How would you change the following Galilean transformation equations in the case where a frame...

Question:

How would you change the following Galilean transformation equations in the case where a frame was moving both in the {eq}x {/eq} and the {eq}y {/eq} direction? (Use the following as necessary: {eq}x, y, z, V_x, V_y, t {/eq})

{eq}x' = x ? V \ t \\ y' = y \\ z' = z \\ t' = t {/eq}

Galilean Transformations:

A Galilean transformation is a transformation of coordinates between two reference frames which move with constant speed one with respect to the other, valid within the limits of Newtonian mechanics where space and time are absolutes and velocities add as 3-vectors.

Answer and Explanation:

The Galilean transformation provided

{eq}x \to x' = x - V_x t\\ y \to y' = y\\ z \to z' = z\\ t \to t' = t {/eq}

represents the Galilean transformation between two reference frames (say S and S' ) with all three coordinate axes respectively parallel (no rotation) and with coincident origins at t = 0 (no translations), V being the velocity of S' with respect to S along the x-axis. If S' were to acquire a velocity {eq}V_y {/eq} along the y-direction, the transformation would become

{eq}x \to x' = x - V_x t\\ y \to y' = y - V_y t\\ z \to z' = z\\ t \to t' = t \, , {/eq}

since the motion along each spatial direction is completely independent on the motion along the other two spatial directions.

The new transformation can be obtained a composition of two consecutive transformations of the kind provided, the first one being for two systems S and S" moving along the x-direction with velocity {eq}V_x {/eq}

{eq}x \to x" = x - V_x t\\ y \to y" = y\\ z \to z" = z\\ t \to t" = t {/eq}

and the second one being between S" and S' with S ' moving along the y-direction of S" with velocity {eq}V_y {/eq}

{eq}x" \to x' = x"\\ y" \to y' = y" - V_y t"\\ z" \to z' = z" \\ t" \to t' = t" \, . {/eq}


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