P11 Kinematics 1 Scalars and Vectors Bundle.notebook 1 July 27, 2013 1. Solve for x. 19.2 13.7 x 2. Solve for θ. θ 7.4 6.2
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1. Solve for x.
19.2
13.7x
2. Solve for θ.
θ
7.4
6.2
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Directions in Two Dimensions
23o north of east 55o west of south
17o south of east 88o west of north
[E23oN] [S55oW]
[E17oS] [N88oW]
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Unit 1-Kinematics
Kinematics is the study of how things move.
Physics 112
We will study such terms as:scalars, vectors, distance, displacement, speed, velocity and acceleration.
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Module 1.1- Position, Distance, Displacement
This module is an introduction to some basic kinematics terminology that will be important throughout the entire unit, as well as in future units.
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Types of Quantities: Scalars vs. Vectors
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Scalars and Vectors
Scalar quantities are quantities that do not have a direction associated with them. They have a magnitude (size) only.
Vector quantities have two parts to them:A Magnitude, or size of the quantity, andA direction
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A scalar is a physical quantity that is described by a single number (with its appropriate unit).
distance, time, temperature, speed
Types of Physical Quantities
A vector is a physical quantity that has both size and a direction.
The size or length of a vector is called its magnitude.
Examples
Many quantities have a directional quality and cannot be described by a single number.
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Examples
force, position, velocity, acceleration
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Drawing Vector Quantities
We can graphically represent a vector as an arrow.
The arrow is drawn to point in the direction of the vector quantity, and the length of the arrow is proportional to the magnitude of the vector quantity.
tip of the arrow
tail of thearrow
20 N
Example
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Adding Vectors Graphically
Suppose that a child walks 200 m east, pauses, and then continues 400 m east. To find the total displacement, or change in position of the child, we must add the two vec tor quantities.
Vectors are added by placing the tail of one vector at the tip of the other vector. The diagram below, drawn to scale, shows the addition of the two segments of the child's walk.
It is very important that neither the direction nor the length of either vector is changed during the addition process. A third vector is then drawn connecting the tail of the first vec tor to the tip of the second vector. This third vector represents the sum of the first two vectors.
This third vector is called the resultant, R. The resultant is always drawn from the tail of the first vector to the tip of the last vector.
To find the magnitude of the resultant, R, measure its length using the same scale used to draw the first two vectors.
200 m 400 m
Scale: 1 division = 100 m
Scale: 1 division = 100 m
200 m 400 m
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When vectors are added, the order of addition does not matter. The same vector sum will result.
NOTE!!!
200 m400 m
If the child had turned around after moving 200 m east and then walked 400 m west, the change of position would have been 200 m west. Note that the vectors are added tip to tail.
Two vectors can have opposite directions.
200 m
400 m
200 m
400 m
200 m
400 m
R
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Vector Addition: The Order Does Not Matter
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Add the following forces. Sketch a graphical solution. State the magnitude and direction of the resultant.
2.5 N
A2.0 N
B
1.5 N
C
5.0 N
D+ +
+
A 2.5 N North B 2.0 N EastC 1.5 N [W40oS]D 5.0 N [W55oN]
5.9 N [W70oN]
Example
Let 1 cm = 1 N
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100 Aker Wood
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When a vector is multiplied by a positive scalar, c, the magnitude of the vector will change. The direction of the vector is not affected.
Multiplication of a Vector by a Scalar
Examples
1. If A = 15 m east and c = 0.5, then
A15 m
cA7.5 m
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2. If B = 30 m/s at 52o
and
c = 2
then
cB = 2 x 30 m/s at 52o cB = 60 m/s at 52o
B
cB
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The Negative of a Vector
The negative of a vector is a vector of the same magnitude but opposite direction. In other words, the vector will rotate 180o.
A B
How do we subtract two vectors?
We have to rewrite the expression as a sum of two vectors.
A + (B)
A B
A B
BA
A B=
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Vectors and Trigonometry
Trig Basics
θ
The longest side, opposite to the right angle, is the
hypotenuse.
This is the side adjacent
to the angle.
This is the side opposite
to the angle. H
A
O
The sine, cosine, and tangent of angle θ are defined as ratios of the sides of the triangle:
sin θ = opp hyp
cos θ = adj hyp
tan θ = opp adj
We can determine the magnitude of the resultant vector from the Pythagorean Theorem.
H2 = O2 +A2
Adding Perpendicular Vectors
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Sample Problem 1Earl walks 3.2 m east and then 9.4 m south. Determine his resultant displacement. Include a labelled vector diagram.
Adding Perpendicular Vectors
RR
3.2 m
3.2 m
9.4 m 9.4 mθ α
H2 = O2 +A2
R2 = (3.2m)2 + (9.4m)2
R =
R = 9.9 m
tan θ = opp adj
tan θ = 9.4m 3.2m
θ = 9.4m x 1 3.2m tan
θ = 71o
His resultant displacement is 9.9 m E 71o S or 9.9 m 71o south of east
or 9.9 m S 19o E or 9.9 m 19o east of south.
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A dog walks 23.7 m west and then 14.9 m south looking for his yoyo.
a) How far did the dog walk?b) What is the dog's resultant displacement?
Sample Problem 2
a) Distance = 23.7 m + 14.9 m = 38.6 m
b) 23.7 m
14.9 m θ
The dog's Resultant Displacement = 38.6 m W 32.2o S or 38.6 m 32.2o south of west or
38.6 S 57.8o W or 38.6 m 57.8o west of south,
tan θ = opp adj
tan θ =14.9m 23.7m
θ = 14.9m x 1 23.7m tan
θ = 32.2o
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A camel walks 12.0 km east, 23.0 km northand finally 3.00 km west. What is the resultant displacement of the camel?
Sample Problem 3
12km
3km
23km
(123)km
Distance =
= 24.7 m
θ
tan θ = opp adj
tan θ =23 m 9 m
θ = 23 m x 1 9 m tan
θ = 69o
The resultant displacement of the camel is 24.7 m E 69O N or
24.7 m N 21 O E
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Practice Problems Vector Addition
1. Two vectors A and B are added together to form vector C. The relationship between the magnitudes of the vectors is given by: A2 + B2 = C2. Which statement concerning these vectors is true?
a) A and B must have equal lengths. b) A and B must be parallel. c) A and B must be going in the same direction. d) A and B must be at right angles to each other.
2. An escaped convict runs 1.70 km due east of the prison. He then runs due north to a friend’s house. If the magnitude of the convict’s resultant displacement is 2.50 km, what is the direction of his total displacement? a) 42.8o south of east b) 47.2o north of east c) 42.8o east of south d) 47.2o east of north
P11 Kinematics
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3. A 7 N force and a 8 N force act concurrently on an object. Which of the following forces is not a possible resultant?
a) 0 N b) 1 N c) 11 N d) 15 N
4. A groundhog named Murray reluctantly leaves his hole and walks 11 km north and then 16 km west.
a) How far did Murray travel? b) What is Murray’s resultant displacement from his hole?
19 km [W35oN] or 19 km 35o north of west
19 km [N55oW] or 19 km 55o west of north
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5. Bubba is flying a plane due north at 225 km/h as a wind carries it due east at 65.0 km/h. Find the magnitude and direction of the plane’s resultant velocity.
6. Baby Bear can’t find Mama Bear. What is his resultant displacement if he walks 2.0 m north, 3.0 m east, 1.0 m south, 5.0 m west, 4.0 m south and then 2.0 m east?
234 km/h [E73.9oN] or 234 km/h [N16.1oE]
3.0 m south
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7. Beulah and Bertha kick a soccer ball at exactly the same time. Beulah’s foot exerts a force of 66 N south. Bertha’s foot exerts a force of 88 N west. What is the resultant force on the ball?
1.1 x 102 N [W37oS] or 1.1 x 102 N [S53oW]
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a) The speed of the vehicle along its descent path is 65 m/s.
b) 58o to the right of the vertical
R35 m/s
55 m/s
θ
8. 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 horizon tal velocity of 55 m/s to the right.
a) At what speed does the vehicle move along its descent path? b) At what angle with the vertical is this path?
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9. Mr. Whipple is in a boat traveling 3.8 m/s east straight across a river 2.40 x 102 m wide. The river is flowing at 1.6 m/s south.
a) What is Mr. Whipple’s resultant velocity? b) How long will it take Mr. Whipple to cross the river?
a) 4.1 m/s [E23oS] or 4.1 m/s [S67oE]
b) It will take Mr. Whipple 63 s to cross the river.
3.8 m/s
1.6 m/sRθ
2.40 x 102 m
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Module Summary
In this module, you should have become familiar with some of the basic terminology. At this point, you should know the following:• The difference between a scalar and a vector.• How to draw vector quantities• How to add and substract vector quantites graphically• How to multiply a vector by a scalar• How to use trigonometry to solve vector quantity problems.
Attachments
Adding Perpendicular Vectors
Vector Addition: The Order Does Not Matter