Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 1 / b

Table of contents
- Solution b

Solution

Put

u = 1/ r

. Since

h = r^{2} \dot{θ}

, the radial equation becomes the Binet equation

u^{''} + u = \frac{μ}{h ^{2}},

where primes denote derivatives with respect to the polar angle. Choosing the angular origin at periapsis gives

u = \frac{μ}{h ^{2}} (1 + e cos f) .

Writing the semi-latus rectum as

h^{2} / μ = a (1 - e^{2})

yields the Kepler orbit

r = \frac{a ( 1 - e ^{2} )}{1 + e cos f} .

Hence the specific angular momentum and specific orbital energy are

h = μ a (1 - e^{2}), C = - \frac{μ}{2 a} .

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version