Irodov Problem 1.210

We shall solve this problem exactly as we solved 1.209, using conservation of energy and angular momentum.

The angular momentum of the planet when it was at infinity is given by mv0l and its total energy is only its kinetic energy since at infinity it has no potential energy. Suppose that at the planet's closest approach its distance from the sun is r and its velocity is v, the sun's mass is Ms and the planet mass is m, then we have,

Conservation of angular momentum





Conservation of energy






From (1) and (2) we have,










Now we can ignore the solution with negative sign since a negative value for r does not make any sense.

Irodov Problem 1.209

The problem can be solved using the two basic conservation principles, i) conservation of angular momentum and ii) conservation of energy.

Let r be the distance of closest/farthest approach of the planet to the sun and v be its tangential velocity at this point. So the sum of the planet's kinetic energy and its potential energy must be conserved at all times. In other words,





Here m is the mass of the planet and Ms the mass of the sun.

The angular momentum of the planet is given by m v x r. Since angular momentum must be conserved, the planet's angular momentum at the closest/farthest approach will be exactly equal to its angular momentum when it is located at r0. In other words,






From (1) and (2) we have,














(3) has two solutions corresponding to the farthest (+ sign) and nearest (- sign) approaches in the planet.