-Suppose NASA discovers a planet just like Earth orbiting a...

-Suppose NASA discovers a planet just like Earth orbiting a star just like the Sun. This planet is 35 light-years away from our Solar System. NASA quickly plans to send astronauts to this planet, but with the condition that the astronauts not age more than 25 years during the journey a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during their journey? b) According to the astronauts, what will be the distance of their trip?

-Suppose NASA discovers a planet just like Earth orbiting a star just like the Sun. This planet is 35 light-years away from our Solar System. NASA quickly plans to send astronauts to this planet, but with the condition that the astronauts not age more than 25 years during the journey
a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during their journey?
b) According to the astronauts, what will be the distance of their trip?

Key Concepts

Special Relativity

Special relativity is the theory formulated by Einstein that revolutionized our understanding of space and time. It establishes that the laws of physics are identical in all inertial frames and that the speed of light in vacuum is constant for all observers regardless of their motion. This theory leads to counterintuitive effects such as time dilation and length contraction, which are essential to analyzing high-speed travel scenarios such as interstellar journeys.

Time Dilation

Time dilation is a consequence of special relativity where a moving clock is observed to tick slower compared to a clock at rest in an inertial reference frame. In the context of a spaceship traveling at speeds close to the speed of light, the time experienced by the astronauts (proper time) is less than the time measured by observers on Earth, enabling the possibility that astronauts age less than the duration of the journey as seen from Earth.

Length Contraction

Length contraction is another relativistic effect whereby the length of an object or the distance between two points, as measured in the frame moving relative to the object, is shorter than the length measured in the object's rest frame. For astronauts traveling at relativistic speeds, the distance to the destination will appear contracted compared to the distance measured in Earth's reference frame, making the journey seem shorter from their perspective.

Lorentz Factor

The Lorentz factor, denoted by ? (gamma), is a crucial quantity in special relativity defined as ? = 1/sqrt(1 - v²/c²), where v is the speed of the moving object and c is the speed of light. It quantifies the amount by which time dilation and length contraction occur. In problems involving high-speed travel, such as interstellar journeys with strict aging constraints, calculating the correct Lorentz factor is essential to relate the time and distance measurements between different inertial frames.

Transcript

00:02 So a planet's discovered.

00:03 It's just like earth.

00:04 It's got a sun, just like sun.

00:07 And we are asked to find out what speed the spaceship will have to travel if it's 35 light years away.

00:15 And we don't want the astronauts to age any more than 10 years.

00:19 Okay, so let's jot down some equations here.

00:29 That will equal the change of time over 1 minus v squared over c square.

00:37 And the frame, the speed of the spaceship and earth's frame of resonance will be l over delta t.

00:45 So then v will be l over delta t times one minus v squared over c squared.

00:56 And we want to solve that last equation for v.

01:00 So let's square both sides.

01:12 So that'll give me v squared will equal l squared delta t squared times one.

01:29 Minus v squared over c squared.

01:33 Ok, so this will be v squared times 1 plus l squared over delta t squared c squared equals l squared over delta t prime squared.

02:02 And i get v t equals l t equals l t over delta t prime squared.

02:09 Over 1 plus l squared over delta t squared squared c squared this will get me to v2 equals c2 over 1 plus delta t and v will then equal speed of light times 1 plus delta t c l squared so now we can put our values in...

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