unit 4 review - mr desantis- old site · unit 4 review. kinematics. kinematics •when air...
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Unit 4 Review
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Kinematics
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Kinematics
• When air resistance is not taken into consideration, released objects will experience acceleration due to gravity, also known as freefall.
• Projectile motion can be predicted and controlled using kinematics
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ThetaFiring Angle (θ) is measured in degrees. It is the angle at which the projectile left the cannon.
Initial Velocity (Vi) is the angular speed of a projectile at the start of its flight.
iV
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gi
xV
sin2
iV
g
iV Initial Velocity
Gravitational Acceleration
x Horizontal Dis tance Traveled
Firing Angle
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g
iV Initial Velocity
Gravitational Acceleration
x Horizontal Dis tance Traveled
Firing Angle
g
2iV sin2
x
x
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Angle Firing
Traveled Distance Horizontal
onAccelerati nalGravitatio
Velocity Initial
x
g
Vi
iV
g1
2i
x2 sin
V
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Horizontal Motion:• Velocity is constant!!!•
Kinematics important info
Vix =
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Kinematics important infoVertical Motion:
-Velocity changes with time due to gravity
-Viy = Vi sin theta
-Velocity is zero in the y direction at peak
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Projectile Motion Problem
A ball is fired from a device, at a rate of 160 ft/sec, with an angle of 53 degrees to the ground.
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Projectile Motion Problem
• Find the x and y components of Vi.
• What is the initial vertical velocity?
• What is the ball’s range (the distance traveled horizontally)?
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Projectile Motion Problem
Find the x and y components of Vi.
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Projectile Motion Problem
Find the initial vertical velocity.
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Projectile Motion Problem
What is the ball’s range (the distance traveled horizontally)?
g
2iV sin2
x
Vi = 160 ft/secTheta = 53 degreesG = -32 ft/sec/sec
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Statistics
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Statistics
The collection, evaluation, and interpretation of data
Engineers use statistics to make informed decisions basedon established principles.
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Statistics
Statistics
Descriptive Statistics
Describe collected data
Inferential Statistics
Generalize and evaluate a population based on sample data
Statistics is based on both theoretical and experimental data analysis
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Methods of Determining Probability
• Empirical
• Experimental observationExample – Process control
• TheoreticalUses known elements
• Example – Coin toss, die rolling
• SubjectiveAssumptions
Example – I think that . . .
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ProbabilityThe number of times an event will occur divided by the number of opportunities
Px = Probability of outcome x
Fx = Frequency of outcome x
Fa = Absolute frequency of all events
x
x
a
FP
F
Expressed as a number between 0 and 1fraction, percent, decimal, odds
Total probability of all possible events totals 1
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Probability
x
x
a
FP
F
What is the probability of a tossed coin landing heads up?
How many possible outcomes? 2
How many desirable outcomes? 1
1P
2 .5 50%
Probability Tree
What is the probability of the coin landing tails up?
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Probability
x
x
a
FP
F
How many possible outcomes?
How many desirable outcomes? 1
1P
4
What is the probability of tossing a coin twice and it landing heads up both times?
4
HH
HT
TH
TT
.25 25%
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Binomial ProcessEach trial has only two possible outcomes
yes-no, on-off, right-wrong
Trial outcomes are independent Tossing a coin does not affect future tosses
x n x
x
n! p qP
x! n r !
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Bernoulli Process
x n x
x
n! p qP
x! n x !
P = Probability
x = Number of times an outcome occurs within n trials
n = Number of trials
p = Probability of success on a single trial
q = Probability of failure on a single trial
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Probability DistributionWhat is the probability of tossing a coin three times and it landing heads up two times?
2 23 2 1 0.5 0.5P
2 1 1 1
x n x
x
n! p qP
x! n x !
P 37.50%.375
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Law of Large Numbers
Trial 1: Toss a single coin 5 times H,T,H,H,TP = .600 = 60%
Trial 2: Toss a single coin 500 times H,H,H,T,T,H,T,T,……TP = .502 = 50.2%
Theoretical Probability = .5 = 50%
The more trials that are conducted, the closer the results become to the theoretical probability
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Probability
Independent events occurring simultaneously
Product of individual probabilities
If events A and B are independent, then the probability of A and B occurring is:
P = P(A) x P(B)
AND (Multiplication)
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ProbabilityAND (Multiplication)
What is the probability of rolling a 4 on a single die?
How many possible outcomes?
How many desirable outcomes? 1
6 4
1P
6
What is the probability of rolling a 1 on a single die?
How many possible outcomes?
How many desirable outcomes? 1
6 1
1P
6
What is the probability of rolling a 4 and then a 1 using two dice?
4 1P (P )(P )
1 1
6 6
1.0278
362.78%
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Probability
Independent events occurring individually
Sum of individual probabilities
If events A and B are mutually exclusive, then the probability of A or B occurring is:
P = P(A) + P(B)
OR (Addition)
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ProbabilityOR (Addition)
What is the probability of rolling a 4 on a single die?
How many possible outcomes?
How many desirable outcomes? 1
6 4
1P
6
What is the probability of rolling a 1 on a single die?
How many possible outcomes?
How many desirable outcomes? 1
6 1
1P
6
What is the probability of rolling a 4 or a 1 on a single die?
4 1P (P )(P )
1 1
6 6
2.3333 33 3 %
6. 3