)

For the acceleration vector a(t) = (ax(t), ay(t)), we have

This means that there is no acceleration for the motion so that the velocity of the particle does not vary; the direction and the magnitude of the velocity vector are both independent of time, just as we have found earlier.

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