A test rocket is fired vertically upward from a well. A catapult gives it an ini

Question

A test rocket is fired vertically upward from a well. A catapult gives it an initial speed of 81.0 m/s at ground level. Its engines then fire and it accelerates upward at 3.90 m/s^2 until it reaches an altitude of 930 m. At that point its engines fail, and the rocket goes into free fall, with an acceleration of -9.80 m/s^2. (You will need to consider the motion while the engine is operating separate from the free-fall motion.)

a) How long is the rocket in motion above the ground? ______s
b)What is its maximum altitude? ______km
c) What is its velocity just before it collides with the Earth? _____m/s

thank you!

Explanation / Answer

Initial speed of the rocket u = 81 m/s Accelration a = 3.90 m/s^2 Height reached until its engine fails, S = 930 m Let t be the time taken to cover this distance. From Kinematic relation                              S = ut + (1/2)at^2                       930 m = (81 m/s)t + 0.5 (3.90 m/s^2)t^2                            930 = 81 t + 1.95 t^2 1.95 t^2 + 81 t - 930 = 0 On solving the above quadratic equation, we get                       t = 9.368 s Let v be the speed at  the time of the failure of the engine.                     v = u + at                        = 81 + (3.90)(9.368)                        = 81 + 36.535                        = 117.53 m/s (a) Time of flight T = t + (v/g) + Sqrt[2H/g]                         = 9.368 + (117.53/9.8) + Sqrt[2*1634.82/9.8]      [from part(b), H = 1634.82 m]                         = 9.368 + 11.992 + 18.26                         = 39.62 s (b) Maximum altitude H = S + (v^2/2g)                                 = 930 m + [(117.53)^2/(2*9.8)]                                 = 930 m + 704.82 m                                  = 1634.82 m                                  = 1.63 Km    (c) velocity just before it hits the ground v' = Sqrt[2gH]                                                            = Sqrt[2*9.8*1634.82]                                                            = 179 m/s          

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