**Problem Statement:**

The position vector of a particle with respect to a frame of reference at rest *O* is given by: **r** = 4 t^{2 }**i** – 2 t **j ** + **k **(m). Another frame of reference *O’* is moving at a constant velocity **V** = 5** i** with respect to *O*. Calculate:

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

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**Solution:**

First we are going to determine the velocity vector of the particle with respect to the frame of reference at rest *O* by deriving the position vector:

The frames of reference *O* and *O’* are both inertial and they are in constant relatice motion; therefore we will use the Galilean transformations to determine the velocity of the particle with respect to *O’*:

The relation between velocities is given by:

And substituting the values of **V** and **v** in the equation above we have:

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We derive **v** to calculate the acceleration of the particle with respect to *O*:

Next we derive **v’** to calculate the acceleration of the particle with respect to *O’*:

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

**Do not forget to include the units in the results of the problem.**