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 79.4 m/s at ground level. Its engines then fire and it accelerates upward at 3.90 m/s2 until it reaches an altitude of 1020 m. At that point its engines fail, and the rocket goes into free fall, with an acceleration of -9.80 m/s2. (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?
(b) What is its maximum altitude?
(c) What is its velocity just before it collides with the Earth?

Explanation / Answer

v0 = 79.4 m/s, a = 3.90 m/s2, h = 1020 m.

velocity just when its engines fail = v = (v02 + 2ah) = 119.4 m/s

time = t1 = (v - v0)/a = 10.26 s

g = 9.80 m/s2. displacement = -h = vt2 - gt22/2

4.90t22 - 119.4t2 - 1020 = 0, t2 = 31.07 s

(a) How long is the rocket in motion above the ground?

t1 + t2 = 413 s

(b) What is its maximum altitude

h + v2/(2g) = 1747 m

(c) What is its velocity just before it collides with the Earth?

v' = -(v2 + 2gh) = -185 m/s

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