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 ______.
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.
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.
HW #1 Vectors Problems
Graphical Vectors
1. Add the two vectors at the right graphically
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 plane flies 9 km north, then flies 6 km S 30o E. Draw the plane’s displacement vector from the start of the trip.
4. Use the vectors below to “Graphically” show each of the following.
a) A + Eb) D + Bc) 3Cd) B + 2D
1D Vectors
5. A man walks 9 km east and then 15 km west. Find his
a)distance traveled
b)displacement
2D Vectors
6. 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, “resulting” velocity of the robin?
7. 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”? (Hint: First find both the magnitude and the direction of the “resultant”).
8. 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.
9. 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.
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 object 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 bird is flying at a speed of 10 m/s at an angle of 30o below the horizontal. At what speed is the bird’s shadow moving relative to the ground.
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?
#2 Vectors HW Problems
2D Vector Problem
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? (Hint: Find the resultant worp speed and see if its direction is in the appropriate range)
Resolving Vectors into Components
11. Resolve the vector at the right into its vertical and horizontal components.
12. Resolve the following vectors into their components.
a) 55 m [N 40o W]b) 0.4 km/h at 60o W of Sc) 19N [SE]
13. 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.
14. 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).
15. 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.
16. 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?
Avg Velocity Problems
17. A baseball is hit from homeplate into the outfield bleachers. It lands 135 meters from the plate, horizontally, and 10 meters vertically above the field. If the ball is in the air for 5 seconds, find:
a)the average velocity of the ball during the trip (Both magnitude and direction).
b)the average velocity of the ball’s shadow upon the field during the trip (magnitude only).
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.
d =
or
To give the proper direction (South 21.6o West orWest 21.6oSouth) 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 or
= =
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 5Newtons [S] and 42 Newtons [SW]. Find the 3rd force.
#3 Vectors HW Problems
18. 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. In which direction will the cake travel?
19. 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?
20. Find the resultant of a 120-Newton force North and an 80 Newton force at N45 E.
21. 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.
22. 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?
23. Back to problem #18. If a third ant started pulling on the piece of cake, what would his force have to be in order to keep the cake from moving? In other words, what 3rd force would make both the x and y-components sum to zero.
Equilibrium Problems are problems in which an object ______. In these problems of values of the x-components of all the vectors involve sum to ______. This is also true for the sum of the values of the y-components.
Examples:
a)Four forces pull on an object, and the object doesn’t move. The first three forces are 40 N [E], 65 N [W 50o S], and 80 N [N 10o W]. Find the magnitude and direction of the 4th force.
b)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!!!!)
c) 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.)
#4 Vectors HW Problems
24. Four boys are playing tug-o-war. Three are pulling with forces of 7 Newtons [N], 17 Newtons [S], and 41 Newtons [S]. If the rope is not moving, find the force of the fourth boy.
25. 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.
26. A speaker in the auditorium is supported by two cables that are 120 apart. If the speaker weighs 200 Newtons, find the tension in each of the cables. (Hint: Look at example problem “c” above to get a general problem solving strategy.)
27. What is true of the sum of the x-components of the vectors that act on an object that is in equilibrium?
28. Twenty forces act on an object that is in equilibrium. If the vector sum of the first nineteen forces is 4500 N [E 20 N], find the magnitude and direction of the twentieth force.
29. Three vectors are added together and the resultant is 50 km [W]. The first two forces are 30 km [E] and 65 km [SW]. Find the magnitude and direction of the 3rd force.
When finding the change in velocity or average acceleration in 2D, we need to remember the equations
______
Examples:
a)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.
b)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?
#5 Vectors HW Problems
- A man was travelling West at 4 m/s is now travelling East at 9 m/s. Find his change in velocity during this timer period.
- Do problem #30 again, this time assuming that the man’s final velocity is 9 m/s North.
- 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.
- 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.
- 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 for the entire trip
c)difference between the magnitudes of and .
- 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.
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? _____
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