AP Physics 1 Assignment – Vectors

Reading  Chapter 3, Open Stax College Physics; Chapter Sections 4.1, 4.5, Etkina et.al.

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	Use vector addition or subtraction as a means of solving relative velocity 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 - 38

Homework 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 answers.

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)  Are the answers to (a) and (b) the same?  Should they be?  Explain.

Note:  All of the remaining homework problems should be solved by calculation and do not require construction with ruler and protractor.  Remember to show your work as usual!!!

3. Sketch each of the following vectors with its x and y components, and calculate the value of each component.  (A sketch of vectors does not have to be measured.) 
(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 course 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 thingamajig initially 5.00 m, 90.0° from a three-pronged widget is displaced 4.00 m, 180.0°.  What is the thingamajig's final position relative to the three-pronged widget?

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. 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.  Find its average velocity.

10. After opening her parachute at a height 1200 m, a skydiver has constant velocity 6.0 m/s, along a diagonal path tilted 20° from vertical (as she is carried by a steady horizontal wind).  (a) Find the time to reach the ground.  (b) Determine the speed of the wind. 

11. 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?

12. 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?

13. A policeman is sitting at a railroad crossing as a train goes by along a track that forms a 75° angle with the roadway.  The radar gun pointing forward in his car indicates that the train is moving away at speed 11 mph.  (a) What is the actual speed of the train along the track?   (b) Would the radar gun ever read zero when a train is going across the road?  Would it ever read a value greater than the actual speed?  Explain.

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° through a wind of 30.0 km/hr, 340.0°.  (a) Find the airplane’s groundspeed.  (b) Determine the displacement of the airplane over the earth in 5.00 minutes.

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. An airplane capable of airspeed v makes a roundtrip flight between two cities, one located a distance d directly north of the other.  The wind blows steadily in one direction as the airplane travels there and back.  Is the total time to make this trip affected by wind?  Discuss and explain.  OR, if you prefer, derive an expression for the total trip time in terms of v, d, and u, where u = speed of the wind (assume a wind direction either northward or eastward).

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 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?

23. 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?

24. 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?

25. An airplane traveling 101 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?  (c) With what speed does the box hit the water?

26. A scene in a movie shows a car drive straight off the edge of a cliff, fly through the air for 2.0 s, and then burst into flames on impact.  The car lands at a position 40.0 m from the base of the cliff.  (a) Find the impact speed.  (b) Find the magnitude of the car’s displacement.

27. The bullet from a gun is fired horizontally with speed v toward a target that is a distance d away. Before hitting the target the bullet drops an amount y below the point at which the gun is aimed.  (a) Derive an expression for y in terms of v and d and physical constants, as appropriate.  (b) Calculate y for a bullet with speed 380 m/s fired at a target 33 m away.

28. 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?

29. 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?

30. 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.

31. 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.

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

33. 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)

34. Choose and solve one of these problems from either text:
Open Stax College Physics Problems & Exercises found at the end of chapter for section 3.4
Etkina, pp. 114 – 116, Chapter 4: 58, 64, 65, 66, 69, 86

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 also an infinite number of incorrect responses!)  (If you want to be really impressive derive an equation for any correct initial speed v₀ in terms of the initial angle θ.)

36. A train is traveling west at a constant 14.0 m/s.  A mouse is in the train, initially at rest relative to the floor of the boxcar in which it is riding.  The mouse is against the south wall of the boxcar and is 2.50 m across from an open door in the north wall.  The mouse accelerates 0.500 m/s² northward relative to the floor and flies out of the open door.  (a) Find the velocity of the mouse at the instant it goes out the door.  (b) Supposing the mouse drops vertically 1.50 m before hitting the ground, determine its impact speed.

37. NASA’s microgravity program uses a specially modified jet airplane to simulate zero-g conditions.  To do this the airplane flies in such a way that the objects inside the aircraft have zero acceleration relative to the plane’s body.  In truth, the objects inside the plane are still being affected by gravity and accelerate relative to the earth the same as any object in freefall.  The zero-g conditions last for 25 s and begin with the plane climbing at a 45° angle and end with the plane descending at a 45° angle.  (a) What must be the acceleration of the airplane relative to earth in order to achieve zero-g conditions inside?  (b) Describe the type of path the airplane must follow.  (c) Determine the speed of the airplane at the beginning of the zero-g conditions.  (d) Determine the amount of “airspace” needed for the maneuver – i.e. the range and change in elevation the airplane will undergo.

0.500 s

4.54 s

63 s

210 s

10.0 mm, 180.0°

3.7 cm

27 cm, 207°

117 cm, -31.6 cm

-6.1 m, 6.1 m

31 m

32 m

43.4 m, 31.1° from cannon

45 m

48.1 m, 315.0°

100 m

108 m

158 m

210 m

496 m

671 m

range: 3.1 km, elevation: 770 m

3350 m, 90.0°

4.23 km

12.1 km

24.1 km, 39.6°

78 km, 47°

150.0 km, 90.0°

193 km north, 230 km east

460 km, 318.5°

510 km, 259°

0.0 km, -842 km
7.8 cm/s, 71°

0.800 m/s

2.1 m/s

2.62 m/s, 182.5°

13.6 km/h, 0.0 km/h
4.1 m/s, 23° downstream

39 km/h, 159°

12 m/s

12.0 m/s, 32.9 m/s

13 m/s

14.1 m/s, 173.6°

14.6 m/s, 349.6°

15.1 m/s

15.2 m/s, 347.5°

73 km/h, 0°

50 mph, 62°

26.2 m/s, 63.4°

28 m/s

56.4 m/s

230 km/h, 318.5°

65 m/s

289 km/h

88.9 m/s

89.8 m/s

170 m/s
219 m/s, 328.4°

0.523 m/s², 182.5°

-2.1 m/s², -5.6 m/s²

9.8 m/s², down

10 m/s², 143°

16 m/s², 67°

144 N, 16.5°