You are at the top of the Empire State Building on the 102nd floor, which is located 373 m above the ground, when your favorite superhero flies over the building parallel to the ground at 60.0 % the speed of light.

You have never seen your favorite superhero in real life. Out of curiosity you calculate her height to be 1.60 m. If the superhero landed next to you, how tall would she be when standing?
a) 2 m
b) 1.6 m
c) 1.26 m
d) 1.28 m

What is the height of the 102nd floor of the Empire State Building as measured by the superhero while flying above it?
a) 373 m
b) 298 m
c) 236 m
d) 239 m

Both A

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To calculate the height of the superhero when standing, we need to take into account the effects of time dilation and length contraction due to her high speed relative to you.

The formula for time dilation is:
t' = t / √(1 - v^2/c^2)

Where:
t' is the observed time for the superhero, which is the same as the time experienced by you (since you are stationary).
t is the proper time experienced by the superhero.
v is the velocity of the superhero relative to you.
c is the speed of light.

Plugging in the values:
v = 0.60c (60% the speed of light)
c = 3.00 x 10^8 m/s (speed of light)

t' = t / √(1 - (0.60c)^2/c^2)
t' = t / √(1 - 0.36)
t' = t / √(0.64)
t' = 1.26t

Since the superhero's height when flying above the Empire State Building is 1.60 m, the height when standing would be 1.60 m multiplied by the time dilation factor of 1.26:
Height when standing = 1.60 m * 1.26 = 2.016 m

So the correct answer is a) 2 m.

To answer the second question, the superhero would measure the height of the 102nd floor of the Empire State Building as the same height that you measure, since length contraction does not apply in this scenario. Therefore, the correct answer is a) 373 m.

To solve this problem, we need to consider the effects of time dilation due to the superhero's high speed. Let's break down the calculations step by step.

1. Calculate the height of the superhero as measured by an observer on the ground:
Since the superhero's height is 1.60 m, we need to calculate the length contraction factor at her speed. According to special relativity, the length contraction factor (gamma) is given by:

gamma = 1 / sqrt(1 - (v^2 / c^2))

where:
v = velocity of the superhero (60.0% the speed of light = 0.6c)
c = speed of light (3.0 x 10^8 m/s)

Plugging in these values, we can calculate gamma:

gamma = 1 / sqrt(1 - (0.6^2))

gamma ≈ 1.25

Now, we can calculate the superhero's apparent height on the ground:

Apparent height = Actual height / gamma = 1.60 m / 1.25 ≈ 1.28 m

Therefore, the answer to the first question is:

d) 1.28 m

2. Calculate the height of the 102nd floor of the Empire State Building as measured by the superhero while flying above it:
To do this, we can use the concept of length contraction again. The superhero's frame of reference is moving relative to the building, so we need to calculate the length contraction factor from her perspective.

Using the same formula as before, but substituting the superhero's velocity (0.6c), we get:

gamma = 1 / sqrt(1 - (v^2 / c^2))

gamma = 1 / sqrt(1 - (0.6^2))

gamma ≈ 1.25

The superhero will perceive the height of the 102nd floor as the actual height of the building divided by the length contraction factor:

Perceived height = Actual height / gamma = 373 m / 1.25 ≈ 298 m

Therefore, the answer to the second question is:

b) 298 m

In conclusion, the superhero's height when standing would be approximately 1.28 m, and she would perceive the height of the 102nd floor of the Empire State Building as approximately 298 m.