Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9

1. You are the technical advisor for the David Letterman Show. Your task is to design a circus stunt in which Super Dave, who weighs 750 N, is shot out of a cannon that is 40o above the horizontal. The “cannon” is actually a 1m diameter tube that uses a stiff spring to launch Super Dave. The manual for the cannon states that the spring constant is 1800 N/m. The spring is compressed by a motor until its free end is level with the bottom of the cannon tube, which is 1.5m above the ground. A small seat is attached to the free end of the spring for Dave to sit on. When the spring is released, it extends 2.75m up the tube. Neither the seat nor the chair touch the sides of the 3.5m long tube, so there is no friction. After a drum roll, the spring is released and Super Dave will fly through the air. You have an airbag 1m thick for Super Dave to land on. You know that the airbag will exert an average retarding force of 3000 N in all directions. You need to determine if the airbag is thick enough to stop Super Dave safely – that is, he is slowed to a stop by the time he reaches ground level. Consider the seat and spring to have negligible mass. Ignore air resistance.

2. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16 Question 17 Question 18 Question 19 Paired Problem 1

3. 1. Which of the following physics principles are most suited to solve this problem? • A : i only • B : ii only • C : iii and iv only • D : all of the above i) Kinematical considerations ii) Linear momentum conservation iii) Mechanical energy conservation iv) Work-Energy Theorem

4. This is a tedious way because it involves vectors. Also, it only applies to the “projectile motion” part of the problem. Choice: A Incorrect

5. There is only a collision at the airbag. We need to analyze other parts of the problem. Considering the energy of the system at different points will be more helpful. Choice: B Incorrect

6. Using these principles makes the problem easier to solve. The work-energy theorem is useful in situations where you need to relate a body’s speed at two different points in its motion. The energy approach is useful when a problem includes motion with varying forces along a curved path. However, conservation of total mechanical energy requires that only conservative forces do work. Choice: C Correct

7. Choice: D Incorrect All of these principles are applicable. We are asked to find the easiest way. Since we are not dealing with time explicitly, we should be applying the work-energy theorem and conservation of mechanical energy.

8. 2. Which form of energy does the spring-Dave-earth system possess just before the spring is released? Assume the reference height is the floor. • A : i only • B : ii only • C : iii only • D : ii and iii only i) Kinetic Energy (K) ii) Spring Potential Energy iii) Gravitational Potential Energy

9. There is no kinetic energy before the spring is released because Super Dave has zero initial velocity (vo=0). Choice: A Incorrect

10. This is one type of potential energy (PE) associated with this system, but there is another type of PE associated with this system as well. Remember that Super Dave is displaced vertically before the spring is released. Choice: B Incorrect

11. This is one of the forms of potential energy (PE) associated with this system, but there is another type of PE associated with this system as well. Remember that the spring is initially compressed. Choice: C Incorrect

12. Choice: D Correct The system has gravitational and spring potential energy, because Super Dave begins above the floor (where PE=0) and the spring is initially compressed.

13. 3. Which of the following conditions are required for the Law of Conservation of Mechanical Energy to hold for a system?-U is elastic potential energy. -The subscripts i and f stand for initial and final.                                      -W is work done by non-conservative forces                                     -E is total mechanical energy. • A: The work done by non-conservative forces must be zero. • B: Energy is not created or destroyed, but can change forms. • C: • D: Ef=Ei or (Uf+Kf=Ui+Ki)

14. Choice: A Correct The other choices are either statements of conservation of total mechanical energy (not the requirements) or are incorrect.

15. Choice: B Incorrect This is how the Law can be expressed in words, but there are more requirements.

16. Choice: C Incorrect This is the mathematical expression of the Law, but there are more requirements.

17. Choice: D Incorrect This is the same as choice C, just in a different form. There are more requirements.

18. A: Ei=mgh 4. Since the gravitational and elastic forces are conservative, and we ignore air resistance, we can apply the Law of Conservation of Mechanical Energy to this system. How do we (mathematically) express the initial mechanical energy of the system before the spring is released? h=height x=compression of the spring k=spring constant g=gravitational acceleration (9.8m/s2) m= Dave’s mass B: C: D: Ei=0

19. Think about the spring, which also adds to the initial mechanical energy since it is compressed. Choice: A Incorrect

20. What about the gravitational potential energy? Recall that the system begins above the floor. Choice: B Incorrect

21. Gravity and the compressed spring contribute to total mechanical energy. These forms of energy are expressed correctly. Choice: C Correct

22. Choice: D Incorrect Kinetic energy is the only type of mechanical energy that is initially zero.

23. 5. What happens to the energy that was stored in the spring right after the spring is released and Super Dave is launched? • A: It transforms to gravitational PE only. • B: It transforms to K only. • C: Some of it transforms to gravitational PE and some to K. • D: It is lost.

24. Dave gains kinetic energy as the spring is released, because he gains a non-zero velocity.Recall : where v is Super Dave’s velocity. Choice: A Incorrect

25. Since Super Dave is shot at an angle  above the horizontal, he gains height which increases his gravitational PE (mgh). Choice: B Incorrect

26. The elastic PE transforms partly to gravitational PE and partly to kinetic energy, because Super Dave gains height and velocity. Choice: C Correct

27. Choice: D Incorrect Since we are ignoring air resistance, there is no non-conservative force involved. Therefore, there is no energy loss.

28. 6. Dave’s energy right at the instance of impact with the airbag consists of which form of energy?: • A: only K • B: only gravitational PE • C: both gravitational PE and K • D: none of the above

29. Choice: A Incorrect The top of the airbag is 1m above the floor, so there should be some gravitational potential energy at the point where Super Dave first makes contact with the airbag.

30. Choice: B Incorrect If Super Dave had no kinetic energy, we wouldn’t have to worry about him getting hurt. Super Dave is traveling with a non-zero velocity. Therefore, he has potential energy.

31. Choice: C Correct Dave’s energy at this point consists of both kinetic energy and gravitational PE, because he is traveling with a non-zero velocity and is still displaced vertically.

32. Choice: D Incorrect He has both kinetic energy and gravitational PE. Remember that he is traveling with a non-zero velocity and is not quite at floor level at this point.

33. 7. Can we still use the principle of mechanical energy conservation after Dave hits the air bag? • A: yes • B: no • C: we don’t have enough information to decide

34. Since the airbag softens Dave’s landing, there is a retarding force, which is a non-conservative force. Therefore, we can not use the principle of conservation of mechanical energy. Choice: A Incorrect

35. The retarding force (non-conservative force), which comes from the airbag resisting Dave’s motion, does work. Thus, we can not use this principle. Choice: B Correct

36. We do have enough information. The airbag causes a non-conservative force to do work on Dave. This retarding force disallows the use of the principle of mechanical energy conservation. Choice: C Incorrect

37. 8. After the impact with the airbag, which physics principles should we use to solve the problem? • A: kinematics • B: work-energy theorem • C: Impulse-Momentum Theorem • D: all are possible ways

38. Kinematical considerations will not be useful here. Choice: A Incorrect

39. Choice: B Correct Using the work-energy theorem will be the best method to solve the problem.

40. Choice: C Incorrect This method is not useful due to insufficient information.

41. Choice: D Incorrect Kinematics and the impulse-momentum theorem will not be helpful here. This problem can be solved by the work-energy theorem.

42. A: 9. Let’s put the concepts that we have considered in questions 1-8 into mathematical form. Which of the following equations correctly describes our application of conservation of total mechanical energy from the point just before the release of the spring to the point just before impact? B: C: D:

43. There is gravitational PE at the points just before release and just before impact, because Dave is above floor level (which we are defining as the zero of gravitational potential energy), in both cases. Choice: A Incorrect

44. What about the gravitational PE just before impact when Dave is still at least 1 m above the floor? Choice: B Incorrect

45. This expression contains all of the forms of energy that are involved at these two points. Choice: C Correct

46. Choice: D Incorrect In addition to the gravitational PE just before release, remember that the compressed spring stores elastic potential energy which can change into other forms of mechanical energy.

47. 10. From the equation we found in question 9, which one of the following expressions is true for v2 after simplification? • A : • B : • C :

48. Choice: A Incorrect Check your algebra. Pay attention to the mass and the factor of 2.

49. Choice: B Correct Correct algebra.

50. Choice: C Incorrect Spring PE is set to zero in this expression, which is incorrect.