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Chapter 26: Problem 19

A spaceship moves at a constant velocity of \(0.40 c\) relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be \(1.0 \mathrm{m}\) long. (a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer? (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?

Short Answer

Expert verified
Answer: (a) 1.0 m (rod held perpendicular), (b) The length of the rod when held parallel can be calculated using the Lorentz contraction formula and the given values for L_0 and v.

Step by step solution

01

Determine the Lorentz factor

Calculate the Lorentz factor: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), using the given velocity \(v = 0.40c\).
02

Calculate length when the rod is perpendicular

Since the rod is held perpendicular to the direction of motion, there is no length contraction, and the length of the rod according to the Earth observer is the same as the proper length: \(L = L_0 = 1.0 \mathrm{m}\).
03

Calculate length when the rod is parallel

When the rod is held parallel to the direction of motion, we need to apply the Lorentz contraction formula: \(L = L_0 \cdot \sqrt{1 - \frac{v^2}{c^2}}\). Plug in the values for \(L_0\) and \(v\) to find the length of the rod according to the Earth observer. The answers to the problem are: (a) 1.0 m (rod held perpendicular), (b) the length of the rod when held parallel can be calculated using the formulas mentioned above.

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Most popular questions from this chapter

Electron A is moving west with speed \(\frac{3}{5} c\) relative to the lab. Electron \(\mathrm{B}\) is also moving west with speed \(\frac{4}{5} c\) relative to the lab. What is the speed of electron \(\mathrm{B}\) in a frame of reference in which electron \(\mathrm{A}\) is at rest?
The fastest vehicle flown by humans on a regular basis is the Space Shuttle which has a mass of about \(1 \times 10^{5} \mathrm{kg}\) and travels at a speed of about \(8 \times 10^{3} \mathrm{m} / \mathrm{s} .\) Find the momentum of the Space Shuttle using relativistic formulae and then again by using nonrelativistic formulae. By what percent do the relativistic and nonrelativistic momenta of the Space Shuttle differ? [Hint: You might want to use one of the approximations in Appendix A.5.]
An object of mass \(0.12 \mathrm{kg}\) is moving at $1.80 \times 10^{8} \mathrm{m} / \mathrm{s}$ What is its kinetic energy in joules?
A plane trip lasts \(8.0 \mathrm{h} ;\) the average speed of the plane during the flight relative to Earth is \(220 \mathrm{m} / \mathrm{s}\). What is the time difference between an atomic clock on board the plane and one on the ground, assuming they were synchronized before the flight? (Ignore general relativistic complications due to gravity and the acceleration of the plane.)
Show that each of these statements implies that \(v<c\) which means that \(v\) can be considered a nonrelativistic speed: (a) $\gamma-1<<1 \text { [Eq. }(26-14)] ;\( (b) \)K<<m c^{2}\( [Eq. \)(26-15)] ;(\text { c }) p<m c[\text { Eq. }(26-16)] ;\( (d) \)K \approx p^{2} /(2 m)$.
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