The X-15 rocket-powered plane holds the record for the...

The $X-15$ rocket-powered plane holds the record for the fastest speed ever attained by a manned aircraft, at $2020 \mathrm{m} / \mathrm{s}$. At this speed, by how much is the $15.5-\mathrm{m}$ -long aircraft length contracted?
Hint: Use the binomial approximation.

Key Concepts

Binomial Approximation

A mathematical technique used to simplify expressions involving powers or roots when a variable is small. In contexts like relativistic speed calculations, it is applied to approximate terms like ?(1-v²/c²), where v²/c² is small compared to 1, making the calculations more manageable.

Special Relativity

A fundamental theory in physics that describes how the laws of physics are the same for all inertial observers, particularly when objects move at speeds close to the speed of light. It introduces non-intuitive concepts such as time dilation and length contraction, which become significant at high velocities.

Length Contraction

This concept states that an object moving relative to an observer will appear shorter in the direction of its motion compared to its proper length (the length measured in the object's rest frame). The contraction is quantified by the factor ?(1-v²/c²), where v is the object's velocity and c is the speed of light.

Lorentz Factor

Also known as gamma (?), the Lorentz factor is defined as 1/?(1-v²/c²) and is central to the mathematics of special relativity. It quantifies the effects of time dilation and length contraction, linking an object's speed to the relative changes seen in time intervals and spatial dimensions.

Transcript

00:01 We're going to talk about length contraction.

00:02 So consider that we have an object that has a proper length, l, meaning that this is the length as measured in the reference ring where the object is at rest.

00:15 And imagine that this object is moving with a speed v relative to an observer.

00:21 I'm going to call this observer s.

00:26 So what special relativity tells us is that the length as measured by an observer that is at rest, relative to reference frame s is equal to the proper line divided by gamma, where gamma is 1 over a square root of 1 minus v squared over c squared.

00:47 And what the exercise tells us is that there is a plane that broke the record for the fastest aircraft that exists.

01:00 And this plane travels with a speed of 2 ,00020.

01:05 And 20 meters per second.

01:09 And our goal is to find out by how much the length of this aircraft contracted when it's traveling at this speed relative to an observer at rest in the earth.

01:25 Okay, so all we have to do is to use this formula that i wrote here.

01:34 So let's do it.

01:35 And we have to use this formula, but we also have to notice that v over c, that is 2 ,020 divided by 3 times 0 .8, is much smaller than 1.

01:48 So there is an approximation that we can use.

01:51 What i'm going to do is to use the approximation that 1 minus x to the n is approximately 1 minus an x as long as x is much smaller than 1.

02:02 And i'm writing this because notice the gamma can be written, i'm going to write it here.

02:10 Gamma is 1 minus v squared over c squared to the minus 1 half...

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