- #1

Kelli Van Brunt

- 11

- 3

- Homework Statement
- A particle with speed v0 and impact parameter b starts far away from a planet of mass M. Starting from scratch, find the distance of closest approach to the planet.

- Relevant Equations
- E = (mv^2)/2 + L^2/2mr^2 - GMm/r

Here were my assumptions: Energy and angular momentum are both conserved because the only force acting here is a central force. The initial angular momentum of this particle is ##L = mv_0b## and we can treat E as a constant in the homework equation given above. I solved for the KE (1/2 mv^2) in the above equation: $$\frac{mv^2}{2} = E + \frac{GMm}{r} - \frac{L^2}{2mr^2}$$

Then I set the derivative with respect to r equal to zero in order to find r at the instant at which KE is maximum, ie. the instant at which the particle is closest to the planet. This left me with: $$\frac{L^2}{mr^3} = \frac{GMm}{r^2}$$

When this is simplified and L replaced with the expression for L given above, I got my final answer, $$r = \frac{(v_0b)^2}{GM}$$

My book does not technically give a solution to this problem, but in the second part of the exercise, it asks to show that my expression for r is equivalent to ##\frac{k}{e+1}##, where ##k = \frac{L^2}{GMm^2}## and e = eccentricity of the orbit. Plugging in what I had for L, I got ##\frac{(v_0b)^2}{GM(e+1)}##. This is only equal to my answer if the eccentricity is zero, ie. if the orbit is circular, which is clearly not the case, since the particle is coming in from far away. Where did I make my mistake here? How would I implement the eccentricity of the orbit into my original solution "starting from scratch"?

Then I set the derivative with respect to r equal to zero in order to find r at the instant at which KE is maximum, ie. the instant at which the particle is closest to the planet. This left me with: $$\frac{L^2}{mr^3} = \frac{GMm}{r^2}$$

When this is simplified and L replaced with the expression for L given above, I got my final answer, $$r = \frac{(v_0b)^2}{GM}$$

My book does not technically give a solution to this problem, but in the second part of the exercise, it asks to show that my expression for r is equivalent to ##\frac{k}{e+1}##, where ##k = \frac{L^2}{GMm^2}## and e = eccentricity of the orbit. Plugging in what I had for L, I got ##\frac{(v_0b)^2}{GM(e+1)}##. This is only equal to my answer if the eccentricity is zero, ie. if the orbit is circular, which is clearly not the case, since the particle is coming in from far away. Where did I make my mistake here? How would I implement the eccentricity of the orbit into my original solution "starting from scratch"?