USAAAO — 2017 First Round

The MIT students have found out about the plans of the Caltech students of sending a rocket to fly a parachute above the “Great Dome” of MIT. They know the rocket will follow an elliptical orbit of semiaxis a with the center of the Earth at one of its foci. To counter this prank, they will aim another rocket at the original rocket, which upon collision will transfer enough energy to the Caltech rocket to make it reach the escape velocity of the Earth at an early point in its trajectory and make it unable to reach back to the Earth’s surface again. What is the energy that needs to be transferred to the Caltech rocket in order for the MIT students to reach their goal? Assume that the Earth has mass M, and the Caltech rocket has mass m.