Relative Motion - Galilean transformations - Relative velocity

Problem Statement:

The position vector of a particle is: r = 4 t i – 2 t + 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 + k  (m).

Calculate:

  1. The velocity vector of O’ with respect to O.
  2. 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.

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