Answers
a. The sketch graph of U(x) versus x is in the attachment.
b. The force function that associate with the potential energy function is F(x) = 8.0 (2.0 + x)⁻²
c. The speed of the particle at x = 2 is 2 m/s.
The full question is in the attachment. To create a graph, substitute some values of x into the equation of the function to determine the pair of point coordinates.
a. Graph of U(x) = 4.0/(2.0 + x) versus x making a curve because the function is a fraction function and the curve will not intersect the x-axis.
If x = 0U(0) = 4.0/(2.0 + x) = 4.0/(2.0 + 0) = 4.0/2.0 = 2.0
(0 , 2.0) If x = 2
U(2) = 4.0/(2.0 + x) = 4.0/(2.0 + 2) = 4.0/4.0 = 1.0
(2 , 1.0)If x = 3
U(3) = 4.0/(2.0 + x) = 4.0/(2.0 + 3) = 4.0/5.0 = 0.8
(3 , 0.8)
From these coordinates a graph is created in the image.
b. A conservative force is a force that works on an object depends only on the initial and final position of the object and not on the path of the object. The total work from a conservative energy is always the same with the changes in potential energy
W = - ΔU
F × Δx = - ΔU
W = the total work (J)Δx = the distance (m)ΔU = the potential energy (J)The conservative force acting on the object is the derivate of the potential energy function for the objects
F = - dU/dx
U(x) = 4.0/(2.0 + x) = 4.0 (2.0 + x)⁻¹
F = - 4.0 × - 2 (2.0 + x)⁻² × 1
F(x) = 8.0 (2.0 + x)⁻²
c. The kinetic energy Ke = 0.5 mv²
m = the massv = the speedKe = kinetic energyWhen the object is released from rest at the origin to x = 2
The initial speed = v₁ = 0Kinetic energy = Ke₁ = 0.5 mv₁² = 0 J
Potential energy = U(0) = 4.0/(2.0 + x) = 4.0/(2.0 + 0) = 4.0/2.0 = 2.0 JAt x = 2
Potential energy = U(2) = 4.0/(2.0 + x) = 4.0/(2.0 + 2) = 4.0/4.0 = 1.0 J
According to law of conservation of energy
Ke₁ + U(0) = Ke₂ + U(2)
0 + 2.0 = Ke₂ + 1.0
Ke₂ = 2.0 - 1.0
Ke₂ = 1.0 J
0.5 mv₂² = 1.0
0.5 × 0.5 × v₂² = 1.0
v₂² = 1.0 ÷ 0.25
v₂² = 4
[tex]v_2 \:=\: \sqrt{4}[/tex]
v₂ = 2 m/s
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Related Questions
Elastic and (for our purposes) Nearly Elastic Collisions:
Select all that apply
conserve the momentum of each object
conserve the total kinetic energy of the system
conserve the total momenta of the system
conserve the kinetic energy of each object
Answers
For elastic collision: conserve the total kinetic energy of the system and conserve the total momenta of the system.
option B and C.
What is elastic collision?
This is a type of collision in which the two objects move with a different velocities after the collision.
In elastic collision both momentum and kinetic energy is conserved unlike the inelastic collision where only momentum is conserved and the kinetic energy of the system is not conserved.
Thus, we can conclude that in an isolated system during elastic or inelastic collision the sum of the initial momentum of the system is equal to the sum of the final momentum of the system.
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A train starts from rest and speeds up for 15 minutes until it reaches a constant velocity of 100 miles/hour. It stays at this speed for half an hour. Then it slows down for another 15 minutes until it is still. Which of the following correctly describes the position vs time graph of the train’s journey?
A: The first 15 minutes is a curve that is concave upward, the middle portion is a straight line with slope 100 miles/hour, and the last portion is a concave downward curve.
B: The first 15 minutes is a curve that is concave downward, the middle portion is a straight line with slope 100 miles/hour, and the last portion is a concave upward curve.
C: The first 15 minutes is a curve that is concave upward, the middle portion is a straight line with slope zero, and the last portion is a concave downward curve.
D: The first 15 minutes is a curve that is concave downward, the middle portion is a straight line with slope zero, and the last portion is a concave upward curve.
Answers
Answer:
The correct answer is: B: The first 15 minutes is a curve that is concave downward, the middle portion is a straight line with slope 100 miles/hour, and the last portion is a concave upward curve.
Explanation:
The first 15 minutes of the journey, the train is accelerating from rest to reach a constant velocity of 100 miles/hour. The position vs time graph of this portion of the journey would be a curve that is concave downward, as the acceleration is positive and the velocity is increasing.
The middle portion of the journey, the train is moving at a constant velocity of 100 miles/hour. The position vs time graph of this portion of the journey would be a straight line with slope 100 miles/hour, as the velocity is constant.
The last 15 minutes of the journey, the train is slowing down to come to a stop. The position vs time graph of this portion of the journey would be a curve that is concave upward, as the acceleration is negative and the velocity is decreasing.
What term describes speeding up,
slowing down, and changing
direction?
A. acceleration
B. neither
C. speed
Answers
Answer:A: Acceleration
Explanation: I hope this helps!
Need the answers can someone help me please
Answers
An update can be caused by changes to props or state. These methods are called in the following order when a component is being re-rendered.
what is the co ordinates of initial and terminal points?
The length of the arrow indicates the magnitude of the vector, and the direction of the arrow is the direction of the vector. The point at the tail of the arrow is called the initial point of the vector, and the tip of the arrow is called the terminal point.Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at (0,0) and is identified by its terminal point ⟨a,b⟩.Zero vector A vector whose initial point and terminal point both lie at the origin; denoted by 0 = 〈0, 0〉.
The terminal point p(x, y) on the unit circle determined by the giving value of t = -3π/4 is determined to be P (1/√2, -1/√2 ).To learn more about coordinates refers to:
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The radius and length of a steel cylinder are 40 mm and 110 mm, respectively. If the mass density of steel is 7850 kg/m3 , determine the weight of the cylinder in pounds.
Answers
The weight of the cylinder in pounds is 26.45 lbs.
Solving for the weight of the cylinder in pounds:To determine the weight of the cylinder in pounds, we first need to calculate its volume. The volume of a cylinder is given by :
V = πr²h
where,
r = the radius
h = the height (or length) of the cylinder.
Given the radius and length, we can substitute these values into the formula to get:
V = π * (0.04 m)² * (0.11 m) = 0.001539 m³
Since the mass density of steel is 7850 kg/m3, we can multiply the volume by the mass density to find the mass of the cylinder:
mass = 7850 kg/m³ * 0.001539 m³
mass = 12 kg
To convert this mass to pounds, we can multiply it by 2.20462 (1kg = 2.20462 lbs)
12 kg * 2.20462 = 26.45 lbs
Hence, the weight of the cylinder in pounds is 26.45 lbs.
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consider the 0/1/2 knapsack problem. unlike 0/1 knapsack problem which restricts xi to be either 0 or 1, 0/1/2 knapsack problem allows xi to be either 0 or 1 or 2 (that is, we assume that 2 copies of each object i are av
Answers
In the 0/1/2 knapsack problem, the objective is to select the set of things that maximises the overall value of the items while staying within the limits of the knapsack. Each item may be selected up to twice.
Similar to the 0/1 knapsack problem, the issue can be written as a linear programming problem and resolved using dynamic programming or the branch and bound approach. The 0/1/2 knapsack problem and the 0/1 knapsack problem are fundamentally different from one another because in the latter, each item may be selected up to twice, but in the former, each item may only be selected once. Accordingly, depending on the particular issue instance, the 0/1/2 knapsack problem is more adaptable and might result in different solutions than the 0/1 knapsack problem. Using a modified version of the conventional 0/1 knapsack method is a common strategy for solving the 0/1/2 knapsack issue.
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A cart travels 4 meters east and then 4 meters north.
Determine the magnitude of the cartâs resultant displacement
Answers
Answer:
Displacement = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2 meters
Explanation:
To determine the magnitude of the cart's resultant displacement, we can use the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse (the displacement) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
We can imagine the cart's displacement as the hypotenuse of a right triangle, where one side is 4 meters (the eastward displacement) and the other side is 4 meters (the northward displacement). Using the Pythagorean theorem, we can calculate the magnitude of the displacement as follows:
Displacement = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2 meters
So the magnitude of the cart's resultant displacement is 4√2 meters.
A 24kg is accelerated by a constant force up a frictionless plane that makes an angle of 37° with the horizontal. Starting from rest at the bottom, it covers a distance of 18 metres in 3 seconds. What is the average power required to accomplish the process
Answers
Explain the direct comparison method of measurement
Answers
Answer:
The direct comparison method of measurement is a method of comparison in which an unknown measurement is compared to a known measurement. This method is used to determine the size, weight, or volume of an unknown object by comparing it to an object whose size, weight, or volume is already known. For example, if you need to determine the weight of an object and have a known weight that is similar in size, you can compare the two objects and estimate the unknown weight. This method of measurement is often used in situations where a more precise measurement is not possible or practical.
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suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to the following formula. y= 32/ (3p+1 (-2/5)) , p>0 (a) What is the rate of change in sales volume when the price is $20? (Round your answer to three decimal places.)
(b) Interpret your answer to part (a). (Round your answer to the nearest whole number.)
If the price increases $1, the sales volume will decrease by _____ units.
Answers
When the price is set at $20, the rate of change in sales volume is -0.086. The number of units sold will fall by 86.
What is volume?Volume is defined as the space occupied by an item inside the confines of three-dimensional space. It is also known as the object's capacity. Volume is a measurement of an object's capacity. For example, the capacity of a cup is 100 ml if it can carry 100 ml of water up to the brim. Volume is also the amount of space filled by a three-dimensional object. Volume and mass are the two fundamental qualities of matter. Volume is simply the amount of space occupied by a thing. There are several methods for measuring volume depending on the physical condition of an item.
Here,
dy/dp = -174/5 [3p + 1]^-7/5
Substitute p = 20 in the above equation, we get
dy/dp = -0.086
(b) The sales volume will decrease by 86 units.
The rate of change in sales volume when the price is $20 is -0.086. The sales volume will decrease by 86 units.
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I NEED HELP AND PLS BE QUICK
A positively charged object hovers a few
centimeters above a positively charged
platform. The object is not moving because
the repulsive force of the platform balances
the object's weight. What will happen if the
object is pushed down and then released?
Answers
Answer:It will shoot back up
Explanation:It is not going to change from positive to negative because you pushed it! It will stay positive and repel away from the positive.
Drag the correct arrow to the diagram.
The diagram shows a series of volcanic islands and a hot spot. Determine the direction of movement of the tectonic plate that helped form the islands.
2023 Edmentum. All rights reserved.
ocean
series of volcanic islands
motion of plate
hot
spot
Answers
Convergent boundaries occur when two plates move in the opposite direction until they collide and create mountains and volcanoes. In the image, the arrows direction is horizontal and oposite to each other.
What are convergent boundaries?
Collision area between two plates. The collition might occur between,
Two oceanic platesOne oceanic plate with a continental one, Two continental plates.When two oceanic plates collide, magma rises to the surface by crevices, creating volcanic arches. In these boundaries occur vertical movements and might originate from protruding areas. There is compressional stress between plates.
In the image, the arrows direction is horizontal and oposite to each other.
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two lightweight metal spheres are suspended near each other from insulating threads. one sphere has a net charge; the other sphere has no net charge. the spheres will exert no net electrostatic corce on each other. attract each other. repel each other. do any of these things depending on the sign of the net charge on the one sphere.
Answers
From insulating threads, two thin metal spheres are suspended close to one another. One sphere has a net charge while the other sphere has no net charge. The spheres will exert no net electrostatic force on each other. The correct option is A.
The uncharged balloon in an experiment advances toward the charged rod. The net charge from the charged sphere distributes onto both evenly, so they are each charged with same sign. The positive and negative charges neutralise, any residual charge is spread evenly. So, both spheres are charged with same sign. If there is no residual charge, both remain neutral.
Suppose the sphere is positively charged, the neutrally charged sphere becomes is negatively charged. The other sphere is polarised by the positively charged one, with a negative charge on the side that is nearest to the first sphere. Now the second sphere moves towards the first sphere.
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A glass cube of index 1.5 lies on a table. A coin is embedded in the glass cube near the surface of the table. When you look at the coin from above, which of the following is true?
The coin appears below the bottom of the cube.
The coin apparent location is the same as it’s real location.
The coin appears to be closer to the top of the cube.
Answers
Due to the refraction of light when we look the coin embedded in a glass cube it appears to be closer at the top of the cube.
Refraction is the bending of a light ray when it goes from an optically rare medium to an optically denser medium. It is determined by the refractive index. Refractive index of a substance is the ratio the speed of light in a vacuum to the speed of light in the substance.
So refractive index 1.5 tells that the speed of light in the glass will be 1.5 times slower than the speed of light in vacuum. That's why it will gets bent towards the normal when it will pass through the glass. So the coin will appear closer to the top surface of the cube.
Hence the option C is correct.
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a block is at rest on a disk that spins about its center, as shown. a student must determine the centripetal acceleration of the block when the block spins with the disk as a result of the force of static friction. what measuring devices, when used together, could be used to determine the centripetal acceleration of the block? select two answers.
Answers
The student can use a meterstick and stopwatch to measure the centripetal acceleration.
Acceleration of an object on a circular orbit. Velocity is a vector quantity (that is, it has both magnitude, velocity, and direction), so as an object moves along a circular path, its direction changes constantly, and so does its velocity, causing acceleration. increase. Centripetal acceleration is defined as the property of motion of a body through a circular orbit. An object that moves in a circle and has an acceleration vector pointing to the center of the circle is called centripetal acceleration. Spinning a ball on an overhead string causes it to experience centripetal acceleration.
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using conversions and data in the chapter, determine the number of hydrogen atoms required to obtain 1.0 kg of hydrogen. a hydrogen atom has a mass of 1.0 u.
Answers
The number of hydrogen atoms required to obtain 1.0 kg of hydrogen is 6.022 x 10^30.
1 kilogram is equal to 1000 grams and 1 atomic mass unit (u) is defined as one twelfth of the mass of a carbon-12 atom, which is 1.6605 x 10^-27 kg.
So, 1 u = 1.6605 x 10^-27 kg
Therefore, 1.0 kg of hydrogen is equal to 1.0 x 10^3 / 1.6605 x 10^-27 = 6.022 x 10^30 hydrogen atoms.
So, the number of hydrogen atoms required to obtain 1.0 kg of hydrogen is 6.022 x 10^30.
A conversion factor converts one unit to another by utilizing your knowledge of unit relationships. For example, if you know that there are 2.54 centimeters in an inch (or 2.2 pounds in a kilogram or 101.3 kilopascals in an atmosphere), converting between those units becomes simple algebra. It is critical to understand some common temperature, size, and pressure conversions, as well as metric prefixes.
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suppose a car's velocity is to the left, and its acceleration is to the right. which of the following describes this car's motion?
Answers
This car is undergoing negative acceleration, meaning that its velocity is decreasing as it moves to the left. The car's motion can be described as decelerating to the left.
What is acceleration?Acceleration is the rate of change of velocity. It is the rate at which an object's velocity changes over time. It is a vector quantity, meaning it has both magnitude and direction. Acceleration can be caused by a number of forces, such as gravity, friction, and applied forces. In one dimension, acceleration is the change in velocity divided by the time it takes for the change to occur. In two dimensions, acceleration is the rate at which an object's velocity vector changes in magnitude and/or direction. Acceleration can be positive, negative, or zero. Positive acceleration occurs when an object speeds up, negative acceleration occurs when an object slows down, and zero acceleration occurs when an object's velocity remains constant.
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assuming charge 1 is located at the origin of the x axis and the positive x axis points to the right, find the two possible values x3,r (the position of charge 3 to the right of charge 1) and x3,l (the position of charge 3 to the left of charge 1).
Answers
This is a physics problem involving Coulomb's Law, which states that the force between two point charges is given by the equation F = (k * q1 * q2) / r², where k is the Coulomb constant, q1 and q2 are the charges, and r is the distance between the charges.
To find the position of charge 3 to the right of charge 1, we need to find the value of x³,r such that the net force on charge 3 is zero. This is because if the net force on charge 3 is not zero, it will accelerate and move away from its current position.
Similarly, to find the position of charge 3 to the left of charge 1, we need to find the value of x3,l such that the net force on charge 3 is zero.
The exact value of x3,r and x3,l will depend on the values of q1, q3 and the distance from charge 3 to charge 1.
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do your graphs show the trends you expected based on what you know about absorbance and transmittance using a spectrophotometer? why or why not? (2 pts.)
Answers
It depends on the data. If the data matches what is expected based on the known properties of absorbance and transmittance using a spectrophotometer, then the graphs will show the expected trends.
If the data does not match what is expected, then the graphs may not show the expected trends.
A diagram or pictorial representation that organises the depiction of data or values is known as a graph. The relationships between two or more items are frequently represented by the points on a graph.
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Explain the dynamic characteristics of instruments
Answers
Answer:
speed of response and response time , lag , fidelity,and dynamic errors are the dynamic characteristics of instruments.
(a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat part (a), assuming the moving particle is an electron
Answers
As the curved path is confined to the XY plane the component of magnitude.
Find the direction of the force on a proton ?
we can see the direction of magnetic force on the proton is towards right and on electron is towards left. Hence, the magnitude of force on electron and proton under the same conditions is equal but the direction of magnetic force is opposite due to opposite signs of these charges.The right hand rule states that: to determine the direction of the magnetic force on a positive moving charge, point your right thumb in the direction of the velocity (v), your index finger in the direction of the magnetic field (B), and your middle finger will point in the direction of the the resulting magnetic force .Electron will move anticlockwise and proton moves clockwise.We need the direction angle of F, i.e. the counterclockwise angle F makes with the positive x axis. Hence, the magnitude is 658 N and the direction angle is 233.9°.
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based on this, which of the following materials would your state need in large quantities to expand future biopower production?
Answers
In order to expand future biopower production, states would need to acquire large quantities of biomass materials such as wood and agricultural crops like corn, switchgrass, and other grasses.
What is materials?
These materials are used in the production of biofuels, which is a type of renewable energy generated from organic matter. Biomass materials are typically converted into liquid fuels such as ethanol or biodiesel for use in vehicles, or burned for electricity generation. Additionally, states may need to acquire large quantities of organic waste materials like animal manure and food scraps for use in anaerobic digestion systems that generate biogas. Biogas is a type of renewable energy that can be used to produce electricity or heat.
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What is the coefficient of static friction if it takes 44 N of force to move a box that weighs 86 N?
Answers
Answer:
0.5
Explanation:
The coefficient of static friction (μs) can be calculated using the formula:Friction Force = μs x Normal ForceWhere Normal force is the weight of the object = 86 NThe force required to move the box is 44 N, which is the Friction forceSo,44 N = μs x 86 NTherefore, the coefficient of static friction (μs) = 44 N / 86 N = 0.5The coefficient of static friction between the box and the surface is 0.5.
Tim, with mass 76.6 kg, climbs a gymna- sium rope a distance of 4.3 m. The acceleration of gravity is 9.8 m/s². How much potential energy does Tim gain? Answer in units of J.
Answers
Answer: Tim gains a potential energy of 3227.93 Joules.
Explanation:
Tim, with mass 76.6 kg, climbs a gymnasium rope a distance of 4.3 m. The acceleration of gravity is 9.8 m/s². How much potential energy does Tim gain?
Data:
m = 76.6 kg
h = 4.3 m
a = 9.8 m/s²
Ep = ?
To calculate potential energy, we have the formula
Ep = m × g × hWhere
Ep = potential energym = massg = acceleration due to gravityh = heightSince it asks us to calculate Tim's potential energy, we should not do formula clearance.
Therefore, we substitute data in the formula and solve, then
Ep = 76.6 kg × 9.8 m/s² × 4.3 m
Ep = 3227.93 kg × m/s² × m
Ep 3227.93 N × m
Ep = 3227.93 Joules
Tim gains a potential energy of 3227.93 Joules.
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a 130 g air-track glider is attached to a spring. the glider is pushed in 11.2 cm and released. a student with a stopwatch finds that 14.0 oscillations take 9.00 s . Part A What is the spring constant
Answers
The spring constant can be calculated as: 12.51 N.
The formula for calculating the period of oscillation is written as the following:
[tex]T=2\pi \sqrt{\frac{m}{k} } \\[/tex]
m is the mass of the spring.
k is the spring constant.
Making the spring constant "k" the subject of the formula will give us the following expression;
[tex]k=4\pi ^{2} \frac{m}{T^{2} } \\[/tex]
Get the period of oscillation "T"
T= 1/f
frequency "f" can express the number of oscillations completed in one second.
If a student with a stopwatch finds that 12 oscillations take 15.0 s, the number of oscillations in one sec will be 9/14 = 0.64 osc.
Period T = 1/0.64 = 1.56sec
Get the required spring constant
[tex]K=4\pi ^{2} \frac{m}{T^{2} } \\k= 4(3.14)^{2}\frac{0.13}{0.64^{2} } \\[/tex]
K= 12.51 N
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a phenomenon is modeled using n dimensional variables with k primary dimensions. the number of non-dimensional variables would be question 2 options: a. k b. n c. n-k d. n k
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A phenomenon is modeled using n dimensional variables with k primary dimensions, so the number of non-dimensional variables would be option c. n-k.
Measurements with quantitative data and distinct sizes are referred to as dimensional variables. Variables with dimensions and yet no constant value are called dimensional variables. Non-dimensional variables, on the other hand, are things that lack both dimensions and a variable value.
Quantities with dimensions but no constant value are called dimensional variables. Non-dimensional variables, on the other side, do not have the value of any variable or even the dimensions. Physical quantities with changing values but no dimensions are typically considered to be dimensionless variables.
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it is a scale which intensifies 10 times over as it increase by a value?
Answers
It is not a proper scale to measure which intensifies 10 times over as it increases by value. The scale which have a short and regular intervals are more accurate.
What are measuring scales ?There are various kinds of measuring gadgets which can be used in appropriate occasions. For example in a chemical lab, there are graduated beakers, measuring vessels, electronic balances, burettes, pipettes etc.
Each type of measuring vessels are used in different situations. In all cases, measuring devices with small regular interval of scale is very easy to use and helps for an accurate measurement especially in the case small volume samples.
Therefore, it is not a proper scale which intensifies 10 times over as it increase by a value. We can't take the intermediate values properly with this scale.
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emission (bright) lines are produced when electrons move to higher energy levels by absorbing photons.
Answers
When an excited electron from a higher energy level returns to a lower energy level by releasing energy in the form of light, a bright-line spectrum of sodium is created.
What is energy?In physics, energy is the quantitative quality that is imparted to a body or a physical system and is visible in the form of heat and light. The law of conservation of energy holds that energy can be transformed in form but cannot be generated or destroyed. Energy is defined by scientists as the ability to do tasks. People have learnt how to convert energy from one form to another and then utilize it to conduct labor, which makes modern civilization feasible. Energy is the ability to do tasks. It can take the form of a potential, kinetic, thermal, electrical, chemical, nuclear, or other energy.
Here,
The sodium bright-line spectrum is formed when an excited electron from a higher energy level returns to the lower energy level by releasing energy in the form of light, resulting in a sodium bright-light spectrum.
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Water flows inside a horizontal pipe so that at the beginning of the pipe its velocity vi is lower than the velocity at the end of the pipe v2. Compare the pressures at the beginning P1 and at the end of the pipe P2. Pi < P2 P1 = P2 P1 > P2 Not enough information.
Answers
The pressure at the beginning and the end of the pipe cannot be determined without additional information.
What is velocity?Velocity is a vector quantity that measures the rate of change of an object’s position in a given direction. It is the rate of displacement over time, or the speed at which an object is moving. Velocity is the speed of an object in a particular direction. It is measured in meters per second (m/s) or kilometers per hour (km/h). Velocity is a combination of both speed and direction, and it can also be represented by an arrow pointing in the direction of motion.
The pressure at the beginning of the pipe (P1) will be determined by the pressure of the source supplying the pipe, such as a pump, as well as the friction losses that occur along the pipe. The pressure at the end of the pipe (P2) will be determined by the available pressure of the source supplying the pipe, the friction losses that occur along the pipe, and the velocity at the end (v2). As such, without knowing the source pressure, the friction losses, and the velocities, it is impossible to determine whether P1 is less than, equal to, or greater than P2.
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you have been hired to design a spring-launched roller coaster that will carry two passengers per car. the car goes up a 12-m -high hill, then descends 20 m to the track's lowest point. you've determined that the spring can be compressed a maximum of 2.1 m and that a loaded car will have a maximum mass of 400 kg . for safety reasons, the spring constant should be 11 % larger than the minimum needed for the car to just make it over the top.
a) what spring constant (k) should you specify ?
b) What is the maximum speed of a 350 kg car if the spring iscompressed the full amount.
Answers
An object can encounter some resistance when moving through the fluid medium of air.
The drag force is the oppositional force applied by the air to a moving object. The size of the cross section A of the front face of the item, the density of the air, and the square of the object's velocity v all contribute to the drag force.
With motion exclusively in the vertical direction and letting
downward be positive, we have that
dv/dt = g - pv, where g = 32 ft/, and so
dv/dt + pv = g. The integrating factor is , so
d/dt( × v) = g× ----->
v = (g/p)× + C ----->
v(t) = (g/p) + C×
Now v(0) = (g/p) + C = 0, and so C = -g/p, giving us
v(t) = (g/p)×(1 - ).
Now we are given that lim(t->infinity)(v(t)) = 99 mi/h = 145.2 ft/s,
so since -> 0 as t -> infinity we have g/p = 145.2 ft/s ------>
p = 32 ft/s^2 / (145.2 ft/s) = 0.2204 s^-1 .
Next, since we have defined downward as positive, the expression
we have for v(t) will equal dy/dt where y(0) = 0 and y increases as
the paratrooper falls. So
dy/dt = v(t) = (g/p)×(1 - )), so
y(t) = (g/p)×(t + (1/p)×) + K.
Now y(0) = (g/p)×(0 + (1/p)) + K = 0 ----->
K = -(g/p^2) = 32 / 0.2204^2 = -658.76 ft, so
y(t) = (145.2)×(t + (4.5372)×e^(-0.2204×t)) - 658.76.
Next, y(6) = (145.2)×(6 + (4.5372)×e^(-0.2204×6)) - 658.76 = 444.84 feet.
The computed drag coefficient is substantially higher than anticipated.
The free fall distance is the longest distance an individual can travel in a predetermined amount of time. However, the distance covered in the time frame indicated in the news report is significantly greater than the greatest distance that could be covered. Consequently, the news story is exaggerated.
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