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prismaticcore
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Homework Statement
A particle of mass m moves from infinity along a straight line that, if
continued, would allow it to pass a distance b/Sqrt[2} from a point P . If the particle is
attracted toward P with a force which is: [tex]\frac{-k}{mr^5}[/tex] radially inwards. If the angular momentum is Sqrt[k]/b, show that the trajectory is given by r=b*coth(theta/Sqrt[2]).
Homework Equations
Angular momentum conservation
Energy conservation
Equation theta in terms of an integral over r
The Attempt at a Solution
Since this is a central force, angular momentum is conserved, so that I can find the velocity when the mass is a distance bSqrt[2] from point P. I will also need the total energy, having these two, I can plug it into the equation for theta in terms of an integral over r. The total energy is just the initial kinetic energy, however the velocity we use there must be different from the velocity of the mass when it's a distance bSqrt[2] away, right? In other words, I picture that the mass has a nonconstant velocity along the line from infinity to bSqrt[2] away from P. Is there something wrong with this reasoning? the particle is subject to a central force, with components of the force along its velocity, so it has to experience an acceleration. Why does the solution say we can just plug in v of the mass when it's a distance bSqrt[2] away as if it were the initial velocity when the particle is at infinity?