# 2.2 Vector addition and subtraction: graphical methods  (Page 5/16)

 Page 5 / 16
Height (m) Time (sec)
0.490 0.1
0.882 0.2
1.176 0.3
1.372 0.4
1.470 0.5
1.470 0.6
1.372 0.7

A ball is launched at an angle of 60 degrees above the horizontal, and the vertical position of the ball is recorded at various points in time in the table shown, assuming the ball was at a height of 0 at time t = 0.

1. Draw a graph of the ball's vertical velocity versus time.
2. Describe the graph of the ball's horizontal velocity.
3. Draw a graph of the ball's vertical acceleration versus time.

The graph of the ball's vertical velocity over time should begin at 4.90 m/s during the time interval 0 - 0.1 sec (there should be a data point at t = 0.05 sec, v = 4.90 m/s). It should then have a slope of -9.8 m/s 2 , crossing through v = 0 at t = 0.55 sec and ending at v = -0.98 m/s at t = 0.65 sec.

The graph of the ball's horizontal velocity would be a constant positive value, a flat horizontal line at some positive velocity from t = 0 until t = 0.7 sec.

## Summary

• The graphical method of adding vectors $\mathbf{A}$ and $\mathbf{B}$ involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector $\mathbf{R}$ is defined such that $\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{R}}$ . The magnitude and direction of $\mathbf{R}$ are then determined with a ruler and protractor, respectively.
• The graphical method of subtracting vector $\mathbf{B}$ from $\mathbf{A}$ involves adding the opposite of vector $\mathbf{B}$ , which is defined as $-\mathbf{B}$ . In this case, $\text{A}–\mathbf{\text{B}}=\mathbf{\text{A}}+\left(\text{–B}\right)=\text{R}$ . Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector $\mathbf{R}$ .
• Addition of vectors is commutative    such that .
• The head-to-tail method    of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
• If a vector $\mathbf{A}$ is multiplied by a scalar quantity $c$ , the magnitude of the product is given by $\text{cA}$ . If $c$ is positive, the direction of the product points in the same direction as $\mathbf{A}$ ; if $c$ is negative, the direction of the product points in the opposite direction as $\mathbf{A}$ .

## Conceptual questions

Which of the following is a vector: a person's height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth's population, the acceleration of gravity?

Give a specific example of a vector, stating its magnitude, units, and direction.

What do vectors and scalars have in common? How do they differ?

Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?

If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in [link] . What other information would he need to get to Sacramento?

Suppose you take two steps $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point $\mathbf{\text{A}}+\mathbf{\text{B}}$ the sum of the lengths of the two steps?

current economic plans (MDGS) needs
I don't know what is happening
surajkumar
What is economic
What is the importance of study economics
Wilma
Economic is the study of how humans make decisions in face of sacristy
Wilma
economics is the study of how humans makes decision in the face of scarcity
Kpienta
economics is the study of human behaviour when faced with difficult situation example when goods and services are scarcity.
Sydney
what is Economic
what is 4ps of economic?
production place Price product
Benedict
Criticism of elasticity
what is unemployment
ohk thanks
Gyamfi
why is unemployment rapid in the country
Gyamfi
I need more explanation
Odo
what is unemployment
not working
Bethel
some one who is willing qualified to work but can't find job
jackie
Abubakar
some one who is willing to work but can't find job
Hawa
Yes true
Brian
Hawa
unemployment refers to the ability for someone who is capable and willing to work but could not find a job..
Mnoko
some one who not able to find a job
Dennis
please what is the secret of learning?
thomas
What is stock market?
explain the various types of cost curve
Short-run average fixed cost (SRAFC) Short-run average total cost (SRAC or SRATC) Short-run average variable cost (AVC or SRAVC) Short-run fixed cost (FC or SRFC) Short-run marginal cost (SRMC) Short-run total cost (SRTC)
Romy
what's economic development and growth
what do you understand by Ceteris Paribus?
the external factor will remained constant, except the price
Hasib
everything being equal
Chenwi
explain the uses of microeconomics
uses of microeconomics
Nikita
what is economic deficit
this is a situation whereby a nation's outcome or available resources are not enough to the people thereby causing scarcity
Ariel
prices of Quality demanded is equal to Quality supplied
it's quantity demand and quantity supplied that's called equilibrium
Romy
no
NABUBOLO
they deal With prices
NABUBOLO
define the elasticity
NABUBOLO
explain different types of elasticity
NABUBOLO
oops 😬 you are right you talk about quality I tell about quantity
Romy
elasticity is the measurement of the percentage change of one economic variable in response to a change in another
Romy
Cross Elasticity of Demand (XED) Income Elasticity of Demand (YED) Price Elasticity of Supply (PES)
Romy
anything else?
Romy
I need to know everything about theory of consumer behavior
Grace
Romy, what is microeconomic?
thomas
Dauda
Thomas, microeconomics is the study of how consumers, workers, and firms interact to generate outcomes in specific markets
Kieran
Dauda, economics is the study of people and choices. it is on one side the study of wealth and on the more important side, a part of the study if man
Kieran
How does one analyze a market where both demand and supply shift?
That's equilibrium market
Ramon
but an equlibrum can appear twice on the same market... both in Movement along the Demand/supply curve of shift in the Curve
Gabriel
I Mean on the same curve..
Gabriel
how can consumer surplus be calculated
Franklyn
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Got questions? Join the online conversation and get instant answers!