A catapult launches a test rocket vertically upward from a well, giving the rock

Question

A catapult launches a test rocket vertically upward from a well, giving the rocket an initial speed of 79.4 m/s at ground level. The engines then fire, and the rocket accelerates upward at 3.90 m/s^2 until it reaches an altitude of 1070 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 and the free-fall motion separately.) For what time interval is the rocket in motion above the ground? What is its maximum altitude? What is its velocity just before it hits the ground?

Explanation / Answer

initial velocity v1= 79.4 m/s

acceleration a= 3.90 m/s^2

height reached= 1070 m

velocity v2 at height 1070 m = sqrt( 79.4^2 + 2 x 3.9 x 1070) = 120.9 m/s

height reached after engine fail = 120.9^2/2x9.8=745.8 m

b.so maximum altitude = 1070 +745.8 = 1815.8 m

a. time of acceleration = v2-v1/a= (120.9 - 79.4)/3.9 = 10.64 s

time in air after engine fail to reach maximum height = 120.9/9.8= 12.33 s

time taken to reach ground= sqrt( 2 x 1815.8/9.8) = 19.25 s

so total time = 10.64 + 12.33 + 19.25 = 42.22 s

c. velocity just before hittig ground= 9.8 x 19.25 = 188.65 m/s

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