# Vectors - Council Rock School District

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Day #1: Intro to vectors, graphically adding vectors

Adding vectors at right angles and finding magnitude and direction

HW: “Day #1 Vectors HW Problems” (from this packet)

Day #2: Displacement and Average Velocity in 2D (no crooked vectors yet)

Vector Components (for pure component and crooked vectors)

Using Velocity Vectors in problem solving

HW: “Day #2 Vectors HW Problems” (from this packet)

Do Buffa, pp. 95-96, problems 4, 10, 11, 14, 20

Day #3: Adding crooked vectors (both graphically and with the component-method)

HW: “Day #3 Vectors HW Problems” (from this packet)

Do Buffa, pp. 95-96, problems 21, 22, 23, 25- 28

Day #4: Equilibrium Problems and F = (3 x + 4 y ) notation

HW: “Day #4 Vectors HW Problems” (from this packet)

Do Buffa, pp. 96-97, problems 17, 30, 44, 45

Day #5: Average Velocity, and Average Acceleration in 2D

HW: “Day #5 Vectors HW Problems” (from this packet)

Do Buffa, pg. 97, problems 40, 47, 48, 49, 53

Day #6: Relative Velocity A

HW: “Days #6 & #7 Relative Velocity HW Problems” 29-40 (from this packet)

Day #7: Relative Velocity B

HW: “Days #6 & #7 Relative Velocity HW Problems” 41 - 46 (from this packet)

HW: Do Buffa, pg. 101, problems 102, 105

Day #8: Review A

HW: “Day #8 Vectors HW Problems” (from this packet)

Ultimate River Problem Worksheet

Ultimate Plane Problem Worksheet

Day #9: Review B

Day #10: Unit 3 Test

Answers to BUFFA book problems (pp. 95-98)

|4) no, no |40) 16 m/s at 79o above the –x-axis |

|10) 6.0 m/s, 3.6 m/s |44) 3.04 N at 9.46o below +x-axis |

|11) N of East (2), 1.1 E 2 m, 27o N of E |45) 8.5 N at 21o below –x-axis |

|14) 2.5 m at 53o above +x-axis |47) parallel 30 N, perpendicular 40 N |

|17) 56.3o below the horizontal (BH), |48) _____ |

|18.0 m/s |49) 27 m at 72o above –x-axis |

|20) 33.5 m/s @ 10.5o above the horiz (AH) |53) 42.8 south of west, 0.91 m, |

|32.93 m/s, 6.1 m/s |might travel in a curve |

|21) C |54) 242 N at 48o below –x-axis |

|22) D |86) B |

|23) D |87) D |

|25) yes, when the vector is in the y-direction |88) D |

|26) no |94) -30 km/h, + 30 km/h |

|27) yes, if equal and opposite |98) 4.0 min |

|28) yes, all are equal |99) 146 sec = 2.43 min |

|30) yes, ______ |102) also increases (1), 4.7 m/s |

|31) 4.9 m, 59o above –x-axis |105) 1.21 m/s |

|32) 35 m, 5m, -5m | |

|33) 113 mi/h | |

|34) 145 N at 50.1o N of E | |

|36) mag. doubles but dir. stays same (1) | |

|30 m at 45o | |

|39) -4 x + 8 y, 8.9 m/s | |

A vector is a quantity with both _____________ (a size or number) and a _____________ (which way it is pointing). A scalar is a quantity with ___________ only.

For example, Velocity is a vector v = 40 m/s [North 30o EAST].

Speed is a scalar s = 40 m/s.

How are vector “directions” given?

When vectors are drawn “graphically”, notice that the length of the arrow used is _____________ to the magnitude of the vector (40 m/s is twice as long as 20 m/s). The table at the right lists some common vectors and scalars.

Vectors can be added or subtracted. When they are, the result is another vector, known as the __________________ vector. We always add vectors __________________. To subtract vectors, we simply add the opposite.

Consider a person walking 40 km North and then 60 km East.

To determine the displacement of the walker, add the two vectors ____________. The “resultant” will be from the start to the finish of the added vectors. This is known as the resultant vector, “R.” I usually EMBOLDEN this vector so that it stands out.

The distance covered by the walker is _______. Note that the distance is a ________ and does not get a direction, only a magnitude. The displacement is another story. Using the ___________________, we can determine the length of the displacement (resultant). It will come out to _____________________. This is the magnitude of the displacement vector. Now we need to find the direction of the new vector. We will always describe the angle in which the vector is pointing in the ____________corner of the drawing. This is where the tail of the resultant meets a tail of one of the original vectors. This angle gives the direction of the new “Resultant” vector.

θ

Find the value of theta in the space below.

If you got .9827 for your answer, your calculator is in ___________ mode. Be careful of this. We will report this answer for displacement as follows:

______________________________________

This has now, both magnitude and direction. It is a complete vector. The walker was displaced this much and in this direction.

Vectors may also be used to describe a summation of velocities. Consider if you jumped into a river with a current flowing to the right at 2 m/s and you swam pointing directly to the other shore, moving through the water at 6 m/s. Yes you will be blown downstream. There are two vectors involved. Your water speed 6m/s, and the current speed 2m/s.

You will move North East as shown here with respect to the shore. The angle in here will be ______________ and the hypotenuse will be __________________________. This value is known as the ___________________. Every vector can be broken down into its ____________, which are in the x & y directions. In the previous examples, the vectors used were relatively simple vectors since they were pointing in purely N,S,E, or West directions. These vectors still had both x & y components, but either their x component or their y component were _______.

Example:

However, the answer to the “swimming” problem (the resultant vector) was “crooked”. It had both an x & a y component.

Day #1 Vectors HW Problems

Graphical Vectors

1. Add the two vectors at the right graphically. Don’t worry about finding numbers. Only a PICTURE answer is expected.

2. A stray dog wanders 4 km west, 2 km North, 3 km west, and finally 3 km south. Draw his journey using vectors and then draw his displacement vector and find its magnitude.

3. A quarter is spun on a table top. It follows the path shown below. Draw the displacement vector of the spinning quarter (assuming that it starts at the left and ends on the right).

4. CHALLENGING PROBLEM! An object is moving in a circular path at a constant speed. Is the velocity constant as well? His velocity at two points is shown below. Draw the change in velocity from point 1 to point 2. (Hint: [pic] and this problem will involve SUBTRACTING vectors).

5. A plane flies 9 km north, then flies 6 km S 30o E. Draw the plane’s displacement vector for the entire trip.

6. Use the vectors below to “Graphically” show each of the following.

a) B – A b) A + D

c) C – B d) 3C

e) B + 2D f) 2A - E

1-D Vectors

7. A man walks 9 km east and then 15 km west. Find his

a) distance traveled

b) displacement

2-D Vectors (at right angles)

8. A robin is flying south for the winter at a rate of 40 miles per hour when it runs into a hurricane blowing due west at 100 miles per hour. What is the new velocity (magnitude and direction) of the robin?

9. Tarzan is hanging from one of his vines. His weight (his force) is 210 pounds, directed Southwards. A boy sees that Tarzan wants to be pushed on his vine. The boy pushes with a force of 90 pounds West. The rope can withstand a 225-lb force. What happens to the “King of the Apes”? (Find both the magnitude and the direction of the resultant).

10. The world is again being attacked by hostile aliens from outer space. This looks like a job for Superman! With his super strength, he is able to fend off the attack. However, he wants to rid the earth of these dangerous aliens forever. As they fly away, they must pass close to our sun. They are heading on a bearing of East with a speed of Worp 8. If Superman can change their course to a bearing of between [N 61o E] and [N 64 o E], they will crash into the sun. He gives them a velocity of Worp 4 on a bearing North. What happens? Has Good triumphed again?

11. On a television sports review show, a film clip is shown of the great tackle that ended the 1960 championship football game in Philadelphia. Jim Taylor of Green Bay was running forward down the field for a touchdown with a force of 1000 newtons (direction = South). Check Bednardek of Philadelphia runs across the field “at right angles to Taylor’s path” (direction = West) and tackles him with a force of 2500 newtons. Find the resultant of this crash.

12. A kite weighs 5 newtons (downward force). A girl throws the kite straight up in the air with a force of 20 newtons. If the wind is blowing horizontally with a force of 20 newtons EAST, find the resultant force acting upon the kite. Find both its magnitude and its direction

13. A clock has a minute hand that is 15 in long. Find:

a) the average speed of the minute hand’s tip.

b) the instantaneous speed of the minute hand’s tip as it passes the 3 on the clock.

c) the displacement of the minute hand’s tip in moving from 1:30 to 1:45.

When finding displacement or average velocity in 2D, we need to remember the equations

__________ ___________ ___________

Example:

a) A certain man’s initial position is 40 m [N], as measured from his house. After 30 seconds of walking, his new position is 50 m [S]. Find:

i) the distance travelled by the man during the trip.

ii) the man’s displacement.

iii) the man’s average velocity.

b) A radar station is tracking a jet. Its location at a certain moment on time is 40 km [W] at an altitude of 5000 m, relative to the station. 2 minutes later, its location is 53 km [E], this time at an altitude of 3000 m. Find the jet’s

i) displacement.

ii) average velocity.

Resolving a vector into components

Every vector can be written as a combination of only x-direction and y-direction vectors. These vectors are called the __________________ of the original vector. When a vector is broken into its components, we call this action “_____________ “ the vector into components.

For each vector below, 1st name the vector. Then resolve the vector into its components.

a) b) c)

d) A vector of magnitude 17 yds has an x-component of 12 yds directed to the right. Find the possible values of the y-component, as well as the possible directions of the original vector.

e) A small airplane takes off at a constant velocity of 150 km/h at an angle of 37o above the ground. How high above the ground is plane after 3 seconds of flight? What horizontal distance does it undergo during this time?

f) A car travelling at a constant speed of 60 km/h travels 700 m along a straight highway that is inclined 4o to the horizontal. An observer notes only the vertical motion of the car. What is the car’s vertical velocity, as well as the vertical distance that it travels.

Day #2 Vectors HW Problems

Resolving Vectors into Components

14. Resolve the vector at the right into its vertical and horizontal components.

15. Wonderwoman, in a feat of super strength, pulls a sled loaded with 20 kids over the sand at the beach (she’s a little early for the winter but can’t wait) with a force of 2,000 lbs. The rope makes a 20 o angle with the horizontal. Find the components of the rope.

16. A VW is parked on a hill with a slope of 30o. The car has a downward force of 9,000 newtons acting on it (otherwise known as its weight). Find the component forces. (Hint: One component is parallel to the slope and the other is perpendicular to the slope).

17. A man pushes a lawn mower with a 500-newton force. The handle makes a 40o angle with the ground. Find the horizontal and vertical components of his applied force.

18. Another man pulls a wagon with his son sitting in it. The man pulls the wagon with a force of 200 N at an angle of 40o with the ground. The sidewalk provides 30 N of friction in the opposite direction of the man’s motion. What is the net force on the wagon in the direction parallel to the ground?

When vectors are “crooked”, they become slightly more complicated to add together. For example, consider a car driving 20 km [West] and then 50 km [West 30o South]. They are still added head to tail. Vectors add ________________. You could add the 20 to the 50 and get the same resultant as if you added the 50 to the 20. An efficient practice is to draw any pure N,S, E, or W vector first and then add the “crooked” vector. We have to break the vectors down into components. The 20 km is already fine. The 50 km must be broken down before we try to use the Pythagorean Theorem.

The 50 km vector is now represented in pure x and y values. The 20 km vector is still fine. Now we have to add up all of our x values (+ right, - left) and all of the y values (+ up, - down).

x: y:

Now, recombine the x and y:

Note that the –63.3 means the vector should point left and the –25 means the vector should point down. Next, the resultant may be drawn in.

θ ’

Δd =

or

To give the proper direction (Is it South 21.6o West or West 21.6o South???) you must follow the component of the vector whose direction you are giving. Notice that we could have added the components in either order, thereby getting two different answers. The angle included in the directions is always the angle included between the “starting” component and the resultant.

Practice: Name each resultant vector below in two (2) different ways.

Go back to the example where we started with the resultant drawing of

We will now enclose the drawing to the right in a rectangle.

Calculating the magnitude of the resultant, we get _________________.

We have a choice for describing the angle of direction, either α or θ.

α = θ =

We could report our answer one of two ways: ____________________

or ________________________

In-Class Examples

When adding crooked vectors, we use the following procedure:

1) Draw all the vectors appropriately.

2) Resolve each vector into its components.

3) Add the x-components together (making sure to keep track of signs), yielding SUPER-X

4) Add the y-components together (making sure to keep track of signs), yielding SUPER-Y

5) Use SUPER-X and SUPER-Y to construct a SUPER-TRIANGLE

6) Find the resultant (magnitude and direction) of the SUPER-TRIANGLE

a) A box is pulled with forces of 45 N [W], 100 N [W 40o N] and 200 N [E 20 o S]. Find the net force on the box.

b) A stationary quarterback is hit simultaneously by 3 defensive players. They hit him with forces of 300 lb [W 20o S], 400 lb at 50 o North of East, and 200 lb [N]. Which direction will the quarterback move after this collision?

c) Three vectors are acting on a box, yielding a resultant force of 80 Newtons [N]. Two of the vectors acting on the box are 5 Newtons [S] and 42 Newtons [SW]. Find the 3rd force.

Day #3 Vectors HW Problems

19. Two ants at a picnic find a small piece of cake. Each picks up the cake and tries to carry it home. The first ant’s home is at a bearing of [W 15o N] and he pulls with a force of 10 dynes. The second ant’s home is at a bearing of [S 15o W] , and he pulls with a force of 13.5 dynes. What do you predict will happen? In which direction will the cake travel?

20. Linda is pulling a sled with a force of 20 pounds and is heading on a bearing of N30( E. Elaine begins to pull the sled with a force of 15 pounds on a bearing of N5( W. What net force is exerted on the sled and in which direction does it go?

21. Find the resultant of a 120-Newton force North and an 80 Newton force at N45( E.

22. Two water skiers are being pulled by a boat. The first skier exerts a force of 300 pounds North on the boat; the second skier exerts a force of 400 pounds at N75( E. Find the resultant of the two forces.

23. Three brothers are playing at the local playground with a frisbee. Each grabs it at the same time. Robbie grabs it at the N63( E point and exerts a force of 100 pounds on it. Chip grabs it at the N27( W point and exerts a force of 100 pounds on it. Ernie pulls with 141.4 pounds at the S18( W point. Who gets control of the frisbee?

Equilibrium Problems are problems in which an object _________________________. In these problems the values of the x-components of all the vectors involved sum to _______. This is also true for the sum of the values of the y-components.

Examples:

a) The “Great Houdini”, in one of his death-defying acts, was suspended from a flagpole while tied in a strait jacket. The flagpole is supported by a cable which makes a 60o angle with the pole. What is the actual weight that the pole is supporting if the force on the cable is 750 lbs. (Maybe Harry should go on a diet!!!!)

b) A traffic light is supported by two cables that are 140( apart. Each cable exerts a force of 150 Newtons on the light. How much does the light weigh? (Think of the weight of the light, the downward force, as the equilibrant.)

c) A True HONORS Problem (! A box is hung from the ceiling as shown. It weighs 50 N and each cable provides a certain tension to keep the box stable. Find the value of each of the tensions in the cables.

Day #4 Vectors HW Problems

24. Three forces act on an object, which is equilibrium. The first two forces are 60 lbs [NW] and 33 lbs [E]. Find the 3rd force.

25. The crane below consists of a long rod with two ropes attached to its end. One of the ropes is being pulled down with a force of 4000 Newtons. If the crane is in equilibrium, find the force in the other rope as well as the force in the boom (the long rod).

TOUGH PROBLEM! Be careful and take your time.

When finding the change in velocity or average acceleration in 2D, we need to remember the equations

____________ ____________

Examples:

a) An object once moving at a velocity of 15 m/s [S] is now moving at a velocity of 25 m/s [E], 15 seconds later. Find the object’s change in velocity during the period of time as well as its average acceleration.

b) A ball is thrown at a wall at a speed of 30 mph. It loses some of its energy in the collision, and bounces off the wall at a speed of 25 mph. If the ball deformed during the collision and was in contact with the wall for 30 ms, find the average acceleration of the ball during the collision.

c) A hockey puck hits the boards with a velocity of 10 m/s E20( S. It is deflected with a velocity of 8.0 m/s at E24( N. If the time of impact is 0.03 s, what is the average acceleration of the puck?

Day #5 Vectors HW Problems

26. An object travels west @ 90 m/s. It changes direction (in 5 seconds) and then travels @ 100 m/s [N 30o E]. Find the average acceleration of the turn.

27. A boy travels @ 10 m/s [S] for 1 minute and then @ 5 m/s [W 10 o N] for 2 minutes. Find his average velocity for the trip.

28. A man travels 10 km [N], then 50 km [E 30 o S], then 100 km [W], and finally 30 km [W 40 o S]. If the trip takes 4hrs, find the:

a) distance traveled.

b) displacement fir the entire trip

c) difference between the magnitudes of [pic] and [pic].

Relative Velocity

Example: A man is standing on a riverboat next to his wife. The boat is moving down a river without propelling itself, using only the current’s speed, which is 50 ft/min. The man starts jogging towards the front of the boat with a speed of 100 ft/min. During his jog, a smaller speedboat, traveling at 150 ft/min (relative to the water), passes the boat (in the same direction). A bird is sitting on the shore, watching the whole situation unravel.

• What is the man’s velocity (relative to the bird) when he is running? _____

• What is the man’s velocity (relative to his wife) when he is running? _____

• What is the man’s velocity (relative to the speedboat) when he is running? _____

• What is the wife’s velocity relative to the bird? _____

• What is the wife’s velocity relative to the speedboat? _____

The man’s velocity depends upon the ____________________ of the observer.

Example: Two cars are moving towards each other on a highway. Car A moves East at 60 mph, while car B moves West at 50 mph. Find the velocity of car A with respect to ….

a) car B ______

b) a bird sitting on the side of the highway ______

c) a boy in the backseat of car A ______

The Relative Velocity Equation

Example:

A boy sits in his car with a tennis ball. The car is moving at a speed of 20 mph. If he can throw the ball 30 mph, find the speed with which he will could hit…

a) his brother in the front seat of the car.

b) A sign on the side of the road that the car is about to pass.

c) A sign on the side of the road that the car has already passed.

River Problems

A 20 m wide river flows at 1.5 m/s. A boy canoes across it at 2 m/s relative to the water.

a) What is the least time he requires to cross the river?

b) How far downstream will he be when she lands on the opposite shore (assuming he tries to cross in the least amount of time)?

c) What will his velocity relative to the shore be as he crosses?

A river is 20 m wide. It flows at 1.5 m/s. If a girl swims at a speed of 2 m/s, find:

a) the time required for the girl to swim 15 m upstream (assuming she points the canoe directly upstream).

b) the time required for the girl to swim 20 m downstream (assuming she points the canoe directly downstream).

c) the angle (between the swimmers path and the shore that the girl should aim when crossing the river if she wants to arrive at the other side directly across from her starting point.

d) How long will it take to cross the river in this case (the case where the girl adjusts for the current by pointing herself upstream)

Days #6 & #7 Relative motion HW Problems

Level I

29. Car A is driving at a speed of 30 mph to the right. Car B is driving at a speed of 40 mph to the left. The car’s are moving toward each other. What is the velocity of ….

a) car B with respect to car A?

b) car A with respect to car B?

c) car A with respect to a police officer parked on the side of the road?

d) Car A with respect to someone in the passenger seat of car A?

30. A boy is sitting on an airplane (made entirely of see-thorough plastic) that is travelling west at 300 mph. He throws a ball toward the back of the plane. If he throws the ball at 30 mph, what is the velocity of the ball relative to …

a) a boy sitting in back of him on the plane?

b) a boy sitting on the ground watching the event from a lawn-chair?

31. A 30 m wide river flows at 1 m/s. A girl swims across it at 2 m/s relative to the water.

a) What is the least time she requires to cross the river?

b) How far downstream will she be when she lands on the opposite shore?

c) If the girl puts her head down and tries to swim straight across the river, at what angle

relative to the shore will she actually travel? What will be her speed relative to the shore?

32. A plane is flying through the air at a speed of 400 mph [E] when it encounters a 50 mph wind that is pointing North. Find the plane’s velocity with respect to the ground while it fights the wind.

33. Delivering a paper, a paper boy is riding his bicycle with a velocity of 12 m/s [E]. If he throws the paper with a velocity of 20 m/s [N], what will be the velocity of the paper with respect to the ground?

34. A field goal kicker attempts a field goal. He kicks the ball with a velocity of 25 m/s directly North toward the goal post. However, there is a wind blowing in the stadium from east to west with a velocity of 6 m/s. What will be the velocity of the ball with respect to the ground?

Level II

35. A 70 m wide river flows at 0.80 m/s. A canoeist (who can paddle the canoe at 2.4 m/s in still water) sets out from shore. At what angle to the shore would the canoe have to aim, in order to arrive at a point directly opposite the starting point? How long would this trip take?

36. An airplane maintains a heading due West at an air speed of 900 km/hr. It is flying through a hurricane with winds of 300 km/hr [S45( W].

a) Find the plane’s velocity (magnitude and direction) relative to the ground.

b) How long would it take the plane to fly 500 km along the path in part “a”?

37. A river is 50 m wide. It flows at 2 m/s. If a canoeist can paddle at a speed of 5 m/s, find:

a) the time required for the canoeist to paddle 20 m upstream.

b) the time required for the canoeist to travel 100 m downstream.

c) the angle (between his canoe tip and the shore) that a canoeist should aim when crossing the river if he wants to arrive at the other side directly across from his starting position.

38. A plane has an airspeed of 100 m/sec. The pilot notices that although he was headed due East, a wind of 80 m/s North is pushing the plane. What is the plane’s velocity relative to the ground?

39. A canoeist paddles “north” across a river at 3.0 m/s. (The canoe is always kept pointed at right

angles to the river.) The river is flowing east at 4.0 m/s and is 100 m wide.

a) What is the velocity of the canoe relative to the river bank?

b) Write the relative velocity equation that was used to solve part “a”.

c) Calculate the time required to cross the river.

d) How far downstream is the landing point from the starting point?

40. A boat which can travel at 5 m/s in still water attempts to cross a river by aiming straight across. It

takes the boat 20 seconds to cross the river and the boat lands 10 meters downstream. Find….

a) the width of the river.

b) the speed of the current.

c) the speed that the boat’s speedometer will read during the trip.

d) the angle relative to the shore that the boat should point itself so that it actually travels straight across.

e) the time the trip in part “d” will take to cross the river.

Level III

41. While standing in the pocket in Sunday’s big game, Donovan McNabb throws a pass downfield. He

can throw the ball with a velocity of 22 m/s. If he wants the ball to go directly South to the end

zone, which way (a specific direction) should he throw the ball if there is a wind blowing from east

to west with a velocity of 8 m/s?

42. A kayak can move at a speed of 5 m/s in still water. How long will it take the kayak (roundtrip) to travel 100

meters upstream and then back to its starting position if the speed of the current is 3 m/s?

43. A plane is flying with an airspeed of 200 mph [N]. It encounters wind that changes its course,

causing it to travel at a speed of 230 mph in a direction [N10oW]. Find the magnitude and

direction of this wind’s velocity.

Level IV

44. A pilot wishes to make a flight of 300 km [NE] in 45 minutes. On checking with the meteorological

office, she finds that there will be a wind of 80 km/h from the north for the entire flight. What

heading and airspeed must she use for the flight?

45. A pilot maintains a heading due West with an air speed of 200 km/h. The wind is blowing 30 km/h in the direction [S 40o E]. Find:

a. the plane’s velocity relative to the ground.

b. the direction that the pilot must fly the plane if he wants to fly due west.

46. A “moving walkway” at an airport moves at 0.5 m/s. A man jumps on the walkway and stands still for 10

seconds. He then begins walking forward at a speed of 1 m/s relative to the walkway until he covers the

same distance that he covered while standing still. Suddenly, he remembers that he forgot his luggage, and

he turns around and runs backwards to the start of the walkway at a speed of 4 m/s relative to the walkway.

Find his total time on the walkway.

Day #8 Vectors HW Problems – Miscellaneous Review

47. A sunbather, drifting downstream on a raft, dives off the raft just as it passes under a bridge and swims against the current for 15 min. She then turns and swims downstream, making the same total effort and overtaking the raft when it is 1.0 km downstream from the bridge. What is the speed of the current in the river?

48. A train moving at a constant speed of 100 km/hr travels East for 40 min., then 30( East of North for 20 min. and finally West for 30 min. What is the train’s average velocity for the trip?

49. A ball is thrown from the top of a building with a speed of 20 m/s and at a downward angle of 30( to the horizontal. What are the horizontal and vertical components of the ball’s initial velocity?

50. A helicopter traveling horizontally at 150 km/hr [E] executes a gradual turn, and eventually is moving at 120 km/hr. [S]. If the turn takes 50 s to complete, what is the average acceleration of the helicopter?

51. A puck sliding across the ice at 20 m/s [E] is struck by a stick and moves at 30 m/s, S30(W. Find its change in velocity (hint: Δv = v2 – v1).

52. A batter strikes a baseball moving horizontally towards him at 15 m/s. The ball leaves the bat horizontally at 24 m/s, 40( to the left of a line from the plate to the pitcher. The ball is in contact with the bat for 0.01 s. Determine

a. the change in velocity of the ball

b. its average acceleration while being hit by the bat

53. A slightly disoriented homing pigeon flies the following course at a constant speed of 15 m/s:

i. 800 m, 37( East of North ii. 300 m due West iii. 400 m, 37( South of East

A crow flies in a straight line (as the crow flies) between the same starting and finishing points as the pigeon in the previous problem. At what speed must the crow fly, if the birds leave and arrive together?

54. The pilot of a light plane heads due north at an air speed of 400 km/h. A wind is blowing from the west at 60 km/h.

a. What is the plane’s velocity with respect to the ground?

b. Write the relative velocity equation that was used to solve part “a”.

55. A train is heading due West at 40 miles/hr. A person is walking towards the rear of the train at 5 miles/hr. Find the velocity of the person relative to a bird that is sitting next to the railroad tracks.

56. Two forces, 60 N East and 80 N at South, respectively, act as right angles to each other. What is the magnitude and direction of the resultant force?

57. An airplane is moving toward the North at a velocity of 600 Km/hr. A wind is blowing towards the East at 60 Km/hr. What is the plane’s actual velocity and direction?

58. A pair of forces, one 40 N East and the other 80 N at E 30o S interact. What is the magnitude and direction of the resultant force?

59. A police cruiser chasing a speeding motorist traveled at 60 km [S], then 35 km [NE], and then 50 km [W].

a) Calculate the total displacement of the cruiser.

b) If the chase took 1.3 hours, what was the cruiser’s… i) average speed for the trip

ii) average velocity for the trip

60. An express bus travels directly from A-town to B-ville. A local bus also links these two towns, but it goes west 30 km from A-town to C-ville, then 30 km South to D-ville, and finally 12 km West to E-town and 30 km Northeast to B-ville.

a) What is the shortest distance from A-town to B-ville?

b) In what direction does the express bus travel?

c) If the express bus took 0.45 hours to go from A-town to B-ville, and the local bus took 3.0 hours, calculate the average speed and the average velocity for both buses.

61. A hiker walks 10.0 km [NE], 5.0 km [W], and then 2.0 km [S] in 2.5 hours.

a) What is the hiker’s displacement?

b) In what direction must the hiker set out in order to return by the most direct route to the starting point?

c) If the hiker walks at a constant speed for the entire trip and returns by the most direct route, how long will the total walk take?

62. Two boys (Juan and Eric) are at point X on one side of a river, 40 m wide and having a current of 1.0 m/s, flowing as shown. Simultaneously, they dive into the water in an attempt to reach point Y, directly opposite X. Both swim at 2.0 m/s relative to the water, but Juan directs himself so that his net motion corresponds to XY, while Eric keeps his body perpendicular to the current and consequently lands at point Z. After landing, Eric runs along shore to point Y at a speed of 6.0 m/s. Which boy arrives at Y first, and how much time does he beat the other?

The ULTIMATE river problem

A river is 100m wide. The current is flowing at 3 m/s. Canoeist Bob paddles at 5 m/s, canoeist Joe paddles at 4 m/s, and canoeist Sally paddles at 6 m/s (all relative to the water).

a) If Bob tries to cross the river perpendicularly, how long will it take him?

b) How far downstream will Bob land after crossing the river?

c) What is Bob’s velocity relative to the shore?

d) Write the relative velocity equation.

e) If Joe wanted to cross the river and land at a point directly across from his starting point, what direction (relative to the shore) would he aim his canoe?

f) How long would it take Joe to cross the river (and land at a point directly across from him)?

g) How long will it take Sally to canoe 50 m upstream?

h) What is Sally’s velocity relative to a bird sitting on the shore?

i) If Bob paddles for 50 seconds downstream, how far will he travel (relative to the shore)?

j) If Sally (paddling upstream) passes Bob (paddling downstream), what is Sally’s velocity relative to Bob?

k) What is Bob’s velocity relative to Sally?

l) If Joe and Sally set out together on a trip downstream, what would Sally’s velocity be relative to Joe?

m) What would Joe’s velocity be relative to Sally?

n) If Joe wanted to canoe 200m upstream and then back to his starting point, how long would it take him?

o) CHALLENGE (don’t get discouraged….its not easy): If Sally wanted to cross the river (in a straight line) and land at a point 20 m downstream, what is the angle that she should point herself relative to the shore?

The ULTIMATE plane problem

On a certain night, the wind is blowing from the west at 50 km/h. A Boeing 747 flies with an airspeed of 800 km/h due North while a small private plane flies at 300 km/h at a heading of [W 40o S].

a) Find the velocity of the Boeing 747 with respect to the ground.

b) Write the relative velocity equation for this situation.

c) What distance will the plane cover if it flies for 2 hours?

d) If it wanted to head due North, how far off course will it be after these 2 hours?

e) What angle should the pilot redirect the plane in order for the plane to head due North?

~~~~~~

f) Find the velocity of the private plane with respect to the ground.

g) How long will it take for the plane to fly 1000 km (relative to the ground) in a straight line?

~~~~~~

h) Challenge (again, don’t get discouraged….its not easy): Find the velocity of the Boeing 747 relative to the private plane.

(Hint: Start off by writing a relative velocity equation. Then, use your answers from parts “a” and “f” above)

|7) 24 km; 6 km [west] |37) 6.67 sec; 14.29 sec; U 66.4 o A |

| | |

|8) 108 mi/hr @ [S 68( W] or [W 22( S] |38) 128.06 m/s [E 38.7 o N] |

| | |

|9) 228 lb. @ [S 23( W] or [W 67( S] |39) 5 m/s [A 53.1 o DS]; 33.3 sec; 133.3 m |

| | |

|10) 8.94 worps @ [N 63( E] or [E 27( N] |40) 0.5 m/s; 100m; 5.03 m/s [A 5.7 o DS]; |

| |[U 84.3 o A]; 20.1 sec |

|11) 2693 N @ [W 22( S] or [S 68( W] | |

| |41) [S 21.3 o E] |

|12) 25 N @ 53( from vertical or 37( from horizontal | |

| |42) 62.5 sec |

|13) 0 026 in/sec; 0.026 in/sec since the speed is constant; | |

|21.21 in [left 45 up] |43) 47.9 mph [N 56.4 o W] |

| | |

|14) x: 34.47 km; y: 28.93 km |44) 460.1 km/h [E 52 o N] |

| | |

|15) Horizontal = 1,879 lbs Vertical = 684 lbs |45) 182.2 km/h [W 7 o S]; [W 6.6o N] |

| | |

|16) Parallel = 4,500 N Perpendicular = 7,794 N |46) 16.2 sec total |

| | |

|17) 321.4 N [downward], 383 N [in motion direction] |47) 2.0 km/hr |

| | |

|18) 123.2 N [in motion direction] |48) 30 km/hr [E40( N] or 30 km/hr [N 50( E] |

| | |

|19) [W 39o S] |49) 17 m/s, -10 m/s |

| | |

|20) 33.4 lbs. @ [N 15( E] |50) 13,824 km/h2 [S51(W] or 1.07 m/s2 [S51(W] |

| | |

|21) 185 N @ [N 18( E] |51) 44 m/s[W37( S] |

| | |

|22) 559 lbs. @ [N 44( E] |52) a) 37 m/s [25( L of pitcher] |

| |b) 3.7 x 103 m/s2 [25( L of pitcher] |

|23) Nobody get the frisbee. It doesn’t even move! | |

| |53) 6.4 m/ |

|24) 43.46 lbs at 77.5o S of E | |

| |54) 404.5 km/h [N 8.5o E] |

|25) in boom: T = 6928.5N in 2nd rope: T = 4000N | |

| |55) 35 mi/hr. W |

|26) 32.9 m/s2 [E 32( N] | |

| |56) 100 N [E 53o S] |

|27) 4.3 m/s [W 40( S] | |

| |57) 603 Km/hr. at N 6o E |

|28) 190 km; 86.74 km [W 23.3( S]; 47.5 km/h; 21.7 km/h; 25.8 km/h | |

| |58) 116.4 N [ E 20o S] |

|29) 70 mph [L]; 70 mph [R]; 30 mph [R]; 0 mph | |

| |59) a: 43 km [S 36o W] |

|30) 270 mph [forward] |b: 1.1x102 km/hr |

| |33 km/hr [S 36 o W] |

|31) 15 sec; 15 m; 2.24 m/s; DS 63.4o Across | |

| |60) a: 23 km |

|32) 403.1 mph [E 7.13o N] |b: [W 24 S] |

| |c: 51 km/hr, 51 km/hr [W 24 o S], |

|33) 23.3 m/s [E 59 o N] |34 km/hr, 7.7 km/hr [W 24 o S] |

| | |

|34) 25.7 m/s [N 13.5 o W] |61) a: 5.5 km [E 68 o N] |

| |b: [S 22 o W] |

|35) U 70.53 o A; 30.97 sec |c: 3.3 hrs |

| | |

|36) 1132.2 km/h [S 79 o W]; 0.44 h |62) Juan beats Eric by 0.2 s |

-----------------------

HONORS Physics

Unit 3: Vectors

Vectors – Day #1

30o

40 m/s

60o

20 m/s

10o

10 m

Vectors Scalars

-

=

+

=

5 km

3 km

10 m/s

5 m/s

40 km

60 km

40 km

60 km

40 km

60 km

Resultant vector

We use _______ alphabet symbols to represent angles as variables and regular lowercase letters such as x, y, z, etc to represent numbers as variables. To find θ, we must use trigonometry ( ____ ____ ____ )

Current speed

(2 m/s)

Water speed

(6 m/s)

Notice how the vectors were arranged

__________.

θ

Resulting

velocity

6 m/s

2 m/s

[pic]

20 m/s

50 m/s

v1

1

2

v2

A B C D E

Vectors – Day #2

50o

20o

50 m/s

60 km

3 N

40o

45 km

30o

Vectors – Day #3

20 km

50 km

30o

+

Method I

Component Method

(the zero is to remind you that the first vector had no donation to the y)

63.3 km

25 km

63.3 km

25 km

[pic]

68.1 km

20o

46.2o

7 m/s

12 ft

Method II

Graphical Method

20 km

+

300

=

50 km

20 km

50 km

50

50 sin 30

50 cos 30

20

300

R

y

x

α

θ

Solving: x = ________________________

y = ________________________

Vectors – Day #4

60o

flagpole

cable

BUILDING

30o

60o

30o

60o

4000 N

Vectors – Day #5

10 m/s

8 m/s

20(

24(

Vectors – Day #6

[pic]

river boat relative to bird

[pic]

[pic]

man relative to bird

man relative to riverboat

Package relative to airplane Ground relative to airplane

Package relative to ground

Z

Y

(

(

1.0 m/s (

X

(

[pic]

[pic]

[pic]

[pic]

Ultimate Plane Problem Answers (on next page)

a) 802 km/h [N 3.6o E] b) [pic]

c) 1603 km d) 100 km too far east

e) [N 3.6o W] f) 264 km/h [W 47o S]

g) 3.792h (3 hrs, 48 min) h) 1,019 km/h [E 77o N]

[pic]

Ultimate River Problem Answers (on previous page)

a) 20 sec

b) 60 m

c) 5.8 m/s [across 31o downstream]

d) [pic]

e) [across 49o upstream] or [upstream 41o across]

f) 37.8 sec

g) 16.7 sec

h) 3 m/s upstream

i) 400 sec

j) 11 m/s upstream

k) 11 m/s downstream

l) 2 m/s downstream

m) 2 m/s upstream

n) 229 sec

o) [across 18o upstream] or [upstream 72o across]