chapter 3 kinematics in two dimensions; vectors. units of chapter 3 vectors and scalars addition of...
TRANSCRIPT
Chapter 3
Kinematics in Two Dimensions; Vectors
Units of Chapter 3
• Vectors and Scalars
• Addition of Vectors – Graphical Methods
• Subtraction of Vectors, and Multiplication of a Vector by a Scalar
• Adding Vectors by Components
• Projectile Motion
• Solving Problems Involving Projectile Motion
• Projectile Motion Is Parabolic
• Relative Velocity
ObjectivesThe students will be able to:1.Identify scalar and vector quantities.2.Perform vector addition geometrically.3.Determine the components of a vector.4. Use the methods of graphical analysis to determine the magnitude and direction of the vector resultant in problems involving vector addition or subtraction of two or more vector quantities. The graphical methods to be used are the parallelogram method and the tip to tail method.
What do the arrows represent?
3-1 Vectors and Scalars
A vector has magnitude as well as direction.
Some vector quantities: displacement, velocity, force, momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time, temperature
SCALAR
A SCALAR quantityis any quantity in physics that has MAGNITUDE ONLY
Number valuewith units
ScalarExample Magnitude
Speed 35 m/s
Distance 25 meters
Age 16 years
VECTOR
A VECTOR quantityis any quantity in physics that hasBOTH MAGNITUDE and DIRECTION
Vector Example
Magnitude and
Direction
Velocity 35 m/s, North
Acceleration 10 m/s2, South
Force 20 N, East
An arrow above the symbolillustrates a vector quantity.It indicates MAGNITUDE andDIRECTION
Vectors are represented by arrows.
• The length of the arrow reflects the magnitude of the measurement
The head of the vector must point in the direction of the quantity
3-2 Addition of Vectors – Graphical Methods
For vectors in one dimension, simple addition and subtraction are all that is needed.
You do need to be careful about the signs, as the figure indicates.
VECTOR APPLICATIONADDITION: When two (2) vectors point in the SAME direction, simply add them together.
EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started.
66.5 m, E
MAGNITUDE relates to thesize of the arrow and DIRECTION relates to the way the arrow is drawn
46.5 m, E + 20 m, E
Leading to 3-2 Addition of Vectors – Graphical Methods
Vectors can be combined (added)
Resultant Vector
• The resultant is the vector sum of two or more vectors.
• It is the result of adding two or more vectors together.
+ =
5 5 15
+
5
Vectors can be combined (added)
Vectors can be combined (added)
Vectors can be combined (added)
Vectors can be combined (added)
Vectors can be combined (added)
What is the resultant vector?
What do the arrows represent?
When an object is tossed, it travels in a “parabolic” path.
Gravity always points downward
There is a velocity component horizontally (vx)
There is a velocity component vertically (vy)
The velocity of the toss is actually the sum of the two vectors
Steps for Graphical Addition
1. Choose an appropriate scale (e.g.. 1cm = _____ m/s)
2. Draw all vectors with tail starting at origin
3. Redraw vector from “head to tail” while maintaining original direction of vector.
4. From tail of first vector to head of last connect lines (this is resultant) direction is towards head of last original vector
5. Measure length and convert back using scale.
6. Measure resultant from 0 degrees.
What happens now?
Graphical:Scale: 1 box = 50 km/h
Head to Tail Method Tail to Tail Method
Finding the Resultant Vector
• Pythagorean Theorem
c2 = a2 + b2
Head to Tail Method
Direction can be measured in degrees
1. 3.0 m/s, 45 deg + 5.0 m/s, 135 deg
5.83 m/s, 104 deg
5.0 m/s, 45 deg + 2.0 m/s, 180 deg
3.85m/s, 66.5 deg
Elaboration Vector Activity1. You take a walk in the park for 15 steps using a compass that points 25º North of East. •How would you use the simulation to represent your path? •Explain why the same representation works for illustrating this different scenario: You drive at 15 miles/hour using a compass that points 25º North of East. •Write another scenario using different units that could also be represented the same.
2.In the simulation, a vector is described by four measurements: R, Ө, Rx, and Ry. Put a vector in the work area, and then investigate to make sense of what these four things represent. In your investigation, use a wide variety of vector measurements and all three styles of Component Displays. Then, describe in your own words what the measurements represent and what “component” means.
3. Suppose you are driving 14 miles/hour with a compass reading of 35°north of east. •Represent the vector using the simulation. How fast is your car traveling in the north direction? How fast in the east direction? •Figure out how the components could be calculated using geometry if you couldn’t use the simulation. •Check your ideas by testing with other vectors and then write a plan for finding the components of any vector.
4. To get to the sandwich shop, you left home and drove 6 miles south and then 10 miles west. •If a bird flew from your house to the sandwich shop in a straight line, how far do you think the bird would fly? Use the simulation to check your reasoning.•What direction should it fly from your house to get to the shop? •Explain how you could use the simulation to answer these questions. •Explain how you could use geometry equations to answer these questions.
5.Suppose you and a friend are test driving a new car. You drive out of the car dealership and go 10 miles east, and then 8 miles south. Then, your friend drives 8 miles west, and 6 miles north. •If you had the dealer’s homing pigeon in the car, how far do you think it would have to fly to get back to the dealership? Use the simulation to test ideas.•The distance that the bird has to fly represents the sum of the 4 displacement vectors. Use the simulation to test ideas you have about vector addition. After your tests, describe how you can use the simulation to add vectors.
6. A paper airplane is given a push so that it could fly 7m/s 35° North of East, but there is wind that also pushes it 8 m/s 15° North of East. •Use the simulation to solve the problem. How fast could it go and in what direction would it travel? •Think about your math tools and design a way to add vectors without the simulation. •Check your design by adding several vectors mathematically and then checking your answers using the simulation.
3-2 Addition of Vectors – Graphical MethodsIf the motion is in two dimensions, the situation is somewhat more complicated.
Here, the actual travel paths are at right angles to one another; we can find the displacement by
using the Pythagorean Theorem.
3-2 Addition of Vectors – Graphical Methods
Adding the vectors in the opposite order gives the same result:
ExplorationCommutative Property
3-2 Addition of Vectors – Graphical Methods
Even if the vectors are not at right angles, they can be added graphically by using the “tail-to-tip” method.
3-2 Addition of Vectors – Graphical Methods
The parallelogram method may also be used; here again the vectors must be “tail-to-tip.”
NON-COLLINEAR VECTORS
When two (2) vectors are PERPENDICULAR to each other, you mustuse the PYTHAGOREAN THEOREM
Example: A man travels 120 km eastthen 160 km north. Calculate his resultant displacement.
120 km, E
160 km, N
the hypotenuse iscalled the RESULTANT
HORIZONTAL COMPONENT
VERTICALCOMPONENT
FINISH
FINISH
3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.
Then we add the negative vector:
Opposite of a Vector
v
- v
If v is 17 m/s up and to the right, then -v is 17 m/s down and to the left. The directions are opposite; the magnitudes are the same.
3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
A vector V can be multiplied by a scalar c; the result is a vector cV that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.
Scalar Multiplication
x
-2x
3x
Scalar multiplication means multiplying a vector by a real number, such as 8.6. The result is a parallel vector of a different length. If the scalar is positive, the direction doesn’t change. If it’s negative, the direction is exactly opposite.
Blue is 3 times longer than red in the same direction. Black is half as long as red. Green is twice as long as red in the opposite direction.½ x
VECTOR APPLICATIONSUBTRACTION: When two (2) vectors point in the OPPOSITE direction, simply subtract them.
EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started.
26.5 m, E
46.5 m, E
-
20 m, W
Homework
Chapter 3 Questions on page 651, 2, 3, 5, 6, 7, 9Problem 3 on page 65
Projectile Lab Report due Monday (full and Complete lab report).
ObjectivesThe students will be able to:1.Use the trigonometric component method to resolve a vector components in the x and y directions.2. Use the trigonometric component method to determine the vector resultant in problems involving vector addition or subtraction of two or more vector quantities.
Vector Components – leading to Section 3-4
150 N
Horizontal component
Ver
tical
co
mpo
nent
A 150 N force is exerted up and to the right. This force can be thought of as two separate forces working together, one to the right, and the other up. These components are perpendicular to each other. Note that the vector sum of the components is the original vector (green + red = black). The components can also be drawn like this:
3-4 Adding Vectors by Components
Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.
Finding Components with Trig
v
v cos
v si
n
Multiply the magnitude of the original vector by the sine & cosine of the angle made with the given. The units of the components are the same as the units for the original vector.
Here’s the correspondence:
cosine adjacent side
sine opposite side
Note that 30.814 + 14.369 > 34. Adding up vector components gives the original vector (green + red = black), but adding up the magnitudes of the components is meaningless.
Component Example
34 m/s
30.814 m/s 2514.369 m/s
A helicopter is flying at 34 m/s at 25 S of W (south of west). The magnitude of the horizontal component is 34 cos 25 30.814 m/s. This is how fast the copter is traveling to the west. The magnitude of the vertical component is 34 sin 25 14.369 m/s. This is how fast it’s moving to the south.
Pythagorean Theorem
34 m/s
30.814 m/s 2514.369 m/s
Since components always form a right triangle, the Pythagorean theorem holds: (14.369)2 + (30.814)2 = (34)2.
Note that a component can be as long, but no longer than, the vector itself. This is because the sides of a right triangle can’t be longer than the hypotenuse.
3-4 Adding Vectors by Components
If the components are perpendicular, they can be found using trigonometric functions.
Other component pairs
There are an infinite number of component pairs into which a vector can be split. Note that green + red = black in all 3 diagrams, and that green and red are always perpendicular. The angle is different in each diagram, as well as the lengths of the components, but the Pythagorean theorem holds for each. The pair of components used depends on the geometry of the problem.
v
v cos
v si
n
vv
v cos v
sin
v cos
v sin
WHAT ABOUT DIRECTION?In the example, DISPLACEMENT asked for and since it is a VECTOR quantity,we need to report its direction.
N
S
EW
N of E
E of N
S of W
W of S
N of W
W of N
S of E
E of S
NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE.
N of E
NEED A VALUE – ANGLE!
Just putting N of E is not good enough (how far north of east ?). We need to find a numeric value for the direction.
N of E
160 km, N
120 km, E
To find the value of the angle we use a Trig function called TANGENT.
200 km
So the COMPLETE final answer is : 200 km, 53.1 degrees North of East
What are your missing components?
Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components?
65 m25
H.C. = ?
V.C = ?
The goal: ALWAYS MAKE A RIGHT TRIANGLE!
To solve for components, we often use the trig functions since and cosine.
EmCHopp
NmCVadj
hypopphypadj
hypotenuse
sideopposite
hypotenuse
sideadjacent
,47.2725sin65..
,91.5825cos65..
sincos
sinecosine
ExampleA bear, searching for food wanders 35 meters east then 20 meters
north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.
35 m, E
20 m, N
12 m, W
6 m, S
- =23 m, E
- =14 m, N
23 m, E
14 m, N
3.31)6087.0(
6087.23
14
93.262314
1
22
Tan
Tan
mR
The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST
R
ExampleA boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.
15 m/s, N
8.0 m/s, W
Rv
1.28)5333.0(
5333.015
8
/17158
1
22
Tan
Tan
smRv
The Final Answer : 17 m/s, @ 28.1 degrees West of North
ExampleA plane moves with a velocity of 63.5 m/s at 32 degrees South of
East. Calculate the plane's horizontal and vertical velocity components.
63.5 m/s
32
H.C. =?
V.C. = ?
SsmCVopp
EsmCHadj
hypopphypadj
hypotenuse
sideopposite
hypotenuse
sideadjacent
,/64.3332sin5.63..
,/85.5332cos5.63..
sincos
sinecosine
ExampleA storm system moves 5000 km due east, then shifts
course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement.
NkmCVopp
EkmCHadj
hypopphypadj
hypotenuse
sideopposite
hypotenuse
sideadjacent
,2.96440sin1500..
,1.114940cos1500..
sincos
sinecosine
5000 km, E
40
1500 km
H.C.
V.C.
5000 km + 1149.1 km = 6149.1 km
6149.1 km
964.2 kmR
The Final Answer: 6224.2 km @ 8.92 degrees, North of East
3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
Homework for Chapter 3
• Problems on pages 65 - 66 #s 7, 10, 13, 14, 15, 16
Force Table Lab
• Analytical Method– Break each vector to be added into its
components.– Add the components.– Find the resultant from the sum of the
components.
Force Table Lab
• If the resultant has zero length, then the net force acting on an object is zero and the object is said to be in equilibrium.
• A vector that is equal in magnitude but opposite in direction to another vector is called an equilibrant, since the sum of the two vectors is zero and an object acted upon by these two forces would be in equilibrium.
