Relative Motion - Galilean transformations

Problem Statement:

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

  1. The velocity vector of the particle with respect to both frames of reference.
  2. 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.

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