PROBLEMS
1. Using the definition of kinetic energy and rest energy, how many times greater is the rest energy of a 1300 kg automobile than its kinetic energy while traveling at 40 m/s down the freeway?
2. Under what circumstances is it bad to describe kinetic energy as
3. As a ball falls under the influence of gravity, does gravity do positive work or negative work? Provide a mathematical explanation.
4. Set up the preceding problem starting from the definition of work as a path integral (with two vectors under the integral with a dot product), and evaluate it showing all steps. Assume the box falls from height to where
5. As I carry a box up a flight of stairs, am I doing positive work or negative work on the box? Provide a mathematical explanation.
6. In the previous problem, assume the path up the stairs can be described by y=0.5x. Start from the definition of work described as a path integral, and solve for the work that I do in lifting the box, starting at the origin, and ending 12.0m horizontally displaced from the starting point. Assume the box's mass is 10kg.
7. As a box is lifted against gravity and placed on a shelf, how does the work done by the lifter compare with the work done by gravity? What is the net work done on the box? What does this imply about its change in kinetic energy? Use definitions and mathematics from this chapter to answer these questions.
8. An applied horizontal force of 100 N pushes against a shopping cart that is initially at rest. Assume this is the only force acting horizontally. Over what distance must this force act in order for the 40 kg shopping cart to reach a speed of 2.5 m/s? You must use concepts from this chapter and not kinematics from an earlier chapter.
9. How does the previous answer change if there is a friction force of 20N that opposes the 100 N applied force? Again, use concepts from this chapter, and not kinematics.
10. What power is required (at the wheels) for a 1400 kg automobile to climb a 4% grade at a constant speed 30 m/s while it is opposed by drag and rolling resistance forces totaling 500 N?
11. If the car in the previous problem increases its power output by 10% (by pressing the gas pedal farther down), at what rate will the car accelerate? Hint: Consider the net force.
12. What is the angle between two unit vectors if their dot product is 0.5?
13. Given and , what is the scalar projection of on The idea is to find how much of the force vector is directed along the displacement vector.
14. What relative orientation between two vectors leads to a maximum value for the dot product? Minimum? Zero?
15. Given a vector field that represents the force of the wind on a sailboat, what is the work done by the wind on a boat traveling along a path defined by from a position (1,1) to a position (5,25)? Assume positions are given in kilometers and that force is in newtons. Note: To make units easy, a newton times a meter is a joule. A newton times a kilometer is a kilojoule (kJ). But F returns newtons.
16. What fuel economy should be expected from a gasoline powered car that encounters a total of 443N of resistive forces while driving down the road? (Those forces are from air drag, rolling resistance and bearing
losses.) Assume a 30% thermodynamic efficiency.
17. What would the MPGe of an otherwise identical vehicle be if running on an electric motor, if you pay 15 cents per kilowatt-hour to charge it? Assume the price of gasoline is $2.79/gallon.
Answers
1. 1.125x1014 times greater.
2. When v is within 10% speed of light.
3. Positive work
4. where is negative.
5. positive work
6. 600 J
7. equal magnitude but opposite signs; zero; zero
8. 1.25m
9. 1.56m
10. 31.8kW
11. 0.076 m/s2
12. 60 degrees
13. 15.7 N.
14. Parallel, anti-parallel, perpendicular
15. 3.77 MJ
16. 50.5mpg
17. 93.9mpge