### Video Transcript

A spaceship has a proper length of
2.00 times 10 to the two meters moves at 0.970๐ relative to the Earth. What is the spaceshipโs length as
measured by an observer on the Earth?

Letโs start by highlighting some of
the important information given. Weโre told that the proper length
of the spaceship is 2.00 times 10 to the two meters. Weโll call that length ๐ zero. And weโre also told that the
spaceship moves at a speed relative to the earth of 0.970๐, where ๐ is the speed
of light. Weโll call that speed simply
๐ฃ. We want to solve for the length of
the spaceship as measured by an observer on Earth. Letโs call that length ๐.

Weโll approach this problem by
recalling that there is a correlation or a connection between length observed in one
inertial reference frame in length observed in another. This correlation says that the
length observed in one inertial reference frame, weโll call it ๐ prime, is equal to
๐พ times ๐, the length observed in the other reference frame. Here, ๐พ is defined as one divided
by the square root of one minus ๐ฃ squared over ๐ squared, where ๐ฃ is the speed of
one reference frame relative to the other.

If we apply this length correlation
relationship to our situation, we find that ๐, the length weโre looking to solve
for, is equal to ๐พ times ๐ sub zero, the proper length of the spaceship. And we can replace ๐พ with its
defined expression. So the spaceshipโs length, as
measured by an observer on Earth, is equal to ๐ sub zero divided by the square root
of one minus ๐ฃ squared over ๐ squared. Weโre given ๐ sub zero and ๐ฃ in
terms of the speed of light ๐. So we can enter those values into
our equation now.

๐ is equal to 2.00 times 10 to the
two meters divided by the square root of one minus the quantity 0.970๐ squared all
over ๐ squared. In this denominator, the ๐ squared
terms cancel out so that our expression inside the square root simplifies to one
minus 0.970 squared.

When we enter these numbers into
our calculator, we find a value for ๐ of 48.6 meters. Compared to its proper length of
200 meters, this is the length of the spaceship as measured by an observer on
Earth.