final_solutions

# final_solutions - Phys 701. Classical Mechanics. Final...

This preview shows pages 1–4. Sign up to view the full content.

Phys 701. Classical Mechanics. Final Exam. 2011. Open book: Goldstein’s textbook can be used. Other texts or notes are not permitted. Please submit a solution of three problems out of four. Choose any three problems you like. If you submit more than three solutions, I will choose the three to be graded. Please explain your reasoning. There will be no partial credit for unexplained calculations with wrong results. Problem 1 A satellite at the height h from the surface at the Earth has velocity v directed perpendicular to the radius connecting the satellite position with the Earth’s center. If v is small, the satellite will crash into the surface of the Earth. For large v it will orbit around the Earth. Find the minimal value of v required to avoid a crash. The radius of the Earth is R , the acceleration of gravity on the surface of the Earth is g . A B h R v Figure 1: for Problem 1. For small v the satellite follows trajectory A and crashes into the Earth. For larger v it follows trajectory B which does not cross the Earth’s surface. Solution: The minimal velocity corresponds to a trajectory that touches the surface in the point opposite to the starting point (see ﬁgure below). We write the energy and angular momentum conservation conditions, denoting the velocity at the touching point as v f , and using the fact that v f is also perpendicular to the radius. v ( R + h ) = v f R mv 2 2 - GMm R + h = mv 2 f 2 - GMm R 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Using g = GM/R 2 rewrite them as v ( R + h ) = v f R mv 2 2 - gmR 2 R + h = mv 2 f 2 - gmR Solving the equations together we obtain v 2 = g 2 R 3 (2 R + h )( R + h ) h R v f v 2
Problem 2 An artillery shell explodes at a height h above the ground. One piece ﬂies directly up with a velocity v 0 = 2 gh and the other directly down with the same velocity. The pieces are displaced form the vertical trajectory due to the Coriolis force. Find the distance

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/13/2011 for the course PHYS 701 taught by Professor Bazaliy during the Fall '11 term at South Carolina.

### Page1 / 7

final_solutions - Phys 701. Classical Mechanics. Final...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online