- #1

CAF123

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## Homework Statement

A particle of mass m is moving in a repulsive inverse square law force ##\mathbf{F}(\mathbf{r}) = (\mu/r^2)\hat{r}##. Given that ##u(\theta) = -\frac{\mu}{mh^2} + A\cos(\theta - \theta_o)##,

1) Determine the paramters of the (far branch of the)hyperbolic orbit: $$\frac{\ell}{r} = -1 + \epsilon \cos(\theta - \theta_o)$$ in terms of ##m,\mu,A, h##.

2)The particle is projected at a distance a from the centre of force with velocity v at an angle ##\beta## wrt the radius vector corresponding to ##\theta = 0##. Use these initial conditions to find h, the semi-latus rectum ##\ell##, ##\epsilon, \theta_o##. Express your results in terms of ##v,\beta, a, m/\mu##.

## The Attempt at a Solution

1)Sub in the expression for u given (the eqn in u was part of a show that) to get $$-\frac{\ell \mu}{mh^2} + \ell A \cos(\theta-\theta_o) = -1 + \epsilon \cos(\theta-\theta_o)$$

I can then identify ##\frac{\ell \mu}{mh^2} = 1 \Rightarrow \ell = \frac{mh^2}{\mu}## and ##\epsilon = \ell A = \frac{mh^2 A}{\mu}##Is this what they mean by parameters?

2) Defining ##\beta## from the radius vector, I arrived at the following two expressions:$$

\mathbf{v} = \sin \beta \hat{\theta} - \cos \beta \hat{r}\,\,\,0< \beta < \pi/2 ,$$or$$\mathbf{v} = \sin \beta \hat{\theta} + \cos \beta \hat{r}\,\,\,\pi/2 < \beta < \pi$$

One IC could be ##u(\theta = 0) = 1/a = -\frac{\mu}{mh^2} + A \cos(\theta_o)##. To get another IC, I thought I could then take the derivative of u and equate the derivative to the radial component of ##\mathbf{v}## I obtained above. (but I have two different radial components depending on ##\beta##)

Many thanks.