An astronaut in a rocket moving at 0.50 c toward the Sun...

An astronaut in a rocket moving at $0.50 c$ toward the Sun finds himsclf halfway betwecn Earth and the Sun. According to the astronaut, how far is he from Earth? In the frame of the Sun, the distance from Earth to the Sun is $1.50 \times 10^{11} \mathrm{~m}$.

Key Concepts

Frame of Reference

A frame of reference is a perspective from which an observer measures and experiences physical phenomena. In the context of relativity, different frames (for instance, that of the astronaut vs. the Sun) can yield different measurements of length, time, and other quantities due to their relative motion.

Lorentz Factor

The Lorentz factor, defined as ? = 1/?(1-v²/c²), quantifies the degree of time dilation and length contraction experienced by moving objects. It provides the mathematical basis for transforming measurements between different inertial frames in special relativity.

Length Contraction

Length contraction is a phenomenon predicted by special relativity, where the measured length of an object moving relative to an observer is shorter than its proper length (the length measured in the object's rest frame). It occurs only in the direction parallel to the motion.

Special Relativity

This is the theory that describes how the laws of physics are the same in all inertial reference frames and how quantities such as time, length, and mass vary for observers in different inertial frames, especially when their relative velocities approach the speed of light.

Transcript

00:01 In this problem, we have an asteroid traveling from the earth toward the sun.

00:08 And we are given that in the sun's frame of reference that the distance between the sun and the earth is 1 .5s, it's in 11 meters.

00:16 So in that frame, half of that ls over 2 will be the distance to the halfway point.

00:22 That's what we're trying to find.

00:24 That's what the sun says the halfway when the asteroid is halfway there.

00:29 What does the astronauts say about that? let's look at how far does the astronauts say that he's traveled from the earth at halfway point.

00:41 So let's look at this from the perspective of the astronauts.

00:44 Here's the astronaut's frame.

00:47 Here is the earth.

00:49 The earth, the astronaut was moving to the left with speed v.

00:55 That means the earth will be moving to the right with speed v.

01:00 And so will the sun frame be moving to the right with speed v.

01:05 Likewise, this halfway point can be seen as moving to the right with speed v also.

01:18 So let's talk about, now we have those pictures, let's talk about this distance here.

01:30 So that is l -s, l -a is equal to what? now, if we think of a connecting rod that has this length between the sun, and the earth.

01:46 That's at rest in the sun frame.

01:49 So this gives you, this is the proper length.

01:53 So the halfway point will be half the proper length, which would be 7 .5 times 10 to 10 or 0 .75 times 10 to the 11 meters...

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