IP Two rocket ships are racing toward Earth, as shown in Figure 29-31 . Ship A is in the lead, a

IP Two rocket ships are racing toward Earth, as shown in...

Key Concepts

Relative Velocity

Relative velocity refers to the speed of one object as observed from another object’s frame of reference. In scenarios involving high-speed objects, using the relativistic velocity addition formula is necessary to accurately determine how fast one object appears to be moving relative to another.

Relativistic Velocity Addition

This concept involves a formula that combines velocities in a way that is consistent with the theory of special relativity. Unlike the classical addition of speeds, the relativistic formula ensures that the resultant velocity never exceeds the speed of light, taking into account time dilation and length contraction effects at high velocities.

Reference Frame

A reference frame is a coordinate system or viewpoint from which measurements such as time, distance, and speed are made. In problems like this, switching between the Earth’s inertial frame and the moving ships' frames is essential for correctly applying the principles of special relativity.

Transcript

00:01 Hello, and in this question here, we're going to investigate how velocities transform between different coordinate frames and special hours here.

00:09 So, in this question here, we've got earth.

00:13 And we assume that earth in this question is addressed.

00:17 We've got this first spaceship, spaceship 1, moving towards the earth with a relative velocity of minus 0 .8c.

00:25 We have a second spaceship, spaceship 2, which moves at a relative velocity relative to the first spaceship of minus 0 .5c.

00:37 So we can ask the question, what is the relative velocity of the second spaceship relative to earth? well, classically, you might expect that you simply add the velocities.

00:48 So it's the velocity of the first spaceship relative to earth plus the velocity of the second spaceship relative to the first spaceship.

00:57 However, this gives a velocity of minus 1 .3c, where minus 1 .3c is greater than the speed of life.

01:08 So this is a big problem.

01:10 So this classical version, where we simply add the velocities, must be incorrect because it gives a speed greater than the speed of light, which is impossible.

01:20 So that means we must find a correct method which correctly predicts that you can't travel faster than the speed of light.

01:28 To do this, we're going to use special relativity.

01:31 And we're going to begin by defining the coordinate system s.

01:36 S is the coordinate system where the observer on earth is at rest.

01:41 And the vertical axis is the time axis and the horizontal axis is the position axis.

01:47 We define a second coordinate system, s prime, and s prime is the coordinate system where the first spaceship is at rest.

01:56 S prime has a relative velocity of s of minus 0 .8c, and the vertical axis is once again t prime, and the horizontal axis is x prime.

02:09 In the coordinate system s prime, so in the coordinate system where the spaceship is at rest, the first spaceship views the second spaceship traveling at a velocity of minus 0 .5c.

02:24 So we want to find out what does an amount, observer in the coordinate frame s view the second space, what velocity does an observer in the coordinate frame s view the second spaceship moving out? to do this, we introduce our lorentz transforms, where x prime is equal to gamma, the lorentz factor, multiplied by x minus u, where u is the relative velocity between s and s prime, and t prime is equal to gamma, t minus minus u v divided by c squared.

03:02 C is the speed of light, u is the relative velocity between s and s prime, and gamma is equal to the square root of 1 minus u squared over c squared.

03:28 So we're going to go to find the relative velocity of the second spaceship relative to earth.

03:35 We introduce the infinitesimal version of these lorentz transforms to get dx prime is equal to gamma x, x, no, dx minus u, dt, and dt prime is equal to gamma dt minus u dx over c squared.

03:57 And i apologize, i incorrectly wrote down the lorentz transform up here.

04:01 This should be u multiplied by x, no fee.

04:04 So we have two equations now, the first equation and the second equation.

04:12 I'm going to divide the first equation by the second equation to get dx prime dt prime.

04:19 Our gambas will cancel dx minus u dt, all divided by dt minus u dx over c squared.

04:30 Now, what is dx prime dt prime dt prime? well dx d t is equal to the velocity of an object in s oh sorry in s so by a similar analogy d t prime divided by d x prime is equal to v prime which is the velocity of an object in s prime so we have now what the velocity of the object is an s prime.

05:19 This is in our situation here, v prime is the relative velocity between the first spaceship and the second spaceship.

05:28 So relative velocity between first and second ship.

05:49 And to simplify this right -hand side of this equation here, we're going to simplify this.

05:53 Pull out a factor of d t from the numerator and the denominator...

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