Question

3. We've primarily looked at closed orbits of planets, but a lot of what we've done can be used to treat projectiles and particle collisions under central forces. For example, a projectile of mass m is launched from a very large distance away with energy E toward a spherical planet of mass M m and radius R. Show that the maximum value of the angular momentum for the projectile to hit the planet is x = V27R(1 + ER/Y), GMm, and is the reduced mass. where E is the energy,

Answer 1

The angular momentum is maximum when the velocity just before the impact and the radius are perpendicular to each other. In this case the angular momentum is max = UVR3v = 'max UR The energy just before the pact is Since it is projected with energy E, from conservation of energy we get bet =E= = V2yRºE+241RY = - = {zur, ( RE +1) This is the proof. |