Understanding Galilean Transformation: A Troubleshooting Guide

[xiankai]

i have trouble taking the equations given, that is the conversion of one coordinate frame to another.

lets assume at the starting point there are two observers (coordinates (x,y,z,t)).

one observer moves in the x direction and the other observer stays still. the observer that moves has the coordinates of x',y',z',t' at any time.

the stationary observer sees an object in the x-direction, n units ahead. for the moving observer, at a particular instant of time, he sees it as n-vt ahead, where v is the relative velocity between the two frames of reference, and t is the time elapsed from the start till the instant.

as the coordinates of object are (x+n,y,z,t) or (x'+n-vt,y',z',t')

hence..

x+n = x'+n-vt

x = x' - vt

x' = x + vt

which isn't the result to be arrived at. can someone help me? i tried doing a search but it seems to elementary a problem to feature as a major stumbling block, couldn't find any topics on it. ><

 

[Doc Al]

the stationary observer sees an object in the x-direction, n units ahead. for the moving observer, at a particular instant of time, he sees it as n-vt ahead, where v is the relative velocity between the two frames of reference, and t is the time elapsed from the start till the instant.

as the coordinates of object are (x+n,y,z,t) or (x'+n-vt,y',z',t')

You seem to be confusing the meaning of x and x'. If the stationary observer sees the object at x = n, the moving observer sees the object at x' = x - vt = n - vt.

The coordinates of the object are simply (x, y, z, t) and (x', y', z', t'), related by the usual Galilean transformation.

 

[charng]

The Galilean transformation is a mathematical tool used to convert coordinates and measurements between different frames of reference. It is based on the principles of classical mechanics and assumes that the laws of physics are the same in all frames of reference. It is commonly used in physics and engineering to analyze motion and other physical phenomena.

It appears that you are having trouble understanding how to apply the equations for the Galilean transformation in a specific scenario. It is important to note that the equations you have provided are simplified versions of the full transformation equations, which may cause confusion.

To better understand the Galilean transformation, it is helpful to visualize the scenario you have described. Imagine two observers, one stationary and one moving at a constant velocity, with an object located at a specific distance from the stationary observer. The goal is to convert the coordinates of the object from the stationary observer's frame of reference to the moving observer's frame of reference.

The first step is to define the coordinate systems for each observer. Let's call the stationary observer's frame of reference (x, y, z, t) and the moving observer's frame of reference (x', y', z', t'). The object's coordinates in the stationary observer's frame of reference can be represented as (x+n, y, z, t), where n is the distance between the stationary observer and the object.

Next, we need to determine the coordinates of the object in the moving observer's frame of reference. This can be done by applying the transformation equations, which are:

x' = x + vt
y' = y
z' = z
t' = t

Using these equations, we can calculate the coordinates of the object in the moving observer's frame of reference:

x' = x + vt = (x+n) + vt
y' = y = y
z' = z = z
t' = t = t

So the coordinates of the object in the moving observer's frame of reference are (x'+n+vt, y', z', t').

It is important to note that the equations you provided in your question were missing the term for the distance n, which is why you were not arriving at the correct result. Additionally, the equations you provided were only for the x-coordinate, but the full transformation equations also include the y and z coordinates.

In conclusion, the Galilean transformation is a useful tool for converting coordinates and measurements between different frames of reference. It is important