The rest radius of the Earth is 6370 km and its orbital speed about the Sun is 29.8 km / s . B

The rest radius of the Earth is 6370 km and its orbital...

The rest radius of the Earth is $6370 \mathrm{~km}$ and its orbital speed about the Sun is $29.8 \mathrm{~km} / \mathrm{s} .$ By how much would the Earth's diameter appear to be shortened to an observer stationed so as to be able to watch the Earth move past at this speed?

Key Concepts

Special Relativity

A fundamental theory in physics introduced by Einstein that redefines the concepts of space and time, particularly when objects are moving at speeds close to that of light. It provides the framework for understanding effects such as time dilation and length contraction, and establishes that the speed of light is constant in all inertial frames.

Lorentz Contraction

A relativistic effect predicted by special relativity, which states that objects in motion relative to an observer will appear shorter along the direction of motion compared to their rest (proper) length. The contraction is quantified by the factor ?(1 ? v²/c²), where v is the object’s speed and c is the speed of light.

Proper Length

The length of an object as measured in its own rest frame, where the object is at rest relative to the observer. In other frames moving relative to the object, the observed length is contracted from this proper length due to the Lorentz contraction effect.

Transcript

0:00 Hi there.

00:01 So for this problem, we are given the information that the rest of radius of the earth is equal to 6 ,370 kilometers, and its orbital speed about descent is equal to 29 .8 kilometers.

00:32 Per second and what we need to determine is by how much will the earth diameter appear to be shortened to an observer that is stationary so as to be able to watch the earth move pass at this speed so we need to find the change in the longitude in this case the change in the diameter now now we are the s, in this case we are the s prime frame because we are moving with the earth.

01:17 Meanwhile, the other observer is addressed with respect to the other frame by seeing the earth passing by.

01:31 So with that said, we will have that the change in the length is l prime.

01:36 Which is the value that we measure minus l sub 0.

01:41 That is the value that is going to be measured by the observer.

01:48 So in that case, since we know we will have in here that l prime is the diameter, and since we are given the radius, we need to multiply that values by 2.

02:03 And here we have this value.

02:07 So passing that to meters, we know that that is equal to 6 .337 times 10 to 6 meters.

02:19 That's the radius and the diameter, sorry.

02:24 And l -0 is going to be given by the contraption formula.

02:30 So that is two times the diameter, again, we put that value in here...

Please give Ace some feedback