**Problem Statement:**

The position vector of a particle is: **r** = 4 t **i** – 2 t^{2 }**j ** + **k **(m) with respect to a frame of reference at rest *O*. The position vector of the same particle with respect to another frame of reference *O’* moving at constant velocity with respect to *O* is:**r’** = 8 t **i** – 2 t^{2 }**j ** + **k **(m).

Calculate:

- The velocity vector of
*O’*with respect to*O*. - The acceleration of the particle with respect to both frames of reference.

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**Solution:**

We are going to use the Galilean transformations to calculate the velocity vector of *O’* with respect to *O*. First we determine the velocity vector of the particle with respect to both frames of reference:

The Galilean transformation for the velocities is given by:

If we substitute **v** and **v’** we can isolate the velocity vector of *O’* with respect to *O*:

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The acceleration vector for eache frame of reference is the derivative of the velocity vector for each one of them:

Since both are **inertial**, the particle has the same acceleration for the two frames of reference.