Consider a meter stick at rest in a reference frame F. It...

Consider a meter stick at rest in a reference frame $F$. It lies in the $x, y$ -plane and makes an angle of $37^{\circ}$ with the $x$ -axis. The reference frame $F$ then moves with a constant velocity $v$ parallel to the $x$ -axis of another reference frame $F$. a) What is the velocity of the meter stick measured in $F$ at an angle $45^{\circ}$ to the $x$ -axis? b) What is the length of the meter stick in $F^{\prime}$ under these conditions?

Consider a meter stick at rest in a reference frame $F$. It lies in the
$x, y$ -plane and makes an angle of $37^{\circ}$ with the $x$ -axis. The reference frame $F$ then moves with a constant velocity $v$ parallel to the $x$ -axis of another reference frame $F$.
a) What is the velocity of the meter stick measured in $F$ at an angle $45^{\circ}$ to the $x$ -axis?
b) What is the length of the meter stick in $F^{\prime}$ under these conditions?

Key Concepts

Inertial Reference Frames

In special relativity, inertial reference frames are coordinate systems in which observers are either at rest or moving at a constant velocity. They form the basis for analyzing and comparing physical events from different perspectives, ensuring that the fundamental laws of physics hold true in every such frame.

Lorentz Transformations

Lorentz transformations are the set of equations used to convert the space-time coordinates of events as observed in one inertial frame to those in another moving at a constant velocity relative to the first. These transformations introduce corrections for time dilation, length contraction, and the relativity of simultaneity, providing the mathematical framework for all relativistic effects.

Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object, as measured by an observer moving relative to the object, is shorter than the length measured in the object's rest frame. This contraction occurs in the direction of relative motion and is a direct consequence of the Lorentz transformations.

Relativistic Velocity Transformation

Relativistic velocity transformation involves converting velocity components from one inertial reference frame to another, taking into account the effects of time dilation and length contraction. This concept explains why different observers may measure different speeds and directional components for the same moving object, particularly when these speeds are close to that of light.

Relativity of Simultaneity

The relativity of simultaneity is the principle that events deemed simultaneous in one inertial frame may not be simultaneous in another frame moving at a constant velocity relative to the first. This concept is crucial when measuring lengths and velocities of objects in different frames, as it affects the timing and spatial separation of events recorded during the observation process.

Transcript

00:01 In this question, we have a metre stick within a reference frame called f.

00:06 It lies in the xy plane, making an angle of 37 degrees within the x axis.

00:11 The reference frame f moves with a constant velocity v parallel to the x axis of another reference frame x prime.

00:18 We want to find the meter stick's velocity measured in the f prime frame at an angle of 45 degrees to the x axis, and we want to find the length of the meter stick in f prime under these conditions.

00:30 We know that the length is equal to the proper length over gamma, where gamma is the lorentz factor and is equal to 1 over the square root of 1 minus v squared over c squared.

00:43 So as the object is moving along the x axis, this y component will not change, but there will be a change in the x component as a result of this equation.

00:53 So what we need to do is we need to know that both of the y components are the same in both reference frames.

00:59 So, ly is equal to ly prime.

01:06 And as a result, l sine theta is equal to l prime, sine theta prime.

01:20 And now we want to look at the x coordinate.

01:23 So we know that the coordinate in the x prime reference frame is equal to l prime cos theta prime.

01:33 And from the above expression, we can see that l prime is equal to l sine theta over sine theta prime.

01:46 So we can substitute this into here.

01:49 And we can say that lx prime is equal to l sine theta over sine theta prime multiplied by cos theta prime...

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