A spaceship travels at constant velocity from Earth to a...

A spaceship travels at constant velocity from Earth to a point 710 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is $0.9999 \mathrm{c}$. A passenger is 20 yr old when departing from Earth. (a) How old is the passenger when the ship reaches its destination, as measured by the ship's clock? (b) If the spaceship sends a radio signal back to Earth as soon as it reaches its destination, in what year, by Earth's calendar, does the signal reach Earth? The spaceship left Earth in the year 2000 .

Key Concepts

Length Contraction

Length contraction is the phenomenon in which the measured length of an object moving at relativistic speeds appears shortened along the direction of motion, as observed from a stationary frame. This effect, also determined by the Lorentz factor, is essential in reconciling measurements of distances in different reference frames.

Signal Propagation Delay

Signal propagation delay refers to the finite time required for light or radio signals to travel between events in spacetime. In relativistic scenarios, particularly those involving vast distances, accounting for the delay due to the constant speed of light is crucial for correctly determining the timing of events, such as when a signal is received on Earth after being sent from a moving spaceship.

Lorentz Factor

The Lorentz factor (often denoted by ?) quantifies the amount of time dilation and length contraction experienced by objects moving at a relativistic speed. It is defined as ? = 1/?(1 ? v²/c²), where v is the relative velocity and c is the speed of light. This factor plays a critical role in converting time intervals and distances between different inertial frames.

Special Relativity

Special Relativity is the physical theory that establishes the constancy of the speed of light in all inertial frames and describes how measurements of time and space vary for observers in different inertial frames. It provides the framework for understanding phenomena such as time dilation and length contraction, which become significant at speeds approaching the speed of light.

Time Dilation

Time dilation is a relativistic effect whereby a moving clock appears to tick slower compared to a clock at rest as measured by an observer in a different inertial frame. This concept is quantified by the Lorentz factor and is crucial when determining how much proper time passes for a passenger traveling at relativistic speeds compared to an observer at rest on Earth.

Transcript

00:01 The question i was asking us how old would the passenger be as it was measured by the ship's clock? well, we know this kind of question is dealing with the time dilation, which is delta t, equal to gamma times delta t 0.

00:17 Okay? gamma is the lawrence factor, which is equal to 1 over square rule, 1 minus v square, over c square, okay? and since the question was asking us the time measured by the spaceship clock which means it's asking delta t zero and delta t here means the time on the earth okay so we have delta t is equal to uh uh oh it's equal to delta t over gamma okay which is equal to square root 1 minus v square over c squared times delta t okay and we know that v the speed of a spaceship is given as 0 .999c.

01:18 Okay? and delta t is given as, remember, the distance from the spaceship to a point is 710 light years.

01:30 So light years is the distance that light travel in one years.

01:34 So since the 710 light years, that means the time takes the light to travel there is 710.

01:43 Years, okay? and therefore, if we plug in the values, we'll have dota t0 is equal to square root 1 minus 0 .999 c square over c squared, okay, n times the time from a measure from the earth, which is 710 years, okay? and it should give us the delta t0 is about 10 years.

02:28 So remember, the passenger was 20 years old.

02:30 So that means during his trip, eventually he will become 30 years old on a spaceship.

02:39 So it's 20 years plus 10 years, which is 30 years.

02:47 So he'll be 30 years old.

02:50 And for question b was asking us, what will be the year on earth's calendar when the signal was sent back, was sending back to the earth...

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