Rocket ship Able travels at 0.400 c relative to an Earth...

Rocket ship Able travels at $0.400 c$ relative to an Earth observer. According to the same observer, rocket ship Able overtakes a slower moving rocket ship Baker that moves in the same direction. The captain of Baker sees Able pass her ship at $0.114 c$. Determine the speed of Baker relative to the Earth observer.

Key Concepts

Special Relativity

Special relativity is a theory established by Einstein that redefines the concepts of space and time for observers in uniform motion relative to each other. It introduces the idea that the laws of physics are invariant in all inertial frames and that the speed of light in a vacuum is the same for all observers, regardless of their relative motion. This theory leads to phenomena such as time dilation and length contraction at speeds close to that of light.

Inertial Reference Frames

An inertial reference frame is one in which an observer is either at rest or moving at a constant velocity, meaning that no acceleration is present. In these frames, the laws of physics are uniform and can be applied consistently, making them ideal for analyzing problems using special relativity, such as calculating relative velocities between moving objects.

Relativistic Velocity Addition

Relativistic velocity addition is a formula used to combine velocities when dealing with objects moving at speeds that are significant fractions of the speed of light. Unlike classical addition of velocities, this formula ensures that the resultant speed never exceeds the speed of light, and it is given by (u + v)/(1 + uv/c²), where u and v are the velocities being combined. This concept is crucial for correctly analyzing the relative motion of objects in different inertial frames.

Lorentz Transformations

Lorentz transformations are the set of equations that relate the space and time coordinates of events as measured in different inertial frames. These transformations account for the effects of time dilation and length contraction and ensure the consistency of the speed of light across frames. They provide the mathematical framework for converting measurements, such as speeds and distances, from one inertial reference frame to another in the context of special relativity.

Transcript

00:01 Okay, so the question was asking us to determine the speed of baker that is relative to the observer.

00:06 While we know we can come out of such an equation, which is vab equal to vao plus bbo over 1 plus val times vbo over c square.

00:17 Okay, so bab is the speed of able that is relative to the baker, and val is the speed of able that is relative to the earth observer.

00:30 So our question is asking us to find out vbo, which is the speed of a baker that is relative to the earth observer.

00:37 So if we plugging the value of vab and vho, we'll have 0 .114 is equal to 0 .4 o 'o plus vbo over 1 plus 0 .4 .4 oc times v0 and then over c squared okay and you can move 1 plus 0 .4 oc times vb o over c square to the left side therefore we have 0 .114c times 1 plus 0 .4 oc times vb o over c square is equal to 0 .40 c plus bbo over c squared is equal to 0 .404 c plus bbo okay, if we expand it, we'll have 0 .114c plus 0 .114c times 0 .40c times vbo over c squared is equal to 0 .404c plus vbo.

02:40 And as you can tell on the left side, you can tell the c can be cancelled in this case.

02:44 Okay.

02:48 And then we'll have 0 .114c plus 0 .114 times 0 .40 times vbo is equal to 0 .40c plus vbo.

03:12 Okay, and then we could have, let's move vbo, ovbo on the left side, and then we'll have 0 .114 times 0 .40 times vbo minus vbo.

03:40 Okay, because i move the vbo from the right side to the left side.

03:45 And then it would be equal to 0 .404 .00c minus 0 .0...

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