Solving Path of Particle Subject to Central Force

[prismaticcore]

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: $$\frac{-k}{mr^5}$$ 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?

Homework Equations

The Attempt at a Solution

 

[jbsteele]

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 plus the potential 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. So that the velocity at infinity is v_0, and I can use the conservation of energy to find the velocity when the mass is a distance b/Sqrt[2] away, v_1 , as: v_1^2 = v_0^2 + 2k/mrThen I can use the conservation of angular momentum to find the relation between the radial coordinate and the angular coordinate, which is given by: Sqrt[k]/b = mr^2*dtheta/dt Integrating both sides with respect to time, I get: theta = Sqrt[k]/b * t + c where c is an integration constant which can be determined by the initial conditions. Now I can use the conservation of energy to relate the radial coordinate to the angular coordinate. To do this, I first write the energy equation as: E = \frac{1}{2}mv^2 - \frac{k}{mr} Integrating both sides with respect to the radial coordinate, I get: E*r = \frac{1}{2}mv^2r - kln(r) + c_1 where c_1 is another integration constant. Now, substituting the expression for the angular coordinate, I get: E*r = \frac{1}{2}mv_0^2r + \frac{1}{2