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Chapter 35: Problem 26

A spacecraft travels along a straight line from Earth to the Moon, a distance of \(3.84 \cdot 10^{8} \mathrm{~m}\). Its speed measured on Earth is \(0.50 c\). a) How long does the trip take, according to a clock on Earth? b) How long does the trip take, according to a clock on the spacecraft? c) Determine the distance between Earth and the Moon if it were measured by a person on the spacecraft.

Short Answer

Expert verified
Answer: a) The trip takes approximately 1.7067 seconds according to a clock on Earth. b) The trip takes approximately 1.477 seconds according to a clock on the spacecraft. c) The distance between Earth and the Moon, if measured by a person on the spacecraft, is approximately \(3.32 \cdot 10^{8}\ \mathrm{m}\).

Step by step solution

01

Find the time taken according to a clock on Earth

To find the time taken according to a clock on Earth, we can use the formula: time = distance / speed. We are given the distance between Earth and the Moon as \(3.84 \cdot 10^{8} \mathrm{~m}\) and the speed of the spacecraft as \(0.50 c\), where c is the speed of light (\(3.00 \cdot 10^{8} \mathrm{\ m/s}\)). Plug in the values, and we get: time = \((3.84 \cdot 10^{8} \mathrm{~m}) / (0.50 \cdot 3.00 \cdot 10^{8} \mathrm{\ m/s})\) time = \(2.56 \cdot 10^{8} / 1.5 \cdot 10^{8}\) time = 1.7067 s So, the trip takes approximately 1.7067 seconds, according to a clock on Earth.
02

Calculate the relativistic gamma factor

Before we move on to find the time taken according to a clock on the spacecraft, we need to calculate the relativistic gamma factor. The gamma factor, represented by \(\gamma\), is given by the formula: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) Plug in the values, and we get \(\gamma = \frac{1}{\sqrt{1 - \frac{(0.50\cdot 3.00 \cdot 10^{8}\mathrm{\ m/s})^2}{(3.00 \cdot 10^{8}\mathrm{\ m/s})^2}}}\) After simplifying, we get \(\gamma = \frac{1}{\sqrt{1 - 0.25}} = \frac{1}{\sqrt{0.75}} = \frac{1}{0.866} = 1.155\)
03

Find the time taken according to a clock on the spacecraft

Now that we have the relativistic gamma factor, we can find the time taken according to a clock on the spacecraft using the time dilation formula: time on spacecraft = time on Earth / \(\gamma\) Plug in the values, and we get the time on the spacecraft = \(1.7067\ \mathrm{s} / 1.155 = 1.477\ \mathrm{s}\) So, the trip takes approximately 1.477 seconds, according to a clock on the spacecraft.
04

Determine the distance between Earth and the Moon if measured by a person on the spacecraft

To find the distance measured by a person on the spacecraft, we will use the length contraction formula: distance on spacecraft = distance on Earth / \(\gamma\) Plug in the values, and we get the distance on the spacecraft = \((3.84 \cdot 10^{8} \mathrm{~m}) / 1.155 = 3.32 \cdot 10^{8}\ \mathrm{m}\) So, the distance between Earth and the Moon, if measured by a person on the spacecraft, is approximately \(3.32 \cdot 10^{8}\ \mathrm{m}\).

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