A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet. A technician on the spacecraft measures the length of the spacecraft as 125 m. An observer on the planet measures the length of the spacecraft as A 36 m B 50 m C 124 m D 314 m E 433 m

Scottish Highers Physics Special Relativity: A spacecraft is travelling at a

A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet - Scottish Highers Physics - Question 4 - 2017

Question 4

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A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet. A technician on the spacecraft measures the length of th... show full transcript

Worked Solution & Example Answer:A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet - Scottish Highers Physics - Question 4 - 2017

Step 1

Determine the relativistic length contraction

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Answer

According to the theory of relativity, the length of an object moving at a significant fraction of the speed of light appears contracted to an observer at rest relative to that object. The formula for length contraction is given by:

$L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$

where:

$L_0$ is the proper length (length measured by the observer in the rest frame, which is 125 m),
$v$ is the velocity of the moving object ( $2.75 \times 10^6 \, \text{m s}^{-1}$ ),
$c$ is the speed of light ( $3.00 \times 10^8 \, \text{m s}^{-1}$ ).

Step 2

Calculate the value of $ \frac{v^2}{c^2} $

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Answer

Let's calculate\n $\frac{v^2}{c^2} = \frac{(2.75 \times 10^6)^2}{(3.00 \times 10^8)^2} = \frac{7.5625 \times 10^{12}}{9 \times 10^{16}} = 8.415 \times 10^{-5}$ \n Next, calculate the square root:\n $\sqrt{1 - \frac{v^2}{c^2}} = \sqrt{1 - 8.415 \times 10^{-5}} \approx \sqrt{0.99991585} \approx 0.999957925$

Step 3

Calculate the contracted length $ L $

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Answer

Now, plug this value into the length contraction formula:

$L = 125 \, ext{m} \times 0.999957925 \approx 124.995 \, ext{m}$

Rounding this gives us approximately 125 m.

Step 4

Select the correct answer from the options

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Answer

The observer on the planet measures the length of the spacecraft as approximately 124 m, which corresponds to option C.

A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet - Scottish Highers Physics - Question 4 - 2017

Worked Solution & Example Answer:A spacecraft is travelling at a constant speed of $2.75 imes 10^6 \, ext{m s}^{-1}$ relative to a planet - Scottish Highers Physics - Question 4 - 2017

Determine the relativistic length contraction

Calculate the value of \( \frac{v^2}{c^2} \)

Calculate the contracted length \( L \)

Select the correct answer from the options

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