in order to receive fiull credi you must show all wark This quiz is open book an

Question

in order to receive fiull credi you must show all wark This quiz is open book and open nales Caleulators are permitted. Put all answers in the provided boxes. A rocket rises vertically, from rest, with an acceleration of 1.8 m/s2 until it runs out of fuel at an altitude of 810 m. After this point, its acceleration is that of gravity, downward. (a) What is the velocity of the rocket when it runs out of fuel? (b) How long does it take to reach this point? (c) What maximum altitude does the rocket reach? (a) How much time (total) does it take to reach maximum altitude? (e) With what velocity does it strike the Earth?

Explanation / Answer

For this multi-part problem, you will need to know the three basic acceleration equations and use the appropriate one depending on the data given:

d = v't + 0.5at^2

v" = v' + at

v"^2 - v'^2 = 2ad

where d = distance (meters)

t = time (seconds)

a = acceleration (meters/sec^2)

v' = original velocity (meters/sec)

v" = final velocity (meters/sec)

a) All you have is the acceleration (1.8), original velocity (0), and distance (810), so use the 3rd equation

v"^2 - v'^2 = 2ad --> v"^2 - 0 = 2(1.8)(810) --> v" = sqrt(2916) = 54 m/s

b) You now have the values to plug into the 2nd equation to solve for t

v" = v' + at --> t = (v" - v')/a = (54 - 0)/1.8 = 30s

c) Maximum altitude will be 810m + the distance d the rocket continues to "coast" upwards until gravitational accelearation (-9.8) brings it to a complete stop. Use the 3rd equation to figure this additional distance, but now v' = 54 and v" = 0

v"^2 - v'^2 = 2ad --> d = (v"^2 - v'^2)/2a = (0 - 54^2)/2(-9.8) = 148.78m Max altitude = 810 + 148.78 = 958.78m

d) Total time for maximum altitude will be the time from part b (30) + the time t to decelerate through the 148.78 m from part c. Using the 2nd equation, and acceleration of -9.8:

v" = v' + at --> t = (v" - v')/a = (0 - 54)/-9.8 = 5.51s

So the total climb time = 30 + 5.51 = 35.51s

e) Presuming no drag or terminal velocity, you now have the data to plug into the 3rd equation again. v' = 0, a = -9.8, and d = -958.78 (negative because you're falling downwards)

v"^2 - v'^2 = 2ad --> v"^2 = 2ad + v'^2 = 2(-9.8)(-958.78) + 0 --> v" = -137.08m/s (again, negative because velocity is downwards)

f) Total air time will equal time rising (30) + time falling t

v" = v' + at --> t = (v" - v')/a = (-137.08 - 0)/-9.8 = 14s

So total time = 30 + 14 = 44s

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