A.5 伽利略相對性與狹義相對論（HL）：練習題

According to the postulates of special relativity, which of the following is true for all observers in inertial frames of reference?

A spaceship has a proper length of \( 100\text{ m} \). It travels past a stationary observer at a constant speed of \( 0.80c \). What is the length of the spaceship as measured by the stationary observer?

A particle of rest mass \( m \) moves with a total energy of \( 3mc^2 \). It undergoes a perfectly inelastic collision with a stationary particle of rest mass \( m \), resulting in the formation of a single composite particle of rest mass \( M \). What is the value of \( M \)?

A boat travels with a speed of \( 5.0\text{ m s}^{-1} \) relative to the water in a river. The river flows at a constant speed of \( 3.0\text{ m s}^{-1} \) relative to the riverbank. According to Galilean relativity, what are the maximum and minimum possible speeds of the boat relative to the riverbank?

Two spaceships, A and B, approach each other. In the frame of reference of a stationary observer on a nearby planet, spaceship A moves to the right at \( 0.90c \) and spaceship B moves to the left at \( 0.90c \). What is the speed of spaceship A as measured by an observer on spaceship B?

State the two fundamental postulates of Einstein's special theory of relativity.

A subatomic particle is accelerated until its total energy is exactly four times its rest mass energy. Calculate the speed of the particle as a fraction of the speed of light \(c\).

A particle of rest mass \(m\) is initially at rest. It is accelerated until its total energy is twice its rest mass energy. Determine the magnitude of the particle's final relativistic momentum in terms of \(m\) and \(c\).

Consider two spaceships, A and B, moving in the same direction relative to an observer on Earth. Ship A moves at \(0.70c\) and Ship B moves at \(0.90c\) relative to Earth.
(a) According to Galilean relativity, what would be the velocity of Ship B relative to Ship A?
(b) Using special relativity, calculate the velocity of Ship B as measured by an observer on Ship A.
(c) Explain why the Galilean result is considered an approximation and under what conditions it is valid.

A rectangular plate has sides \( L_x \) and \( L_y \) in its rest frame S. It moves at a constant velocity \( v \) in the positive x-direction relative to an observer in frame S'.
(a) Write expressions for the dimensions of the plate as measured in S'.
(b) Show that the area \( A' \) of the plate measured in S' is related to its proper area \( A_0 \) by \( A' = \frac{A_0}{\gamma} \).
(c) If the plate is a square in its rest frame and its area in S' is observed to be half of its proper area, calculate the speed \( v \) of the plate.