The matrix of the linear transformation from ℝ2 to ℝ2 that rotates any vector through an angle of 120° in the anticlockwise direction is[tex]\left[\begin{array}{ccc}-1/2&\sqrt{3}/2 \\ \sqrt{3}/2 &-1/2\\\end{array}\right][/tex]
What is a matrix that can be taken for counterclockwise for an angle [tex]\alpha[/tex]
The matrix can be written as [tex]\left[\begin{array}{ccc} cos\alpha &sin\alpha \\ sin\alpha & cos\alpha \\\end{array}\right][/tex]
While putting the value of angle in the matrix, we have
[tex]\left[\begin{array}{ccc} cos120 &sin120 \\ sin120& cos120 \\\end{array}\right][/tex]
for [tex]\alpha[/tex] = 120
and thus, we have the answer as:
[tex]\left[\begin{array}{ccc}-1/2&\sqrt{3}/2 \\ \sqrt{3}/2 &-1/2\\\end{array}\right][/tex]
Therefore, the matrix can be stated as a matrix A of the linear transformation from r2 to r2 that rotates any vector through any angle of 120 in the anticlockwise direction.
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A star that has exhausted its supply of hydrogen is considered _____. group of answer choices
a. depleted
b. eroded
c. mature
d. dead
Answer:
A mature star is a star that has exhausted its supply of hydrogen
Explanation:
For the most part, a star is powered by nuclear fusion. The energy is first produced by the fusion of hydrogen atoms in the core of a main-sequence star. Later, when the helium predominates in the core, stars like the Sun begin to fuse hydrogen along a spherical shell around the core. This allows the star to steadily increase in size, passing through the subgiant stage and eventually reaching the red giant stage.
When they reach this stage, the hydrogen in them nearly completely depletes, and they get brighter and hotter, eventually becoming low-mass white dwarfs.
When a star's supply of hydrogen in its core runs out, leaving only helium, the outward force provided by fusion begins to diminish, and the star can no longer maintain equilibrium. When the force of gravity exceeds the force of internal pressure, the star begins to disintegrate. The outcome of this collapse is determined by the mass of the star.
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After oxygen has been administered, the next priority intervention the nurse would initiate for a patient with a pulmonary embolus is the administration of which of these therapies?
After oxygen has been administered, the next priority intervention the nurse would initiate for a patient with a pulmonary embolus is the administration of IV Heparin.
What is Heparin?Heparin is a prescription medicine used to treat and prevent symptoms of blood clots caused by medical conditions or medical procedures. Heparin may be used alone or with the other medications. Heparin belongs to a class of drugs called the Anticoagulants, Cardiovascular, Anticoagulants, Hematologic.
What are the side effects of Heparin?Heparin may cause serious side effects including:
skin warmth or discoloration,
chest pain,
irregular heartbeats,
shortness of breath,
dizziness,
anxiety,
sweating,
unusual bleeding or bruising,
severe pain or swelling in your stomach, lower back/growing
dark or blue-colored skin on your hands/feet,
nausea,
vomiting,
loss of appetite,
unusual tiredness,
bleeding that will not stop,
nosebleed,
blood in the urine or stool,
black or tarry stools,
coughing up blood or vomit that looks like the coffee grounds,
skin changes where the medication was injected,
fever,
chills,
runny nose,
watery eyes,
easy bruising,
purple or red spots under your skin,
sudden numbness or weakness,
problems with vision or speech, and
swelling or redness in an arm or leg
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Suppose the irregularly
shaped object is a massive whale shark that
is about to be moved into a cube-shaped
aquarium tank. How might you combine the
"Length × Width x Height" formula with the
displacement method to determine the
shark's volume? Explain.
The displacement method can be used to determine the shark's volume by measuring the corresponding volume of the displaced water in the can.
How to measure the volume of irregular shapeIf the object has an irregular shape , the volume can be measured using a displacement can.
The displacement can is filled with water above a narrow spout and allowed to drain until the water is level with the spout. As the irregular object is lowered into the displacement can, the water level rises.
The level of water displaced is the volume of the irregular shape.
Thus, the displacement method can be used to determine the shark's volume by measuring the corresponding volume of the displaced water in the can.
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A high-voltage power line is a catenary described by y = 9cosh(x/9) find the ratio of the area under the catenary to its arc length.
The correct answer for the ratio of the area under the catenary to its arc length is 12 : 1 .
y = 12cosh(x/12) Since the анс length from FGN) on the interval [c, d] is Arc length =
Let y' =f'(x) prime = derivative of 12cosh(x/12) for x = Sinh(x/12) .
. A(x) length = ∫√1+ Sinh² (1/2). dx
= int c ^ d sqrt(cosh^2 (alpha/12)) * dx
= integrate d from c to d cosh( x 12 )* dx
=[12. sinh(x/12) ] e ^ d
= 12[sinh(d/12) - sinh(c/12)]
and area under f(x) on the interval [c, d]
Area = ∫ f(x).dx from [c,d ]
= 144[sinh(d/12) - sinh(c/12)]
the area ratio of the area under the catenary to it's length is
= 144[sinh((d)/12) - sinh(c/12))/ 12[sinh( d/ 12 ) -sinh( c /12 )] .
=12
So, ratio is 12:1
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The density beyond which people cease to be nutritionally self-sufficient is called:________
Physiological nutrition is the density at which people are no longer self-sufficient in terms of nutrients.
How the body absorbs nutrients from food, how we get the energy we require, how we use nutrients, and how all of this relates to health and disease are all topics covered by nutrition physiology.
Why does physiology need nutrition?
Nutritional requirements exist in addition to daily energy requirements in order to keep the body from losing its own proteins, carbs, and lipids. Such molecules must be replenished frequently because they are constantly broken down.An organism uses food to prolong its life through some kind of biochemical and physiological process known as nutrition. It gives living organisms nourishment that can be transformed into energy and chemical building blocks.The ingestion of nourishment, meeting psychological needs, and bodily adjustment are the three purposes of food.To learn more about Physiological nutrition, visit:
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Ordered: vistaril 75 mg im now. used with permission from pfizer, inc how many ml will you give for each dose?
Option c) 1.5 ml of volume will be given for each dose.
How does Vistaril work and what does it do?The prescription drug Vistaril is used as preoperative sedation, to treat anxiety symptoms, and to cure skin hives or itching. Vistaril may be taken either on its own or along with other drugs.
The class of medications that includes Vistaril includes antiemetic agents, first-generation antihistamines, and piperazine derivative antihistamines.
What Vistaril side effects might there be?Serious adverse reactions to Vistaril include:
Heartbeats that are rapid or hammering, headaches accompanied by chest pain, extreme dizziness, fainting, and convulsions (seizures)
Immediately seek medical attention if you experience any of the above symptoms.
The most typical Vistaril side effects include:
Fatigue, headache, dry mouth, and rash on the skin
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If the maximum energy given to an electron during compton scattering is 30 kev, what is the wavelength of the incident photon?
Required wavelength is λ = 0.0119 nm
The Compton effect in physics is defined as the phenomenon that arises when a specific beam is dispersed on a specific particle with an increase in wavelength.
Equation :Given;
The campton scattering is k = 30 kev
The speed of light is c = 2.9979 X 10 ⁸ m/s
The planck's constant is h = 6.626 X 10⁻³⁴
The expression from Campton shift is given by,
λ' - λ = λc (1 - Cos∅)
Here, ∅ is the angle of scattering whose value is 180° for maximum energy transfer and λc is the campton wavelength whose value is
λc = h/mc
substitute the values in the above expression
λ' - λ = λc (1 - Cos 180°)
λ' - λ = 2 (h/mc)
Here, m is the mass of electron
On rearing the above expression
λ' = λ + 2 (h/mc)
The expression from conservation of energy is given by,
E - E' = k
(hc/λ) - (hc/λ') = k
hc((λ' - λ)/λλ') = k
hc ((λ' - λ)/k) = λλ'
Here E and E' are the initial and final energy of electron
substitute the values in the above equation
hc (2(h/mc)/k)) = λ (λ + 2 (h/mc))
λ² + (2h/mc)λ - 2h²/mk = 0
λ² + [tex]\frac{2(6.626X10^{-34}Js)}{(9.11X10^{-31}kg) (2.997X10^{8}m/s ) }[/tex] λ - [tex]\frac{2(6.626X10^{-34}Js)}{(9.11X10^{-31}kg) (30X10^{3}evX\frac{1.6X10^{-19} }{1ev} J) }[/tex] = 0
λ² + (4.85X10⁻¹²)λ - (2X10⁻²²) = 0
On solving the above expression
λ = 1.19 X 10⁻¹¹ m
λ = (1.19 X 10⁻¹¹ m X 10⁹ nm/1m)
λ = 0.0119 m
Thus, required wavelength is λ = 0.0119 nm
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When using higher magnification, you need more light. which parts of the microscope can be adjusted to provide more light, and how would you adjust those parts to increase the light?
Increased lamp voltage is achieved by turning the light intensity dial.
To enlarge the diameter of the hole and let more light through the slide, the iris diaphragm was modified.
Condenser: Position it higher and closer to the slide's bottom to better direct light to the centre of the slide.
How do you adjust the light level on a microscope?Utilize the brightness adjustment knob to change the brightness. Turn the brightness control knob while looking through the eyepieces to make sure there is no glare in the field of view.
Use a daylight balancing filter if your compound microscope has a certain sort of illumination. It typically rests directly on top of the luminator or in a filter holder above the light. This filter is blue.
The daylight balancing filter will correct the colour temperature and produce a higher-quality image if your microscope is lighted by tungsten or halogen (and a better colour image). This blue filter is not necessary if your microscope is an LED.
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Object a is moving at speed v in a circle of radius r. object b is moving at speed 2v in a circle of radius r/2. what is the ratio of the period of object a to the period of object b?
The correct answer for the ratio of the period of object a to the period of object b 4.
we know time for one revolution. around which is the period for one revolution it has to travel distance equal to perimeter.
So, speed of object in circular motion
v = perimeter /t
time period = 2πr/v
for object A ,Ta = 2 π R/ v
for abject B, Tb = (2π R/2 )/2v
now Ta / Tb = ((2πR)/v)/((2π(R/2))/(2V)) = 1/(1/4) = 4
Circular motion is defined in physics as the movement of an object around the circumference of a circle or the rotation of an object along a circular path. It can be uniform, with a constant angular rate of rotation and speed, or non-uniform, with a variable rate of rotation.When the magnitude of an object's velocity in uniform circular motion is v, the velocity is equal to the circumference C of the circle divided by the period. As a result, V = fracCT. The circumference of a circle is equal to pi multiplied by R. So, C = 2Π R.
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In traditional mass media, __________ is the paid media space that sponsors use to persuade audiences
In traditional mass media, advertising is the paid media space that sponsors use to persuade audiences
Advertising is a form of communication aimed at publicizing a product or service in order to influence the public to purchase or use it.
This type of communication is paid and promoted by companies and media with a commercial purpose.
For example, a company specializing in designing and implementing ads for other company's products is an advertising company.
There are four types of advertising:
DisplayVideoMobileNativeWhat is communication?the communication is a process through which information is transmitted through the use of oral or written language.
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Which layer of the osi model ensures the data being sent from the sender is formatted correctly so it can be interpreted correctly by the recipient system?
6th layer of OSI model ensures complete and rebel delivery of data packets.
(presentation layer is the 6th layer.)
OSI - Open system Interconnection is a conceptual model that provides a common basis for the condition.
It is a reference model for how application communicate over a network.
This model describes 7 layer that computer system used to communicate over a network.
It was the model for network communication and adopted by all major computer and companies.
The complications between US computing system are split in several layers.
physical. Datalink. Network. Transport. Session Presentation Application.The purpose of OSI reference model is to guide technology developer.
4th layer (presentation layer) of OSI model ensures complete and rebel delivery of data packets.
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What is the average rate of decomposition of a3 in the time interval 20-30 seconds?
5.0E-6(0.000005) is the average rate of decomposition of a3 in the time interval 20-30 seconds.
what is decomposition?Decomposition, often known as rot, is the breakdown of dead organic matter into more basic organic or inorganic components such carbon dioxide, water, simple sugars, and mineral salts.The procedure, which is a component of the nutrient cycle, is crucial for recycling the limited amount of stuff that takes up real estate in the biosphere.After a live thing dies, its body starts to decay.Animals like worms also contribute to the breakdown of the organic compounds. Decomposers are organisms that carry out such processes.Despite the fact that no two creatures degrade in exactly the same way, they all go through the same sequential phases.Taphonomy, which comes from the Greek word taphos, which means tomb, is the broad name for the science that examines decomposition.To learn more about decomposition, refer to the following link:
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There is a negative surface charge density in a certain region on the surface of a solid conductor. just beneath the surface of this region, the electric field
Just beneath the surface of this region, the electric field is zero.
Surface charge densityCharge density is a measure of the amount of electric charge present in a space per unit volume in one, two, or three dimensions, according to electromagnetism.
The quantity of electric charge per surface area or volume, respectively, is known as the linear surface or volume charge density.
What is the surface charge density equation?Σ = q/A
The formula for surface charge density is = q/A, and the SI unit for surface charge density is coulombs per square meter (cm2). Assuming q is the charge and an is the area of the surface over which it flows
What does surface charge mean?A two-dimensional surface having a positive electric charge is called a surface charge. This 2-D surface and surface are restricted by these electric charges.
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4. Computer programmers in the 1970s thought 8 kilobytes was a lot of memory and the
computers fit the space of about 10 textbooks. Some cell phones today can store 128
Gigabytes. How many 1970 computers would it take to match the memory of a cell
phone today?
Answer:
1.28E11 / 8.0E3 = 1.60E7 computers (16,000,000 computers)
What approach responds to the limitations of the other legacy-measuring methods?
The other-legacy measuring methods' weaknesses are addressed by the bottom-up approach.
An organizational decision-making process known as a "bottom-up approach" begins at the bottom of the hierarchy rather than the top. In actuality, this means that choices won't always be made by the CEO or department head (this is known as a top-down approach).
What distinguishes a bottom-up strategy from a top-down approach?
When choices affecting the entire organization are made, top-down management is used, whereas bottom-up management involves all teams in these types of decisions.The bottom-up approach often focuses its analysis on specific traits and micro features of a single stock. The focus of bottom-up investment is on fundamentals that apply to individual companies or industries.What bottom-up management has to offer:
Promoting cooperation.Enhancing morale. Creating new insights. Enabling your group.To learn more about Bottom-up approach, visit:
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A jetliner goes from rest to a takeoff speed of 183 mi/h in 36.1 s. what is the magnitude of its average acceleration?
A jetliner goes from rest to a takeoff speed of 183 mi/h in 36.1 s.
the magnitude of its average acceleration is 81.80
CalculationTake off speed of the jetliner = 183 mi/h
so time taken = 36.1s
average acceleration = change in velocity / time taken
= (183 x 1609.3)/ 3600
= 81.80 m/[tex]s^{2}[/tex]
What is average acceleration?The rate at which an object’s velocity changes is measured as its acceleration. Since it is a vector, speed and direction are also included. In the SI, acceleration is measured in meters per second squared (m s 2).
An object’s velocity is how quickly it is travelling in a specific direction. A vector, that is. In the SI, velocity is measured in meters per second (m s).
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If, in the figure, you start from the Bakery, travel to the Cafe, and then to the Art Gallery in 1.50 hours. What is the average velocity?
The average velocity is 1 km/hr.
What is average velocity?Velocity is the speed of a body in a given direction.
Velocity can also defined as the displacement of body with time.
The average velocity will be be the ratio of the total displacement with time.
Average velocity = displacement/timeTaking the right direction as positive and the direction to the left as negative:
displacement of the body = (4.0 - 2.5) km
displacement of the body = 1.5 km
Average velocity = 1.5 km/ 1/5 hours
Average velocity = 1 km/hr
In conclusion, the average velocity is obtained from the displacement and time taken.
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1) If students spend more time using TikTok, then they will read fewer books. Independent Variable:
Dependent Variable:
Answer: number of books
Explanation:
Water comes down the spillway of a dam from an initial vertical height of 170 m. what is the highest possible speed of the water at the end of the spillway?
The speed of water at the end of spillway is 57.7 (m/s)
Calculation:
According to the Bernoulli equation:
P/? + v^2/2 + gz = constant
However, the atmospheric pressure and density are essentially the same at the peak and bottom of the falls. Therefore,
v^2/2 + gz = constant , so:
v_bottom^2/ 2 = v_top^2/2 + gz, or:
v_bottom = sqrt(v_top^2 + 2gz) = sqrt(v_top^2 + 2g*170)
If v_top ~ 0, then:
v_bottom ~ sqrt(2gz) = sqrt(2*9.8*170) = 57.7 (m/s)
What is the flow rate of water?
The amount of water that might theoretically flow out of your kitchen faucet or bathtub each minute is measured by your water flow rate, sometimes referred to as your gallons per minute (GPM). Your flow rate is affected by a number of variables, but your household size is the most important one.
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Autotransformers are especially practical if the difference between the primary and secondary voltages is _____.
Autotransformers are especially practical if the difference between the primary and secondary voltages is realtively small.
What is Autotransformer?An autotransformer is a type of electrical transformer that has just one winding. The word "auto" refers to a single coil that runs autonomously, not to a system that runs without human intervention. The primary winding and secondary winding sides of an autotransformer are distinct portions of the same winding. While there is no metallic conducting channel between the primary and secondary windings of a normal transformer, they are separate. The autotransformer winding has at least three taps that are utilized to connect electrical appliances. In terms of size, weight, and cost, autotransformers usually outperform conventional dual-winding transformers, but they also have the disadvantage of lacking electrical isolation between the primary and secondary circuits. Autotransformers also have advantages in terms of reduced leakage reactance, fewer losses, lower excitation current, and higher VA rating for a given size and mass.Learn more about the Autotransformer with the help of the given link:
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As your friend skateboards down a hill, you measure their velocity at 1.0-second
intervals. You record the following data: ((0.0 s, 0.0 m/s), (1.0 s, 2.3 m/s),
(2.0 s, 4.6 m/s)}. Determine your friend's acceleration.
Answer:
what does the acronym VAT stand for
The correct answer for the acceleration is [tex]2.3[/tex] m/s².
Given:
Initial velocity:- [tex]0.0[/tex] m/s
Final velocity:- [tex]4.6[/tex] m/s
Time interval:- [tex]2.0[/tex] seconds
Acceleration calculated here will be average acceleration:
From equation of motion:
[tex]v= u+at[/tex]
[tex]a = \dfrac{v-u}{t}[/tex]
[tex]= \dfrac{4.6 -0.0}{2}[/tex]
[tex]= 2.3[/tex]
Acceleration is [tex]2.3[/tex] m/s².
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The postal service had a special deal. You could ship up to 2 kg for five dollars for a
limited time. For Amy's friend's birthday, Amy wanted to ship her sweets and candy,
and she wanted to get the best bang for her buck. She first made her cookies. They
had a mass of 400 grams. How many kilogram (s) of sweets and candy could Amy
still ship if only the cookies were in the box for the five dollar special?
weight of sweets and candy is 1.6kg
It is given that a total weight of 2 kg is allowed by the postal service
The weight of cookies is given to be 400gm
we know that 1kg =1000gm
400gm= 400/1000 gm =0.4kg
Therefore weight of sweets and candy that Amy can send=
total weight- weight of cookies
=2kg - 0.4kg
=1.6kg
Therefore, weight of sweets and candy is 1.6kg
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___________ measurements of the orbital radii of six of the planets were very accurate compared to our modern values.
Copernicus measurements of the orbital radii of six of the planets were very accurate compared to our modern values.
Describe Copernican heliocentrism.Nicolaus Copernicus created the Copernican heliocentrism astronomical theory, which was first presented in print in 1543. This theory put the Sun in the centre of the cosmos, stationary, and had Earth and the other planets orbit it at uniform speeds in circular orbits modified by epicycles. The geocentric Ptolemaic paradigm, which put Earth at the centre of the cosmos for ages, was replaced by the Copernican model.
Although he had shared a draught of his own heliocentric theory with peers before 1514, he did not choose to publish it until after being persuaded to do so by his student Rheticus.
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A ship travels 20 km to the south and then 40 km to the west. the ship's displacement from its starting point is?
A ship travels 20 km to the south and then 40 km to the west. The ship's displacement from its starting point is approximately 44.7 km. Hence, the displacement is approximately 45 km.
Here, the calculation is done,
a = 20 km (southward movement)
b = 40 km (westward movement)
Now, one can find the displacement (d) using the Pythagorean theorem:
d² = a² + b²
d² = (20 km)² + (40 km)²
d² = 400 km² + 1600 km²
d² = 2000 km²
Now, by taking the square root of both sides to find the displacement (d):
d = √(2000 km²) d ≈ 44.7 km (rounded to one decimal place)
So, the ship's displacement from its starting point is approximately 44.7 km.
Hence, the displacement is approximately 45 km.
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what is nuclear emission?
True or false: in applying the pythagorean theorem to vector addition, the resultant vector is given by the perpendicular vector.
It is False in applying the pythagorean theorem to vector addition, the resultant vector is given by the perpendicular vector
What is Pythagoras theorem ?A useful technique for figuring out the outcome of adding two (and only two) vectors that form a right angle to one another is the Pythagorean theorem. The technique cannot be used to combine more than two vectors or to combine vectors that are not at a 90-degree angle.
The Pythagorean Theorem is named for Pythagoras of Samos, a religious figure and mathematician who held the view that everything in the cosmos is made up of numbers. (There are numerous ways to demonstrate this.) The side opposite the right angle is known as the hypotenuse in a right triangle.The Pythagorean Theorem demonstrates how the side lengths of a right triangle can be calculated from the sum of the areas of three intersecting squares. This theorem is a very helpful tool that serves as the foundation for more intricate trigonometry ideas like the Pythagorean theorem's opposite.Learn more about Pythagoras theorem here:
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The idea that light consists of photons, bundles of pure energy, was first proposed by?
Einstein was the first to postulate that light is made up of photons, which are essentially energy bundles.
What a photon is made of?A photon is a microscopic particle made up of electromagnetic radiation waves. Maxwell demonstrated that photons are merely electric fields moving through space. Photons move at the speed of light and have no charge or rest mass. Charged particles produce photons, though they can also be produced by other processes, such as radioactive decay.
It's common to refer to photons as energy packets. This analogy is excellent because a photon has uncondensable energy. An alternating electric field is how this energy is kept in reserve. The universe's size, which is the longest theoretical wavelength of light, and the Planck length, which some theories claim to be the smallest, have never been measured. These energy packets may travel great distances without losing energy or speed. In vacuum, photons move at the speed of light, which is 2.997 x 108 m/s.
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Astronomers infer that the universe is expanding because distant galaxies all appear to?
Answer:
They appear to recede from our viewpoint.
That is, their emitted spectra are shifted toward the red (red-shift).
The wave-lengths that are received on earth are somewhat longer because of the "Doppler" shift of the receding atoms.
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 6 i 3 j, v(0) = k, r(0) = i
Use the fundamental theorem of calculus.
[tex]\displaystyle \vec v(t) = \vec v(0) + \int_0^t \vec a(u) \, du[/tex]
[tex]\displaystyle \vec v(t) = \vec k + \int_0^t (6\,\vec\imath+6\,\vec\jmath) \, du[/tex]
[tex]\boxed{\vec v(t) = 6t\,\vec\imath + 6t\,\vec\jmath + \vec k}[/tex]
[tex]\displaystyle \vec r(t) = \vec r(0) + \displaystyle \int_0^t \vec v(u) \, du[/tex]
[tex]\displaystyle \vec r(t) = \vec\imath + \displaystyle \int_0^t (6t\,\vec\imath + 6t\,\vec\jmath + \vec k) \, du[/tex]
[tex]\boxed{\vec r(t) = (3t^2 + 1)\,\vec\imath + 3t^2\,\vec\jmath + t\,\vec k}[/tex]
The velocity vector is v(t) = 6t i + 3t j + k, and the position vector is r(t) = [tex]\rm 3t^2 i + (3/2)t^2 j + kt + i.[/tex]
We have to integrate the specified acceleration vector to obtain the velocity vector of the particle before integrating the velocity vector to obtain the position vector of the particle.
So, the acceleration vector: a(t) = 6i + 3j
We will the find the velocity vector,
To find the velocity vector, we integrate the acceleration vector with respect to time 't':
∫a(t) dt = ∫(6i + 3j) dt
when we integrate with respect to time 't' that gives us:
v(t) = 6t i + 3t j + C
where 'C' is the constant of integration.
Using the initial velocity condition to find 'C':
Given initial velocity: v(0) = k
Substitute t = 0 into the velocity vector equation:
v(0) = 6(0) i + 3(0) j + C
k = C
So, C = k
We put the value of 'C' back into the velocity vector equation:
v(t) = 6t i + 3t j + k
Calculate the position vector r(t):
To find the position vector, we integrate the velocity vector with respect to time 't':
∫v(t) dt = ∫(6t i + 3t j + k) dt
Integrating with respect to time 't' gives us:
r(t) = [tex]\rm 3t^2 i + (3/2)t^2 j + kt + D[/tex]
where 'D' is the constant of integration.
We use the initial position condition to find 'D':
Given initial position: r(0) = i
Substitute t = 0 into the position vector equation:
r(0) = [tex]\rm 3(0)^2 i + (3/2)(0)^2 j[/tex]+ k(0) + D
i = D
So, D = i
when we put the value of 'D' back into the position vector equation:
r(t) = [tex]\rm 3t^2 i + (3/2)t^2 j + kt + i[/tex]
Therefore, the velocity vector is v(t) = 6t i + 3t j + k, and the position vector is r(t) = [tex]\rm 3t^2 i + (3/2)t^2 j + kt + i.[/tex]
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