ألبرت اينشتاين والنظرية النسبية [ أمثلة محلولة ]

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Problems on the Time Dilation and the Length Contraction

1- How fast must a meter stick be moving if its length is observed to shrink to 0.5m?
2- With what speed will a clock have to be moving in order to run at rate that is one half the rate of a clock at rest?
3- A spacecraft moves at a speed of 0.9c. If its length is 25m when measured from inside the spacecraft, what is its length measured by a ground observer?
4- Two events are observed in a frame reference of O to occur at the same space points, the second occurring 2sec after the first. In a second frame O' moving relative to O, the second event is observed to occur 3sec after the first. What is the distance between the positions of the two events as measured in O'?
5- Two events are observed in a frame of reference O to occur simultaneously, at point separated by a distance of 1m. In a second frame O' moving relative to O along the line joining the two points in O, the two events appear to be separated by 2m. What is the time interval between the events as measured in O'?
6- The meson, an unstable particle, lives on the average, about 2.6x10^-8sec (measured in its own frame of reference) before decaying.
(a) If such a particle is moving with respect to the laboratory with a speed of 0.8c, what lifetime is measured in the laboratory?
(b) What distance, measured in the laboratory, does the particle move before decaying?
7- Two spaceships, each measuring 100m in its own rest frame, pass by each other traveling in opposite directions. Instruments on spaceship A determine that the front end of spaceship B requires 5x10-6sec to traverse the full length of A.
(a) What is the relative velocity of the two spaceships?
(b) A clock in the front end of B reads exactly one o'clock as it passes by the front end of A. What will the clock read as it passes by the rear end of A?
8- The straight-line distance between the earth and the star Alpha Centauri is about 4.3x10¹⁶m. Suppose a spaceship could be sent to the star with a speed 2x108m/sec.
(a) How long will the trip take according to earth clocks?
(b) How large a time will the spaceship clocks record this journey to take?
(c) How large will the spaceship occupants measure the earth to the star distance to be?
(d) How fast will the spaceship occupants compute their speed to be from the result of (b) & (c)?
9- A rocketship of length 100m, traveling at v/c=0.6, carries a radio station just at its nose. A radio pulse is emitted from a stationary space station just as the tail of the rocket passes by.
(a) How far from the space station is the nose of the rocket at the instant of arrival of the radio signal at the nose?
(b) By space-station time, what is the time interval between the arrival of this signal and its emission from the station?
(c) What is the time interval according to measurements in the rest frame of the rocket?

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