Question

A spaceship travels toward Earth at a speed of \(0.97 c .\) The occupants of the ship are standing with their torsos parallel to the direction of travel. According to Earth observers, they are about \(0.50 \mathrm{m}\) tall and $0.50 \mathrm{m}$ wide. What are the occupants' (a) height and (b) width according to others on the spaceship?

Short answer

Answer: The dimensions of the occupants from their perspective on the spaceship are approximately 0.122 m in height and 0.50 m in width.

Step-by-step solution

Identify the given information

We are given the following data: - Speed of the spaceship: v = 0.97c - Measured dimensions of the occupants according to the Earth observers: height = 0.50m, width = 0.50m

Calculate the Lorentz factor

The Lorentz factor is calculated using the equation: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\). Substituting the given values, v = 0.97c, we get: \(\gamma = \frac{1}{\sqrt{1 - \frac{(0.97c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.9409}}\)

Simplify the Lorentz factor

Simplify the expression under the square root: \(\gamma = \frac{1}{\sqrt{1 - 0.9409}} = \frac{1}{\sqrt{0.0591}}\)

Calculate the Lorentz factor

Calculate the value of the Lorentz factor: \(\gamma = \frac{1}{\sqrt{0.0591}} \approx 4.1\)

Calculate the occupants' height from their perspective

According to special relativity, the length contraction occurs only in the direction of motion, which in this case is the height of the occupants (torsos parallel to the direction of travel). Therefore, we can use the Lorentz factor to find their height in their own reference frame: Height\(_{spaceship} = \frac{Height_{Earth}}{\gamma}\) Height\(_{spaceship} = \frac{0.50m}{4.1} \approx 0.122 m\)

Calculate the occupants' width from their perspective

The width of the occupants does not change as it is perpendicular to the direction of motion. Consequently, their width remains unchanged according to the spaceship's occupants, and we can write: Width\(_{spaceship} = Width_{Earth} = 0.50 m\) Thus, we have the dimensions of the occupants from their perspective on the spaceship: a) Height: Approximately 0.122 m b) Width: 0.50 m