Mechanics
Vectors
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· Use vectors to represent vector quantities.
· Solve perpendicular vector addition problems using trigonometric functions.
· Given a vector, resolve it into its component parts.
· Use vector resolution to add multiple vectors at any angle.
Activity and Notes
¹ Bill walks 200 meters east or 0°. Graphically show the path walked. Show the vectors that represent 200 meters at 95°, 200 meters at 200°, and 200 meters at 250°.
vector – arrow-tipped line segment used to
graphically represent vector quantities; the length of the line represents the
vector’s magnitude; the direction of the arrow represents the direction of the
quantity; represented as variables in bold or with a symbol above the variable;
v or 
.
tail – open end of a vector.
head – arrow part of a vector.
Activities and Notes
vector addition – combining vectors by placing the tail of the second vector to the head of the first vector; the sum of the vectors show a third vector; the directions of the original vectors do not change but are simply moved.
resultant – vector sum of two or more vectors; drawn from the tail of the first vector to the head of the last vector.
¹ Draw the vectors that represent a person walking 100 meters east, pausing, then walking another 100 meters east. Draw another vector if the person went back 50 meters, or 50 meters west.
When drawing vectors to scale, a ruler can be used to
accurately measure the magnitude of the resultant vector when working in
one-dimension.
¹ Draw the vectors that represent a person walking 30 meters east, then 40 meters north.
Drawing two-dimensional vectors graphically follows the
same rules for one-dimensional vectors.
Place the tail of the second vector to the head of the first
vector. Draw the resultant vector by
connecting the tail of the first vector to the head of the last vector. The direction of the resultant vector must be
included. The vector’s direction is show
as an angle, measured from the horizontal.
If drawn to scale, a ruler is used to measure the magnitude of the
resultant vector and a protractor is used to measure the angle q
(direction) of the resultant vector.
horizontal – zero line; used as a point of reference for measuring vector angles.
The direction q of the resultant vector
is found by placing a coordinate system with the origin at the tail of the
resultant vector. From the horizontal
(x-axis), start in quadrant I and move counterclockwise until the angle reaches
the resultant vector. The angle made is
the direction q of the resultant vector.
It is not always convenient to measure the magnitude and
direction of a vector. Using
trigonometry, the magnitude and direction of the resultant vector can be
calculated.
Recall the right triangle with respect to the
trigonometric functions sine, cosine, and tangent: SOH – CAH – TOA.
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To mathematically solve for any unknown angle q use
the trigonometric functions for sine, cosine, and tangent. To calculate an unknown side of the right
triangle, use the Pythagorean Theorem a2 + b2 = c2
where a and b are the legs of the triangle and c represents the hypotenuse.
Determine the magnitude and direction of the resultant
vector that represents a person walking 30 meters east or 0°,
then 40 meters north or 90°.
Using the Pythagorean Theorem the resultant vector has a
magnitude of 50 meters.
The direction q is found using SOH – CAH
– TOA. Since the opposite and adjacent
sides are already known, TOA is used.
The resultant vector is 50 meters @ 53.1°.
The resultant vector of a person walking shows the
displacement of the person, not the total distance traveled. For the above problem, the person walked a
distance of 70 meters, but the displacement, or change in position, was only 50
meters.
Change the order the person walks to see if the
displacement, distance walked, or resultant vector changes. Determine the magnitude and direction of the
resultant vector that represents a person walking 40 meters north or 90°,
then 30 meters east or 0°.
Using the Pythagorean Theorem the resultant vector has a
magnitude of 50 meters.
The angle or direction of the resultant vector q is
not simply the angle inside the right triangle.
The direction is always measured from the horizontal and
counterclockwise to the resultant vector.
Instead, trigonometry can be used to find an angle f
inside the right triangle which can be used to calculate q.
Since the opposite and adjacent sides are already known,
TOA is used to find f.
Since f and q form a right angle of 90°, q is
found from 90°
- f
= 90°
- 36.9°
= 53.1°.
The resultant vector is 50 meters @ 53.1°.
The order you add the vectors does not change the
magnitude or direction of the resultant vector.
The complexity of the problem is affected depending on the order the
vectors are added.
Summary Review – Vector Basics and Adding Perpendicular
Vectors
ü A vector is used to graphically represent vector quantities. The arrow-tip is called the head of the vector and the open end is called the tail of the vector.
ü Vectors are added by placing the tail of a vector to the head of the previous vector.
ü A resultant vector is the vector sum of two or more vectors. They are drawn from the tail of the first vector to the head of the last vector.
ü The direction or angle of a vector is always measured from the horizontal.
ü Trigonometric functions and the Pythagorean Theorem are used to calculate the magnitude and direction of resultant vectors.
Homework – Adding Perpendicular Vectors
1. What does a resultant vector represent?
2. Find the resultant vector if a person walks 22 meters west then 17 meters east. Find the distance the person traveled. What is the person’s displacement?
3. Find the resultant vector if a person walks 34 meters east then 19 meters south. Find the distance the person traveled. What is the person’s displacement?
4. A motorboat heads due east at 16 m/s across a river that flows due north at 9 m/s. What is the resultant velocity (speed and direction) of the boat? If the river is 136 meters wide, how long does it take the motorboat to reach the other side? How far downstream is the boat when it reaches the other side of the river?
Notes
The simple addition and trigonometric rules used to
calculate one-dimensional and perpendicular vectors do not apply to all
vectors. If the vectors are not
one-dimensional or perpendicular, the parts of each vector must be analyzed
individually and then combined to find the resultant vector.
Every vector can be thought of as a resultant vector –
made up of two perpendicular vectors.
These two vectors are the parts or components of the main vector.
component – one of two perpendicular vectors that make up a single vector; can be drawn two ways, for example, north then east or east then north.
vector resolution – process of finding the magnitude of a component in a given direction.
Vector resolution is done using trigonometry. First draw the components, Rx and
Ry and label the angle q. Using SOH – CAH – TOA and the magnitude and
direction of the vector, determine the value of the components.
Since the value of q and R are both know, the
cosine and sine are used to find Rx and Ry.
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Determine the components of the velocity vector 10 m/s @
30°.
Notes
When adding vectors that are not perpendicular, each
vector is first broken into its components.
To keep track of the components, the number of the vector is included as
a subscript.
The components of vector 1 are found using vector
resolution.
and 
The components of vector 2 are found using vector
resolution.
and 
Since all of the x components are known they can be added
together.
Since all of the y components are known they can be added
together.
Since the x and y components are known, the magnitude of
the resultant vector can be found using the Pythagorean Theorem.
The direction of the resultant vector is found using 
.
Find the resultant vector d (includes magnitude and
direction) that represent a person walking 12 meters @ 10° then
8 meters @ 120°.
d1 = 12
meters @ 10°
d2 = 8
meters @ 120°
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The magnitude and direction
of the resultant vector is 11.93 meters @ 49°.
Summary Review – Components of Vectors and Adding Vectors
at Any Angle
ü All vectors can be thought of as a resultant vector that can be broken into components.
ü
The x component is found using 
.
ü
The y component is found using 
.
ü Vector resolution is used to add vectors at any angle.
ü When adding vectors at any angle, each vector is broken into its components individually.
ü The final resultant vector represents the magnitude and direction of the added vectors.
Introduction – What’s Going On?
Vectors are used to describe quantities when direction is important. Velocity, position, weight, and force are a few examples of quantities that can be expressed as a vector. To further understand the significance of a vector, you will become one. Actually, you will make vectors, measuring what you do and using mathematics to verify your work. This lab contains two parts that review the procedures involved when creating vectors and calculating their properties using trigonometry. Equations used in this lab are listed below.
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Percent Error = |
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Your job is simply to create vectors. Each group will consist of four Rope Holders (2 on Team A and 2 on Team B), one Timer, and one Runner/Position Marker. To simplify the data collection process, position measurements will be taken on precise one-second time intervals.
It is the job of the Timer to call out the time at one second intervals. The Runner/Position Marker will be the person doing the actual moving around while dropping markers to mark their position every second. The Rope Holders (2 Teams) carry a stretched rope parallel to each other to help the Runner move in a straight line. Use the attached table for data collection.
Part I – Setting Up the Field of Play
- Mark off an area 30 meters by 30 meters.
- The origin will be located in the South-West corner.
- Team A will move East, creating the x-axis.
- Team B will move North, creating the y-axis.
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Team members will travel
at a constant speed, remaining parallel to each other at all times! |
Part II – Making Vectors
- Begin all trials with a member of each team at the origin.
- The Runner will follow at the intersection of the two ropes.
- The Runner will drop a marker every second.
- Collect data for the four trials shown in the data table.
Part III – Wrap Up
- Calculations
– Using the trial data, draw right triangles and calculate
the change in position (Dd) and direction (q)
of the Runner. Using the calculated
change in position, calculate the Runner’s velocity () in m/s.
Determine the Percent Error of change in position and
direction. (Measured Value =
Trial Data, Accepted Value = Calculated Value) Include at least one sample calculation
from each mathematical operation performed. (3 points/variable, 3 points for
triangle drawings, 2 points for data collection)
- Report Your Findings – Write a minimum of one paragraph explaining your results and any conclusions you can draw from what you experienced. Include an explanation of the sources for error beyond the obvious. (5 points)
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TRIAL (Speed Team A vs. Team B) |
1 (vA = vB) |
2 (vA > vB) |
3 (vA < vB) |
4 (2vA = 2vB) |
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Dt (sec) |
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Dx (cm) |
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Dy (cm) |
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Dd (cm) |
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q (°) |
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Dd (cm) |
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q (°) |
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v (cm/s) |
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Dd Percent Error |
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q Percent Error |
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1. Determine the resultant vector when walking 10 m east then 25 m west. What distance is covered? What is the displacement?
2. Determine the resultant vector when adding 35 m/s @ 0° and 54 m/s @ 90°.
3. Determine the displacement when traveling 44 m @ 270° then 19 m @ 180°.
4. Determine the components of the vector that represents 312 m/s @ 111°.
5. Determine the resultant vector when adding 4 m @ 200° and 11 m @ 16°.
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1. An airplane normally flies at 200 km/hr. What is the resultant velocity of the airplane if it experiences a 50 km/hr tail wind? A 50 km/hr head wind?
2. You head downstream on a river in a canoe. You paddle at 5 km/hr and the river is flowing at 2 km/hr. How far downstream will you be in 30 minutes?
3. A hiker leaves camp and, using a compass, walks 4 km east, 6 km south, 3 km east, 5 km north, 10 km west, 8 km north, and 3 km south. At the end of three days, the hiker is lost. How far is the hiker from camp? What direction should he travel to return to camp?
4. Diane rows a boat at 8 m/s directly across a river that flows at 6 m/s. What is the resultant speed of the boat? If the stream is 240 meters wide, how long will it take Diane to row across? How far downstream will Diane be?
5. A weather station releases a weather balloon. The balloon’s buoyancy accelerates it straight up at 15 m/s2. At the same time, a wind accelerates it horizontally at 6.5 m/s2. What is the magnitude and direction of the resultant acceleration?
6. Dave rows a boat across a river at 4 m/s. The river flows at 6 m/s and is 360 meters across. In what direction does Dave’s boat go? How long does it take Dave to cross the river? How far downstream is Dave’s landing point? How long would it take Dave to cross the river if there were no current?
7. Kyle wishes to fly to a point 450 km due south in 3 hours. A wind is blowing from the west at 50 km/hr. Compute the proper heading and speed that Kyle must choose in order to reach his destination on time.
8. Tammy leaves the office, drives 26 km at 95°, then turns onto a second highway and continues in a direction of 30° for 62 km. What is her total displacement from the office?