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Prove that if a space vehicle travels along a parabolic path with the Earth (or some other planet) at its focus, the total mechanical energy of the vehicle is zero.

Answer»


SOLUTION :Calculate first the total energy of the SPACECRAFT at the perigee (i.e. at the apex of the parabola): `W=(mv_0^2)/2-(gammaM)/r_0` .According to Newton.s second law `(gammamM)/r_0^2=(mv_0^2)/R_0` , where `R_0`is the radius of curvature at this point. Hence `(gammamM)/r_0=(mv_0^2r_0)/R_0`and the - To Ro total energy is `W=mv_0^2(1/2-r_0/R_0)`
But the distance from the FOCUS to the apex is` r_0=f=p//2` (SEE Problem 10.4), and the radius of curvature at this point is Ro=p (see Problem 3.13). Substituting these values we see that the total energy is zero. In compliance with the law of CONSERVATION of energy it will be zero at any other point of the path as well.


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