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S02. Galilean Relativity - SOLUTIONS

 

Box \(\PageIndex{1}\)

Exercise 2.1.1: With the Galilean boost transformation, velocities add in a simple manner.  If \(u' = dx'/dt\) where \(x'(t)\) is the \(x'\) location of some particle, find \(u = dx/dt\) as a function of \(u'\) and \(v\).

Solution \(\PageIndex{1.1}\)

\[\begin{equation*}
  \begin{aligned}
        dx = dx' + vdt'
  \end{aligned}
\end{equation*}\]

dividing by \(dt\) gives

\[\begin{equation*}
  \begin{aligned}
        u= \frac{dx}{dt} = \frac{dx' + vdt'}{dt}
  \end{aligned}
\end{equation*}\]

and since \(dt = dt'\) this simplifies to

\[\begin{equation*}
  \begin{aligned}
        u = u' + v
  \end{aligned}
\end{equation*}\]

 

Box \(\PageIndex{2}\)

Exercise 2.2.1: Show that Newton's Law for a spring with spring constant \(k\) centered at \(x=x_c\), \( -k (x-\ell-x_c)  = m {\ddot x}\), is invariant under the Galilean transformation.

Solution \(\PageIndex{2.1}\)

Starting from the law in the unprimed frame and using the coordinate transformation we get

\[\begin{equation*}
  \begin{aligned}
        -k((x' + vt') - (x_c' + vt')) = m\ddot{x}'
  \end{aligned}
\end{equation*}\]

which simplifies to

\[\begin{equation*}
  \begin{aligned}
        -k(x' - x_c') = m\ddot{x}'
  \end{aligned}
\end{equation*}\]

thus it has the same form and is therefore invariant under the Galilean transformation.

It might be worth pointing out that \( d/dt = d/dt' \) and since \(x'\) only differs by a term linear in \(t\) from \(x\), their second derivatives with respect to time are the same (hence the right-hand side in the first line of equation in this solution).

 

Box \(\PageIndex{3}\)

Exercise 2.3.1: Discuss what this invariance means for applicability of Newton's laws in both the primed and unprimed frames, and the question: "is such a law consistent with the principal of Galilean relativity?"

Example \(\PageIndex{3.1}\)

Because Newton's laws are the same in both coordinate systems, one can not do an experiment to tell whether one is in the moving frame or the unmoving frame. All experiments, in either frame, will be consistent with Newton's laws, giving no indication of any motion with respect to some absolute rest frame.

Newton's laws -- and certainly the very specific one we examined in the above exercise -- are consistent with the principal of Galilean relativity. The principal of relativity is that all motion is relative. The Galilean boost leaves the form of the law invariant, guaranteeing that there is nothing special about either of the two frames; there is no objective meaning to a statement that one is moving and the other is not.