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 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 distance than 4.50 m/s2 or travel
faster than 11.3 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

About 18 seconds.

t_1=time required to reach maximum allowable velocity
t_1=v/a where v is the max allowable velocity and a is the max allowable acceleration
t_1=5.39 s

y_1=distance traveled during acceleration
y_1=y + v*t_1 + (1/2)*a*(t_1)^2 where y is initial position, v is initial velocity, a is accel and t_1 is from above. Let y=0 at ground and we know v=0 so:
y_1=(1/2)*a*(t_1)^2
y_1=48m

y_2=distance traveled during constant velocity (zero acceleration)
y_2= 373 - y_1 = 325m

t_2 = time required to travel 325m at max allowable velocity
t_2 = (y_2)/v where v = max allowable velocity
t_2 = 12.64s

Total time = t_1 + t_2 = 18.03 s

EDIT: when I use underscores think of the number after it as a subscript. So y_1 is y with a subscript of 1. Also, if you want to solve problems like this in the future, you can do most of it with the equations of motion.

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