Within a complex machine such as a robotic assembly line, suppose that one particular part glides along a straight track. A control system measures the average velocity of the part during each successive interval of time ∆t0 =t0 - 0, compares it with the value vc it should be, and switches a servo motor on and off to give the part a correcting pulse of acceleration. The pulse consists of a constant acceleration am applied for time interval ∆tm=tm-0 within the next control time interval ∆t0. As shown in Fig. P2.35, the part may be modeled as having zero acceleration when the motor is off (between tm and t0). A computer in the control system chooses the size of the acceleration so that the final velocity of the part will have the correct value vc . Assume the part is initially at rest and is to have instantaneous velocity vc at time t0.
(a) Find the required value of am in terms of vc and tm.
(b) Show that the displacement ∆x of the part during the time interval ∆t0 is given by
∆x=vc (t0 - 0.5tm). For specified values of vc and t0,
(c) What is the minimum displacement of the part?
(d) What is the maximum displacement of the part?
(e) Are both the minimum and maximum displacements physically attainable?