Question

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 70.0% the speed of light.

You have never seen your favorite superhero in real life. Out of curiosity you calculate her height to be 1.50 m . If the superhero landed next to you, how tall would she be when standing?
A). 2.10 m
B). 1.50 m
C).1.15 m
d). 1.07 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
B) 266
C) 204
D) 190

Answer 1

Hero's height when standing: B) 1.50 m

Height of the 102nd floor as measured by the superhero: C) 204 m

The situation described involves relativistic effects due to the superhero's high speed, as described by Einstein's theory of special relativity. The Lorentz contraction formula is relevant for calculating the apparent height of the superhero from the perspective of an observer on the Empire State Building.

Firstly, let's calculate the height of the superhero when standing still, as observed by the person on the ground. The Lorentz contraction formula is given by:

`\[ \text{Height} = \text{Original Height} * \sqrt{1 - \left((v^2)/(c^2)\right)} \]`

where (v) is the velocity of the superhero and (c) is the speed of light.

Given that the superhero is flying at 70.0% the speed of light
`(\(v = 0.7c\))` and her original height is 1.50 m, the height when standing still
`(\(v = 0\))` is:

`\[ \text{Height} = 1.50 \, \text{m} * \sqrt{1 - \left((0.7^2)/(1^2)\right)} \]`

Calculating this gives the apparent height of the superhero when standing.

Next, the height of the 102nd floor of the Empire State Building as measured by the superhero while flying above it can be calculated using the same Lorentz contraction formula.

Given that the original height of the 102nd floor is 373 m, the apparent height as measured by the superhero is:

`\[ \text{Apparent Height} = 373 \, \text{m} * \sqrt{1 - \left((0.7^2)/(1^2)\right)} \]`

Now, comparing the calculated heights with the given options:

For the superhero's height when standing: Option B is the closest match.

For the height of the 102nd floor as measured by the superhero: Option C is the closest match.