In an attempt to reduce the extraordinarily long travel...

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip. a. How fast must the rocket travel relative to earth? b. How much energy is needed to accelerate the rocket to this speed? c. How many times larger is this energy than the total energy used by the United States in the year $2012,$ which was roughly $1.0 \times 10^{20} \mathrm{J} ?$

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip.
a. How fast must the rocket travel relative to earth?
b. How much energy is needed to accelerate the rocket to this speed?
c. How many times larger is this energy than the total energy used by the United States in the year $2012,$ which was roughly $1.0 \times 10^{20} \mathrm{J} ?$

Key Concepts

Special Relativity

Special Relativity is the theory developed by Einstein which describes how time, space, and energy behave at velocities close to the speed of light. It fundamentally changes the classical mechanics notions of absolute time and space, and introduces phenomena such as time dilation and length contraction, which become significant when objects move at relativistic speeds.

Time Dilation

Time dilation is a consequence of Special Relativity where time measured in a moving frame of reference passes more slowly relative to time measured in a stationary frame. In scenarios involving high-speed travel, such as a journey to a distant star, the proper time experienced by the travelers is less than the time elapsed on Earth, allowing for significantly shorter aging despite long Earth-frame travel times.

Lorentz Factor

The Lorentz factor (commonly denoted as gamma, ?) quantifies the relationship between time, length, and mass at relativistic speeds. It is defined as ? = 1/?(1 ? v²/c²), where v is the velocity of the moving object and c is the speed of light. This factor is essential for calculating time dilation, length contraction, and relativistic kinetic energy, and it increases significantly as the velocity approaches the speed of light.

Relativistic Kinetic Energy

Relativistic kinetic energy extends the classical concept of kinetic energy to regimes where speeds are close to the speed of light. It is calculated using the expression E = (? ? 1)mc², where m is the mass of the object and ? is the Lorentz factor. This formula accounts for the dramatic increase in energy required as an object’s speed approaches light speed, reflecting the nonlinear relationship between energy and velocity in relativistic contexts.

Energy Scale Comparison

Energy scale comparisons involve putting the computed relativistic energy requirements into perspective with familiar large-scale energy values, such as the total annual energy consumption of a country. This helps illustrate the practical challenges and immense energy resources needed for high-speed space travel, highlighting the gap between current technology and the energy demands of relativistic rockets.

Transcript

00:01 Hello everyone.

00:02 In this problem, we are asked to calculate at what speed a spaceship has to travel to a distance star that is 450 light years away, so that during the round trip, so on the trip going there and on the trip coming back, we could only get 20 years older.

00:23 So what we want to do is we want to cover a total distance of 900 light years, and in that time, we want to age 20 years, i .e., as we're traveling in a spaceship, we want the proper time elapsing in that traveling spaceship frame to be 20 years.

00:43 So first of all, what we have to do is we have to figure out what this distance is, or essentially, what is the time that is going to be.

00:53 Dilating that we're going to look at.

00:55 So in the original frame in which the spaceship is traveling like in a let's say the rest frame of planet earth, the speed of the spaceship is going to be v, right? so this variable v and is going to travel to a distance of 450 light years and then back.

01:16 So the total time that it's going to take is going to be moving at this speed it will be d over v, so 2 times l over v.

01:28 So that's a total time taken in hour frame.

01:31 And this time taken in our frame is going to be the dilated time t -prime measured in the spaceship frame.

01:39 So we're going to include the gamma factor next to t -prime in this case.

01:46 So then we have that d over v is equal to gamma times d -prime, where t -prime is tau, which is the proper time in the spaceship.

01:53 Frame which is 20 years.

01:55 So then we just have to do a bunch of algebra.

01:57 So we put in the gamma factor in its full glory.

02:01 So it's 1 over root of 1 minus v squared over c squared.

02:04 We square everything after dividing both sides here by t prime.

02:11 So squaring everything gives us d over v t prime, all squared, is equal to 1 over 1 minus v squared over c squared.

02:19 Then we multiply fruit by 1 minus 2 squared over c squared and multiply 3 by v squared.

02:25 And then we essentially just rearrange, just follower nods.

02:28 And i was like, expand this bracket.

02:31 You're going to have d over t prime squared minus d over t prime squared as v squared over c squared.

02:38 So we move the v squared over c squared term to the other side and factorize v squared.

02:43 That gives us this factor in the bracket over here.

02:46 And then we divide by that factor.

02:50 And we essentially end up with v being equal to d over t prime over the square root of one plus d over t prime c or squared.

02:59 So then now we are ready to plug you the numbers...

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