Twins who are 19.0 years of age leave the earth and travel...

Twins who are 19.0 years of age leave the earth and travel to a distant planet 12.0 light-years away. Assume that the planet and earth are at rest with respect to each other. The twins depart at the same time on different spaceships. One twin travels at a speed of $0.900 c,$ and the other twin travels at $0.500 c .$ (a) According to the theory of special relativity, what is the difference between their ages when they meet again at the earliest possible time? (b) Which twin is older?

Key Concepts

Special Relativity

Special relativity is a foundational theory in physics, developed by Albert Einstein, which describes the behavior of objects moving at significant fractions of the speed of light. It introduces concepts such as the constancy of the speed of light for all observers, the relativity of simultaneity, and how measurements of time and space change for observers moving relative to one another.

Time Dilation

Time dilation is a phenomenon predicted by special relativity where a clock moving relative to an observer ticks slower than a clock at rest with respect to that observer. This effect becomes especially pronounced at speeds close to the speed of light, and it is crucial for understanding how differently aging can occur between individuals moving at different relative speeds.

Lorentz Factor

The Lorentz factor is a key component in the equations of special relativity, quantifying how time, length, and mass transform between different inertial frames. It is defined as ? = 1/?(1 - v²/c²), where v is the speed of the moving object and c is the speed of light, and it plays a central role in calculating time dilation effects.

Twin Paradox

The twin paradox is a thought experiment in special relativity that involves a pair of twins, one of whom travels at a high speed while the other remains on Earth. Upon reunion, the traveling twin is younger due to time dilation effects. This scenario is used to illustrate the non-intuitive consequences of time dilation and the effects of different inertial frames on the passage of time.

Transcript

00:01 Twins who are 19 years of age leave the earth and travel to a distant planet 12 light years away.

00:07 Assume that the planet and the earth are at rest with respect to each other.

00:11 The twins depart at the same time on different spaceships.

00:16 One twin travels at a speed of 0 .9000c and the other twin travels at 0 .5 .0c.

00:24 According to the special theory of relativity, what is the difference between their ages when they meet again at the earliest possible time? and part b, which twin is older? okay, so the distance between the earth and the planet is 12 light years.

00:40 Remember that light years is a unit of distance.

00:45 So the speed of the first twin is 0 .9c.

00:50 The speed of the second twin is 0 .5c.

00:57 So the time traveled for the first twin is the distance over the velocity.

01:10 So 12 light years is 12c times one year.

01:19 So we get 13 .33 years.

01:26 So the first twin in the spaceship, we're looking for the proper time.

01:35 So 1 minus v1 squared over c squared, which comes to 5 .81 years.

01:43 So for the twin that's on earth, sorry, from an observer on earth, the other twin has a time d over v2, which is 24 years.

02:05 Using the same equation as above, except using v2, we get that the proper time is 20 .78 years...