Friday, May 9, 2008

Irodov Problem 1.147

Suppose that the frame K' is one that has a velocity v. Then as seen from K' the velocity vectors of masses m1 and m2 will be v1-v and v2-v respectively. So the total kinetic energy of the two masses as seen from K' will be,

The extrema of E will be at dE/dv = 0, i.e.

Since v is a vectors, to show that this corresponds to the minima, we will have to show that the Hessian matrix of E with respect to v must be positive definite. The Hessian matrix of E is given by (m1 + m2) I which is positive definite. Hence v must be the minima.

The kinetic energy of the system as seen from the frame K' can be obtained by substituting v in E. The kinetic energy is given by,

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