Spacecraft I, containing students taking a physics exam,...

Spacecraft I, containing students taking a physics exam, approaches the Earth with a speed of $0.600 c$ (relative to the Earth), while spacecraft II, containing professors proctoring the exam, moves at $0.280 c$ (relative to the Earth) directly toward the students. If the professors stop the exam after 50.0 min have passed on their clock, for what time interval does the exam last as measured by (a) the students and (b) an observer on the Earth?

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Key Concepts

Special Relativity

This is the framework in which the laws of physics, including the constant speed of light in vacuum, remain the same in all inertial frames. It introduces phenomena that do not arise in classical mechanics, such as time dilation and length contraction, when objects approach the speed of light.

Proper Time

Proper time is the time interval measured by an observer for whom two events occur at the same location in space. It represents the shortest time interval between these events and is fundamental in calculating how time is experienced differently in various inertial frames.

Time Dilation

Time dilation refers to the effect where a moving clock is observed to run slower relative to a clock that is at rest in a given inertial frame. This means that the time interval measured in a moving frame (dilated time) will be longer than the proper time measured in the rest frame of the event.

Lorentz Factor

The Lorentz factor, often denoted as ?, is defined as ? = 1/?(1 ? v²/c²), where v is the relative velocity between observers and c is the speed of light. It quantifies the amount of time dilation, length contraction, and other relativistic effects experienced by objects moving at significant fractions of the speed of light.

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