You are an engineer in charge of designing a new generation of elevators for a p

Question

You are an engineer in charge of designing a new generation of elevators for a prospective upgrade to the Empire State Building. Before the state legislature votes on funding for the project, they would like you to prepare a report on the benefits of upgrading the elevators. One of the numbers that they have requested is the time it will take the elevator to go from the ground floor to the 102nd floor observatory. They are unlikely to approve the project unless the new elevators make the trip much faster than the old elevators. If state law mandates that elevators cannot accelerate at greater than 3.60 m/s^2 or travel faster than 15.8 m/s, what is the minimum time in which an elevator can travel the 373 m from the ground floor to the observatory floor?

Explanation / Answer

lets say elevator starts from ground
calculate distance to reach 15.8 m/s speed
vf^2 = vi^2 + 2*a*d
15.8^2 = 0 + 2*3.60*d
d = 34.7 m

time taken to travel this:
vf = vi + a*t
15.8 = 0 + 3.60*t
t = 4.39 s

So,
distance travelled while accelerating and decelerating = 34.7 m + 34.7 m = 69.4 m
time while accelerating and decelerating = 4.39 s + 4.39 s = 8.8 s

rest of the distance will be covered by 15.8 m
time taken for remaining distance = remaining distance / speed
= (373 m - 69.4 m) / 15.8 m/s
= 19.2 s

Hence,
total time taken = 8.8 s + 19.2 s = 28 s

This is the minimum time since we have used maximum acceleration and maximum velocity in our equation
Answer: 28 s

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