1. Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 22

2. Quiz This is a one-dimensional problem. Suppose a particle is attracted to the origin with a force Find the potential function.

3. Work-energy theorem: Mechanical energy is conserved!

4. Examples Strategy: write down the total mechanical energy, E, E = KE + U at the initial and final positions of a particle:

5. Initial E 1 =KE 1 +U 1 …

6. Final E 2 =KE 2 +U 2

7. Then use or

8. H

9. Water Slide Who hits the bottom with a faster speed?

10. Roller Coaster You are in a roller coaster car of mass M that starts at the top, height H, with an initial speed V 0 =0. Assume no friction. a)What is the speed at the bottom? b)How high will it go again? c)Would it go as high if there were friction? H

11. Roller Coaster with Friction A roller coaster of mass m starts at rest at height y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

12. Conservative Forces If there are only conservative forces in the problem, then there is conservation of mechanical energy Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another –Good examples: Gravity and Springs Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost. –Good example: Friction (like on Roller Coasters)

13. Law of Conservation of Energy Mechanical Energy NOT always conserved If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc. Energy = Kinetic Energy + Potential Energy + Heat + Others… –Total Energy is what is conserved! K 1 +U 1 = K 2 +U 2 +E Heat…

14. Total Energy is what is conserved! K 1 +U 1 = K 2 +U 2 +E Heat …

15. A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude V m. What is magnitude of the velocity when the bullet returns to the ground? How high will it go?