quiz2-solutions

5 7 the reduced gravity aircraft rga lovingly known

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7. The reduced gravity aircraft (RGA) (lovingly known as the “vomit comet”) is a NASA aircraft used to train astronauts in a temporary weightless environment. Here is an altitude vs time graph of a typical trip of the RGA: Just previous to t = 20 s the plane is traveling in a straight line climbing at 45 . At t = 20 s the engines are turned off and the plane coasts until t = 45 s when the pitch is 45 down and at that instant the engines are turned on again. The weightlessness is experienced during the interval t = 20 s to t = 45 s. Although this is labeled “zero- g ” on the figure, the actual value of gravitational acceleration at the extreme altitude of 32,000 feet is g = 9 . 5 m/s 2 . The speed of the plane at t = 20 s is (neglect air resistance) (a) 380 mph 4 (b) 450 mph (c) 270 mph (d) 530 mph (e) 750 mph ANS: The motion between 20 s and 40 s, when the engines are off and air resistance is neglected, is that of a projectile ( i.e. , it’s parabolic motion). From t = 20 to t = 20 + (45 - 20) / 2 = 32 . 5 s the plane goes from the time the projectile trajectory starts until the point it reaches maximum height (this we see from the symmetry of the motion: at t=20 s and 45 s the angle of the trajectory with the horizontal are the same, so the height is the same at those times). Now v y ( t ) = - g ( t - t 0 )+ v 0 y , so with t 0 = 20 s, v (20) = v 0 and 6
v (32 . 5) = 0 , we have v 0 y = v y ( t ) + g ( t - t 0 ) = 0 + (9 . 5)(32 . 5 - 20) m/s = 119 m/s. Finally since v 0 y = v 0 sin(45 ) we get for the speed v 0 = v 0 y / sin(45 ) = 119 / 0 . 707 m/s = 168 m/s, or, going to mph, 168 m/s = 168 m/s · (3600 s / 1 hr) · (mile / 1609 m) = 375 . 8 miles/hr. Answer (a) is this, rounded up. Note that we could have used instead the condition that Δ y = 0 between 20 s and 45 s: 0 = - 1 2 g ( t - t 0 ) 2 + v 0 y ( t - t 0 ), giving the same result for v 0 y . 7
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