An aircraft we will call the \"receiver\" is approaching a secondaircraft in mid-air, which we will call the \"tanker\", forrefueling. The receiver is following the same path as that of thetanker. The tanker is moving at a constant velocity of 500 feet persecond, with its position at time t minutes given by s[t]=500t.

At time t = 0 seconds, the receiver is 2000 feet behind thetanker and has velocity 1000 feet per second. At this time, thereceiver begins to decelerate at a constant rate, so that itsposition at time t (along the path of the tanker) is given by:

r[t]=-2000+1000t+1/2 at^2

where the distance is measured in feet, and a is theacceleration (which is negative). This latter function is carefullycalculated by the crew so that in a short amount of time thereceiver will catch up to the tanker, and at that instant they willmatch speeds. The receiver will then link up with the tanker andbegin refueling.

Observing that both the position and the velocity of thereceiver will equal those of the tanker at the time refuelingbegins, set up a system of equations for a and t (where a is theacceleration of the receiver and t is the time when the twoaircraft meet). Solve your system to determine the following:

(a). How long will it take the receiver to catch up to thetanker?

(b). What is the deceleration (negative acceleration) of thereceiver?

(c). Draw motion curves (position, velocity, and accelerationcurves) for the receiver and the tanker and write a paragraphdescribing their motions in detail.

(d). Create a graph of the receiver's position and the tanker'sposition on the same axes. Also, define a function giving thedistance between the receiver and tanker, and graph that functionas well. Describe what these graphs illustrate.