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Chapter 26: Problem 18

A spaceship is moving at a constant velocity of \(0.70 c\) relative to an Earth observer. The Earth observer measures the length of the spaceship to be $40.0 \mathrm{m}$. How long is the spaceship as measured by its pilot?

Short Answer

Expert verified
Answer: The length of the spaceship as measured by the pilot is approximately 56.19 meters.

Step by step solution

01

Understand the length contraction formula

The length contraction formula is given by: \(L = L_{0} \sqrt{1 - \frac{v^2}{c^2}}\) where \(L\) is the contracted length measured by the observer, \(L_0\) is the proper length measured in the rest frame of the object (in this case, the pilot), \(v\) is the relative velocity of the object, and \(c\) is the speed of light.
02

Rearrange the formula to solve for the proper length

We need to find \(L_0\), the proper length as measured by the pilot. Rearrange the formula to solve for \(L_0\): \(L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}}\)
03

Substitute the given values and solve for the proper length

Plug in the given values: \(L = 40.0\,\text{m}\) and \(v = 0.70c\): \(L_0 = \frac{40.0\,\text{m}}{\sqrt{1 - \frac{(0.70c)^2}{c^2}}}\) Simplify the expression in the square root: \(L_0 = \frac{40.0\,\text{m}}{\sqrt{1 - (0.70^2)}}\)
04

Calculate the proper length

Now, calculate the proper length: \(L_0 = \frac{40.0\,\text{m}}{\sqrt{1 - 0.49}}\) \(L_0 = \frac{40.0\,\text{m}}{\sqrt{0.51}}\) \(L_0 \approx 56.19\,\text{m}\) So, the length of the spaceship as measured by the pilot is approximately 56.19 meters.

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Most popular questions from this chapter

Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of \(0.60 c\) with respect to Earth. The light is moving in the direction of motion of the spaceship. Siu- Ling is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?
As observed from Earth, rocket Alpha moves with speed \(0.90 c\) and rocket Bravo travels with a speed of \(0.60 c\) They are moving along the same line toward a head-on collision. What is the speed of rocket Alpha as measured from rocket Bravo? (tutorial: adding velocities)
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