Assignment - Vectors

Reading

Chapter 3

Objectives/HW

	The student will be able to:	HW:
1	Add or subtract vectors graphically and determine a vector's opposite.	1, 2
2	Calculate the components of a vector given its magnitude and direction.	3, 4
3	Calculate the magnitude and direction of a vector given its components.	5 - 9
4	Use vector components as a means of analyzing/solving 2-D motion problems.	10 - 13
5	Add or subtract vectors analytically (using trigonometric calculations).	14, 15
6	Define and apply “frame of reference” and use vector addition or subtraction as a means of solving relative motion problems.	16 - 20
7	State the horizontal and vertical relations for projectile motion and use the same to solve projectile problems and apply vector properties to projectile motion.	21 - 36

Homework Problems

Note:  Problems 1 and 2 should be solved by drawing and measuring vectors with ruler and protractor.  This is called “constructing the vectors” or “graphical method”.  You need not calculate the answer.

1. An airplane flies the following vectors (legs) of a flight:  26 km, 90° and then 62 km, 30°.  Use the graphical method to determine the resulting displacement of the plane.

2. Three forces act on point P.  Force A equals 80.0 N, 60.0°.  Force B equals 70.0 N, 0.0°.  Force C equals 40.0 N, 315.0°.  (a) Add the vectors in the order A +B + C by the graphical method.  (b) Draw and measure the sum of the same vectors but in a different order (e.g. C + A + B = ?) .  (c)  How do the solutions to (a) and (b) compare to one another?

Note:  All of the remaining problems should be solved by calculation and do not require construction with ruler and protractor.  A sketch of vectors does not have to be measured.  (The graphical method could be used to check your work if you so desire.)

3. Sketch each of the following vectors with their x and y components, and calculate the value of each component.

(a) 35.0 m/s, 70.0°             (b) 8.6 m, 135°                        (c) 842 km, 270.0^°

(d) 13.6 km/h, 0.0°            (e) 122 cm, 345.0°                   (f) 6.0 m/s², 250°

4. A boat travels on a heading of 40.0° north of east for a distance of 300 km.  How far north and how far east of its starting point does the boat sail?

5. Given the components below, sketch the vector and calculate its magnitude and direction.

(a) A_x = 34.0 m, A_y = -34.0 m                   (b) B_x = 0.0 km, B_y = 150.0 km

(c) C_x = -8.0 m/s², C_y = 6.00 m/s²             (d) D_x = -18.0 m/s, D_y = -15.0 m/s

(e) E_x = 2.6 cm/s, E_y = 7.4 cm/s                 (f) F_x = -10.0 mm, F_y = 0.0 mm 

6. A ship leaves its home port expecting to travel to a port 500 km due south.  Before it can move, a severe storm comes up and blows the ship 100 km due east.  How far is the ship from its destination?  In what direction must the ship travel to reach its destination?

7. A plane moves due north at 225 km/h and at the same time a wind carries it due east at 55 km/h.  Find the resulting velocity of the plane.

8. A weather station releases a weather balloon.  The balloon’s buoyancy accelerates it upward at 15 m/s².  At the same time, a wind accelerates it horizontally to the right at 6.5 m/s².  What is the resulting acceleration?

9. A descent vehicle landing on the moon has a vertical velocity toward the surface of the moon of 35 m/s.  At the same time, it has a horizontal velocity of 55 m/s.  (a) At what speed does the vehicle move along its diagonal descent path?  (b) At what angle with the vertical is this path?

10. An airplane and a helicopter leave an airport in Montana at the same time.  The aircrafts' velocities: v_a = 225 mph, 20.0°, v_h = 102 mph, 49.0°.  If these velocities are maintained which craft will cross the border into Canada first?  Or do they arrive at the same time?

11. The high noon sun shines straight down on an airplane that is climbing through the atmosphere at an angle of 8.0° from horizontal.  The plane's climb rate is 750 meters per minute.  This the rate at which its altitude increases.  (a) What is the speed of the airplane?  (b) What is the speed of the plane's shadow crossing the flat desert below?

12. A thingamajig initially 5.00 m, 90.0° from pt. A is displaced 4.00 m, 180.0°.  What is the thingamajig's final position?  Hint: d = s_f – s_i (displacement equals change in position).

13. An airplane with initial position 345 km, 180.0° from Knoxville travels to a position of 305 km, 270.0° from Knoxville in 2.00 hours.  (a) What is its displacement?  (b) What is its average velocity?

14. Perform the following vector operations using numerical calculation(s).  Make a sketch of the addition or subtraction for each problem.

(a) (5348 m, 90.0°) + (2001 m, 270.0°)
(b) (55 km/h, 0°) - (5.0 m/s, 180°)
(c) (18.0 m/s, 40.0°) + (12.0 m/s, 100.0°)
(d) (24 cm, 180°) + (12 cm, 270°)
(e) (18.6 m/s, 60.0°) - (19.4 m/s, 105.0°)

15. A car travels along a curved section of road.  Initially the car has velocity 30.0 m/s, 90.0°.  After 5.0 s the car has a velocity of 30.0 m/s, 95.0°.  (a) Calculate the change in the car's velocity.  (b) Calculate the car's average acceleration.

16. An airplane has airspeed 275 km/hr and heading 45.0°.  If the wind is 30.0 km/hr, 340.0°, what is the plane's velocity with respect to the earth?

17. A cruise liner must keep a velocity of 45 km/hr, 145° with respect to earth to stay on course and schedule.  If the ship encounters a current of 12 km/hr, 90°, what must be its speed through the water and heading?

18. A boat is traveling 3.8 m/s straight across a river 240 m wide.  At the same time the flow of the river carrying it downstream at 1.6 m/s.  (a) What is the resultant velocity of the boat?  (b) How long does it take the boat to cross the river?  (c) How far downstream is the boat by the time it reaches the other side?

19. A river flow toward 90.0°.  A riverboat pilot heads his boat at 297° in order to go directly across the river with velocity 6.0 m/s, 0.0°.  (a) What is the speed of the current?  (b) What is the speed of the boat through the water?

20. Choose and solve one of the following problems from your book, pp. 74 – 76:  42, 63, 65, 72

21. Ignoring air resistance, the equations you learned for projectiles accurately describe such things as baseballs, leaping athletes, javelins, bullets, etc.  However, an object with a trajectory that extends great distances would vary significantly from these equations (even if we still ignore air resistance).  Explain some reasons why this would be so.

22. A dud bomb is released from the bay of an aircraft during a practice maneuver.  Where will the aircraft be relative to the bomb when the bomb hits the ground, assuming the plane’s velocity remains constant?  Explain your answer.

23. A friend who is competing in the long jump asks you to explain the physics of this event.  Does the height of the jump make any difference?  What factors influence the length of the jump?  From a physics standpoint what strategies result in the longest jump?

24. A stone is thrown horizontally at 8.0 m/s from a cliff 78.4 m high.  How far from the base of the cliff odes the stone strike the ground?

25. A toy car runs off the edge of a table that is 1.225 m high.  The car lands 0.400 m from the base of the table.  (a) How much time does it take for the car to fall?  (b) What was the speed of the car as it rolled off the table?

26. An airplane traveling 1001 m above the ocean at 125 km/h is to drop a box of supplies to shipwrecked victims below.  (a) How many seconds before being directly overhead should the box be dropped?  (b) What is the horizontal distance between the plane and the victims when the box is dropped?

27. Divers at Acapulco dive from a cliff that is 61 m high.  If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear the rocks safely?

28. A dart player throws a dart horizontally at a speed of 12.4 m/s.  The dart hits the board 0.32 m below the height from which it was thrown.  How far away is the player from the board?

29. An arrow is shot at a 30.0° angle above horizontal ground.  It has an initial speed of 49 m/s.  (a) How high will the arrow go?  (b) What horizontal distance will it travel?

30. A baseball is hit and leaves the bat with velocity 30.0 m/s, 53.0°.  Immediately an outfielder runs 4.00 m/s toward the infield and catches the ball at the same height it was hit.  What was the original distance between the batter and the outfielder?

31. A human cannonball (a mime, in fact) is launched with initial velocity 25.9 m/s, 55.0°.  (a) Find the mime's velocity 2.50 s after launch.  (b) Using the cannon as a reference, find the mime's position 2.50 s after launch.

32. A bullet is fired from a gun atop a cliff 80.0 m above a plain below.  The muzzle velocity is 775 km/h, 30.0°.  (a) Find the range.  (b) Find the impact velocity.  (c) Find the maximum height above the plain.

33. Find the maximum range over level ground for a gun that has a muzzle speed of 345 m/s.

34. Watching a football game one day you hear the announcers say a 45 yard punt has a "hang time" (time in the air) of 4.0 s.  Determine the initial velocity of the football in mph.  (Try this one in “American units” where g = 32.2 ft/s², 1 mile = 5280 ft)

35. You are trying to rescue a fair princess trapped in a castle tower 25.0 m above the ground.  Because of a moat you can only get to within 30.0 m of the base of the tower.  You wish to catapult an Uzi to the princess in order that she may blast her way to freedom.  Calculate an initial velocity that will get the Uzi to the princess.  (There are an infinite number of correct responses!)  (But there are also an infinite number of incorrect responses!)  (Ain't that just infuriating?)  (If you want to be really impressive find a mathematical description for the set of initial speeds and angles that will solve the problem.)

36. Choose and solve one of the following problems from your book, pp. 72 – 76: 
21, 29, 39, 66, 73

Selected Answers - Vector Assignment

1. 78 km, 47°

2. a. 144 N, 16.5°

    b.

    c.

3. (in order: x,y)
    a. 12.0 m/s, 32.9 m/s

    b. -6.1 m, 6.1 m
    c. 0.0 km, -842 km
    d. 13.6 km/h, 0.0 km/h
    e. 117 cm, -31.6 cm
    f. -2.1 m/s², -5.6 m/s²

4. 193 km north, 230 km east

5. a. 48.1 m, 315.0°

b. 150.0 km, 90.0°

    c. 10 m/s², 143°
    d. 23.4 m/s, 219.8°

    e. 7.8 cm/s, 71°

    f. 10.0 mm, 180.0°

6. 510 km, 259°

7. 230 km/h, 76°

8. 16 m/s², 67°

9. a. 65 m/s

    b. 58

10. Hint: find components

11. a. 89.8 m/s

      b. 88.9 m/s

12. 6.40 m, 128.7° from A

13. a. 460 km, 318.5°

      b. 230 km/h, 318.5°

14. a. 3350 m, 90.0°

      b. 73 km/h, 0°

      c. 26.2 m/s, 63.4°

      d. 27 cm,207°

      e. 14.6 m/s, 349.6°

15. a. 2.62 m/s, 182.5°

      b. 0.523 m/s², 182.5°

16. 289 km/h, 39.6°

17. 39 km/h,159°

18. a. 4.1 m/s, 23° downstream

      b. 63 s

      c. 100 m

19. a. 12 m/s

      b. 13 m/s

20.

21.

22.

23.

24. 32 m

25. a. 0.500 s

      b. 0.800 m/s

26. a. 14.3 s

      b. 496 m

27. 6.5 m/s

28. 3.2 m

29. a. 31 m

      b. 210 m

30. 108 m

31. a. 15.2 m/s, 347.5°

      b. 43.4 m, 31.1° from cannon

32. a. 4.23 km

      b. 219 m/s, 328.4°

      c. 671 m

33. 12.1 km

34. 50 mph, 62°

35.

36.