As measured by an observer on the earth, a spacecraft runway...

As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 m. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of 4.00 $\times$ 10$^7$ m/s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 m. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at
a speed of 4.00 $\times$ 10$^7$ m/s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

Key Concepts

Lorentz Transformation

The Lorentz transformations are the mathematical formulas used to convert the time and space coordinates of events from one inertial frame to another moving at a constant relative velocity. They encapsulate the effects of both length contraction and time dilation, providing a consistent way to relate measurements made by observers in different inertial frames.

Time Dilation

Time dilation is another consequence of special relativity, stating that a clock moving relative to an observer ticks slower than a clock at rest in the observer's frame. This phenomenon implies that the time interval between two events, as measured in a moving frame (the proper time), is shorter than the same interval measured in a frame where the events occur at different spatial locations.

Proper Length

Proper length refers to the length of an object measured in the frame in which the object is at rest. It is the maximum length measured and is used as a reference when calculating length contraction in other inertial frames moving relative to the object.

Inertial Reference Frames

In special relativity, an inertial reference frame is one in which an observer is either at rest or moving with constant velocity, and the laws of physics hold in their standard form. This concept is fundamental since measurements of time and space depend on the relative motion between different inertial frames, as illustrated by the differences between observations made on Earth and those made by a pilot in a moving spacecraft.

Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object moving relative to an observer is measured to be shorter than its proper length. This effect occurs only in the direction of relative motion and is governed by the Lorentz factor, which depends on the relative speed between the observer and the object.

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