# Coordinates transformation and falling body trajectory

As it's well known, assuming the earth fixed and non rotating, the trajectory of a falling body with initial speed $v_0 = \{v_{0x},v_{0y},{v_{0z}}\}$ is contained in a plane. Assuming an observer in the origin of a fixed reference frame, he will measure three coordinates of the falling body: $P=\{\rho,\theta,\phi\}$. We can suppose to be $\rho$,distance, $\theta$, elevation and $\phi$, azimuth. How is it possible to find a reference frame in which the azimuth $\phi$ is zero? Thanks in advance.

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The observer has to be in the plane of the motion. The surfaces where $\phi$ is constant are planes; the observer just has to orient her axes appropriately for one of these planes to be the plane of motion.

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The observer isn't in the plane of the motion. The question is: what is the transormation matrix from the observer reference frame to the frame in which the azimuth doesn't change in time? – Riccardo.Alestra Oct 19 '12 at 15:06
@Riccardo: If by "transformation matrix" you mean only a rotation without translation, the answer is that there is no such transformation. The surfaces of constant azimuth are planes through the origin; thus, for the azimuth to remain constant, the motion has to occur in a plane through the origin, that is, the origin has to be in the plane of motion; and "observer" is just a confusing synonym for "origin". – joriki Oct 19 '12 at 15:24
I mean rotation and translation. – Riccardo.Alestra Oct 19 '12 at 15:27
Obviously there are infinite reference frames in which the condition: $\phi=0$ is satisfyed. One of this, for example, could be the reference frame of the impact point. – Riccardo.Alestra Oct 19 '12 at 15:39
@Riccardo: Then the problem is underspecified -- you need some point on the trajectory. – joriki Oct 19 '12 at 15:40