x = x0 + vet + iatz, v = 170 + at, (172) = (170)2 + Zan , Ax = ( )At, Atrapezoid = (b12b2)hr Atriangle = bh v0+v Introduction to the Problem On December 8, 2005 Southwest Airlines flight 1248, a Boeing 737-700 class jet with 103 persons aboard, attempted to land on a snow-covered runway at Chicago's Midway Airport. Tragically, the attempt resulted in the death of a child on the ground, as the plane slid far enough beyond the runway to leave airport property and collide with a car (in which the child was a passenger) on the road beyond. Both the analysis and prevention of accidents such as this rely on the kinematics of one-dimensional motion. In this problem we will examine how the National Transportation Safety Board (NTSB) arrived at the conclusion that "the probable cause of a fatal runway overrun...was the pilots' failure to use available reverse thrust in a timely manner to safely slow or stop the airplane after landing." (NTSB press release 53-07-48). The NTSB report does not give the full range of data available to the official accident investigators, but it does yield enough details to make a basic check of their findings. According to the information released to the public, the plane touched down with 4576 feet of regular runway remaining and an 82 foot runway safety overrun area beyond. The plane was moving at a ground speed of 131 knots when it touched down. Brakes were employed essentially immediately, but the pilots failed to completely reverse the engine thrust until 18 seconds after touchdown (for comparison, four of the five air carrier airplanes landing in the 25 minutes prior to the accident reversed engine thrust within 4 seconds of touchdown, and all did so within 6 seconds). At approximately 29 seconds after touchdown the plane left the back of the runway safety area moving at a ground speed of about 53 knots. Problem 1: 6) bl dl In this part of the problem, we will find the two accelerations of the plane. The initial velocity of the plane was v0 = 67.4 m/sec. From t = 0 sec to t = 18 sec, the acceleration of the plane was E"; = a1. From L\" = 18 sec to t = 29 sec, the acceleration of the plane was ii\"; = a2. The velocity of the plane at t = 29 sec is 27.3 m/sec. From t = 0 sec to t = 29 sec, the total distance travelled by the plane was 1419.8 m. i. Sketch a velocity vs. time graph for the motion. ii. What is the velocity of the plane at i: = 18 sec? Answer: 45.7 m/sec iii. What is the acceleration :11? Answer: 1.2 m/sec2 iv. What is the acceleration (1;? Answer: 1.7 m/sec2 In this part of the problem, we will find the total distance the plane would have needed to come to rest. The initial velocity of the plane was 120 = 67.4 m/sec. From L\" = 0 sec to t = 18 sec, the acceleration of the plane was ii"; = a1 = 1.2 m/secz. From t = 18 sec until the plane comes to rest, the acceleration of the plane was a; = a2 = 1.7 m/secz. What is the total distance traveled by the airplane, until it comes to rest? Answer: 1635.8 m. Note that this answer is larger than 1419.8 m. In this part of the problem, we will find the total distance the plane would have needed to come to rest, if the reverse thrust was activated immediately on touchdown. The initial velocity of the plane was 120 = 67.4 m/sec. From t = 0 sec until the plane comes to rest, the acceleration of the plane was E; = a2 = l.7 m/secz. What is the total distance traveled by the airplane, until it comes to rest? Answer: 1336.1 m. Note that this answer is smaller than 1419.8 m. In this part of the problem, we will find the latest possible time the reverse thrust could have been activated so that the airplane would have safely come to rest on the runway, without going into the safety runway. The initial velocity of the plane was 170 = 67.4 111/596. From L\" = 0 sec to t = t1, the acceleration of the plane was Eff = a1 = 1.2 11/1/3962. From t = :1 until the plane comes to rest, the acceleration of the plane was a}; = a2 = 1.7 m/secz. The total distance traveled by the airplane is 1394.8 m. Let's define the time that the acceleration a; is present as t2. Thus, the total time the plane needs to come to rest is t = :1 + :2. i. What is the velocity of the plane at t = t1? Answer: 63.75 m/sec ii. What is the time t1? Answer: 3.04 sec. Notice that this time is less than 18 sec. iii. What is the time t2? Answer: 37.5 sec. Notice that t = 1 + 2 = 40.04 sec. This is slightly larger than the time the plane needs to come to rest in part c, where the reverse thrust is activated immediately on touchdown (t = 2\" = 39.65 sec). 2