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## Homework Statement

I'm working my way through

*Classical Mechanics*by J.R. Taylor. I'm stumped by the one-star ("easiest") Problem 5.11: "You are told that, at known positions x1 and x2, an oscillating mass m has speeds v1 and v2. What are the amplitude and angular frequency of the oscillations?"

## Homework Equations

Taylor gives the following equations for simple harmonic motion:

5.5 x(t) = C[itex]_{1}[/itex]e[itex]^{iωt}[/itex] + C[itex]_{2}[/itex]e[itex]^{iωt}[/itex]

5.6 x(t) = B[itex]_{1}[/itex]cos(ωt) + B[itex]_{2}[/itex]sin(ωt)

5.11 x(t) = Acos(ωt - [itex]\delta[/itex])

5.14 x(t) = Re[Ae

^{i(ωt - [itex]\delta[/itex])}]

Taylor gives relationships among the constants A, B's, and C's.

## The Attempt at a Solution

Solutions are given at the back of the book:

A = sqrt( (x[itex]_{2}[/itex][itex]^{2}[/itex]v[itex]_{1}[/itex][itex]^{2}[/itex] - x[itex]_{1}[/itex][itex]^{2}[/itex]v[itex]_{2}[/itex][itex]^{2}[/itex]) / (v[itex]_{1}[/itex][itex]^{2}[/itex] - v[itex]_{2}[/itex][itex]^{2}[/itex]))

ω = sqrt( (v[itex]_{1}[/itex][itex]^{2}[/itex] - v[itex]_{2}[/itex][itex]^{2}[/itex]) / (x[itex]_{2}[/itex][itex]^{2}[/itex] - x[itex]_{1}[/itex][itex]^{2}[/itex]) )

Equations 5.6 and 5.11 immediately give positions at times ωt[itex]_{1}[/itex] and ωt[itex]_{2}[/itex]; differentiating those equations gives speeds. It's easy to get the amplitude A in terms of ωt, e.g.

A = x[itex]_{1}[/itex]/cos(ωt[itex]_{1}[/itex]).

But I can't figure out how to get A in terms of the x's and v's. I don't see how to separate ω from the t's, or how to get ω out of the trig functions.