Which statement best explains why accelerating a car from 20 mph to 40 mph quadruples its kinetic energy? Hint: look at the equation for KE.

B will make it to the finish line first assuming they are all charged objects released from rest at the same time.

This is because the acceleration of the charged object is proportional to the magnitude of the electric field and inversely proportional to the mass of the object. Since all the charged objects have the same mass, the acceleration will depend only on the magnitude of the electric field.

In this case, the electric field is uniform and in the direction of the finish line, which means that the magnitude of the electric field is the same for all the charged objects. Therefore, the acceleration will be the same for all the charged objects.

However, since the magnitude of the force on each object is different due to their different charges, the velocity of the objects will differ. The charged object with the smallest charge, which is object B, will have the highest velocity and make it to the finish line first.

Objects A and D will have the same velocity and reach the finish line simultaneously. Object C will be the slowest and reach the finish line last. Therefore, the correct answer is B.

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This is a quote from the novel Song of Solomon which has been authored by Tomi Morrison.

The narrative comes to an ambiguous end with the vision of flying. Readers are left in the dark regarding the fate of Milkman and Guitar even though they both appear to fly in the conclusion. However, Milkman's ability to fly indicates that he has beyond the limitations of his physical form and has a connection to his great-grandfather, Jake, whom he has spent his entire life trying to understand. Despite the fact that Pilate is dead, the book finishes mystically and with a hopeful attitude.

Song of Solomon is a 1977 novel by American author Toni Morrison, her third to be published. It follows the life of Macon "Milkman" Dead III, an African-American man living in Michigan, from birth to adulthood.

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Answer:

Explanation:a

According to the question, for a car with a force of 18 N and displacement of 8 m, the work done by a car is calculated as 144Nm.

Force may be defined as a process of pushing or pulling on an object that significantly produces acceleration in the body on which it acts. It is an external agent capable of changing a body's state of rest or motion. It has a magnitude and a direction.

According to the question,

The force applied on a car = 18 N

The displacement made by a car = 8m.

Now, the work done is calculated with the help of the given formula:

                          = 18 N × 8m = 144Nm.

Therefore, the work done by a car is calculated as 144Nm.

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(a) The total acceleration of the car in xy components is 16.73 m/s^2.

(b) The linear velocity of the car in xy components is 33.3 m/s.

To find the total acceleration of the car in xy components, we need to find the sum of the centripetal and tangential accelerations.

The centripetal acceleration is given by a = v^2/r

where;

At point P, the velocity of the car is 120 km/h = 33.33 m/s. Therefore, the centripetal acceleration is:

a_c = v^2/r

a_c = (33.33 m/s)^2 / 70 m

a_c = 15.88 m/s^2

The tangential acceleration is given by a_t = dv/dt

where;

v is the velocity of the car and

t is time

The rate of change of velocity is given as 5 m/s^2. Therefore, the tangential acceleration is:

a_t = 5 m/s^2

The total acceleration of the car in xy components is the vector sum of a_c and a_t. Since the two accelerations are perpendicular, we can use the Pythagorean theorem to find the magnitude of the total acceleration:

a_total = √(a_c^2 + a_t^2)

a = √((15.88 m/s^2)^2 + (5 m/s^2)^2)

a = 16.73 m/s^2

(b) The linear velocity of the car in xy components can be found using the formula;

v = rω

where;

The angular velocity is related to the linear velocity by the formula ω = v/r. At point P, the linear velocity of the car is 33.33 m/s. Therefore, the angular velocity is:

ω = v/r = 33.33 m/s / 70 m

ω = 0.476 rad/s

The linear velocity in the x-direction is v_x = v cos(θ),

where;

Since the car is entering the circular portion of the track, the angle θ is 0. Therefore, v_x = v cos(0) = 33.33 m/s.

The linear velocity in the y-direction is v_y = v sin(θ). Since the car is entering the circular portion of the track, the angle θ is 90 degrees. Therefore, v_y = v sin(90°) = 33.33 m/s.

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Explanation:

the displacement is the length of direct line of sight from the starting point to the end point. in other word the baseline of the created triangle.

the legs of that triangle are 10 cm and 6 cm.

the angle between both legs is 90 + 45 = 135°.

because 90° is the angle between South and West. and for southwest we have to add 45°

the law of cosine :

c² = a² + b² - 2ab×cos(C)

c is a side, C is the opposite angle, a and b are the other 2 sides.

so, we have in our case

displacement² = 10² + 6² - 2×10×6×cos(135) =

= 100 + 36 - 120×cos(135) =

= 136 - 120×-0.707106781... =

= 136 - 120×-sqrt(2)/2 =

= 136 + 60×sqrt(2) =

= 220.8528137...

displacement = 14.86111751... cm ≈ 15 cm

The force required to stretch the spring up to 0.5 m is 20 N. Then, the spring constant of the spring is 40 N/m. Then, a force of 50 N will stretch the same spring up to 1.2 m.

The force required to stretch an elastic material is directly proportional to the displacement x of the material.

hence, F = -kx

Where, the proportionality constant k is called spring constant.

Given that, the force required to stretch the spring up to 0.5 m is 20N.

then, spring constant = force/displacement

k = 20 N/0.5 m

  = 40 N/m.

Then, the displacement made by a force of 50 N applied on the same spring is:

x = F/k

  = 50 N/40 N/m

  = 1.2 m.

Therefore, the force of 50 N will stretch the spring up to 1.2 m.

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Your question is incomplete. But your complete question probably was:

A 20 N force stretches a spring to 0.5 m. How far will a 50 N force stretch the same spring?

In order to reach the fishing boat in the smallest amount of time and distance, the speedboat's pilot should point it 15.0 + 13.7 = 28.7° east of true north from the lighthouse.

Typically speaking, if I see the initial line of sight from the lighthouse as being straight north, this is the simplest for me to solve.

The fishing vessel is thus traveling 40.0 – 15.0 = 25.0° east of north.

The fishing boat's eastward speed is 29.0sin25.0, or 12.3 kilometers per hour.

The fishing boat's northward speed is 29.0cos25.0, or 26.3 kilometers per hour.

If the speedboat matches the fishing boat's eastward speed such that the two boats stay on a line that is exactly north/south of one another, the speedboat will go the smallest distance in the quickest time.

The speedboat makes northward progress at

√(52.0² - 12.3²) = 53.4 km/hr

The net northward speed difference is

53.4 – 26.3 = 27.1 km/hr

so the gap between them closes in

16 km/27.1km/hr = 0.6 hr (or 36 min)

The speed boat will be traveling in the direction θ to the right of our artificial north

sinθ = 12.3/52.0

θ = 13.7°

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To help psychologists separate uninformed opinions from examined conclusions, psychologists use the scientific method to conduct research. Thus, the correct option for this question is D.

The scientific method may be defined as the process of objectively establishing facts through testing and experimentation.

The basic process involves making an observation, forming a hypothesis, making a prediction, conducting an experiment, and finally analyzing the results.

Psychologists generally employ the scientific method before stating the question, offering a theory, and then constructing rigorous laboratory or field experiments to test the hypothesis.

Therefore, to help psychologists separate uninformed opinions from examined conclusions, psychologists use the scientific method to conduct research. Thus, the correct option for this question is D.

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Range of values for given condition will depend on magnitude and directions of the three given forces. [tex]|R| < = 2400 N[/tex]

To find the range of values for the magnitude of force P so that the resultant of the three forces does not exceed 2400 N, find magnitude and direction.

We can do this by using vector addition. Adding the three forces together, we get:

R = F1 + F2 + F3

where F1, F2, and F3 are the magnitudes and directions of the three given forces, and R: magnitude with resultant force direction.

To ensure that the resultant force does not exceed 2400 N, we must have:

|R| <= 2400 N

Therefore, we need to find the range of values for the magnitude of force P that satisfy this inequality.

The magnitudes and directions of the three given forces are not specified in the problem, so we cannot provide a specific numerical answer. However, we can provide a general method for solving the problem.

To find the range of values for the magnitude of force P, we can first find the maximum and minimum values of the magnitude of the resultant force for different values of P. We can then find the range of values of P that satisfy the inequality above.

This can be done numerically by using vector addition and trigonometry to find the magnitude and direction of the resultant force for different values of P. Alternatively, we can use graphical methods such as force polygons or vector diagrams to visualize the resultant force and find the range of values of P that satisfy the inequality.

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The unknown weight is 16 N and  the mass of the pulley is 4.26 kg.

The weight of the mass can be found using the formula;

W = T

where;

so W = 16.0 N.

To find the mass of the pulley, we need to take into account its weight and the tension in the cable. Since the tension is greater than the weight of the object, we know that the tension is also supporting the weight of the pulley.

T - m_pulley x g = 0

where;

m_pulley = T/g

= 41.8 N/9.81 m/s^2 = 4.26 kg

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Answer:

Below

Explanation:

Find time of first stone to strike water ....second stone take 1 s less

First stone

d = 1/2 a t^2

43.9 = 1/2 (9.81)(t^2)   shows  t = ~ 3 seconds

Second stone

d = vo t + 1/2 a t^2

43.9 = vo (t) + 1/2 (9.81) t^2       t = 3 -1 = 2 seconds

43.9 = vo (2) + 4.905 (2)^2

   shows vo = 12.1 m/s

The voltage across the element is 24 V where if the energy absorbed by the element is 120 j.

Given data as per the question:

Charges = 5 coulomb

Energy absorbed by the element = 120 J

As per the formula we have,

Energy = Voltage X Charge

120= Voltage X 5

Voltage = 120/5 = 24 V

The voltage in the whole process will be negative because the energy is absorbed.

In a series circuit, the current is the same for all the components. While the circuit reaches its steady state, the capacitor charges and the voltage across its plates increases until it reaches the one on the terminals, and at that point it is in the steady state.

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If the energy absorbed by the element is 120 j, So 5*V1 =120 , V1 =24 Volts.

The voltage distinction be V1 Volts.

When 5C of charge moves from A to B, its energy increments by 120J.

So 5*V1 =120

V1 =24 Volts.

The voltage distinction is hence 24 Volts.

At the point when the charge moves from higher potential to lower, it loses energy and when it moves from lower potential to higher, it retains energy. The energy ingested (or lost) is relative to the potential (voltage) contrast between the two focuses.

The voltage in the entire cycle will be negative in light of the fact that the energy is retained.

In a series circuit, the current is no different for every one of the parts. While the circuit arrives at its consistent express, the capacitor charges and the voltage across its plates increments until it arrives at the one on the terminals, and by then it is in the consistent state.

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The statement consistent with the first law of thermodynamics is the internal energy of the gas will increase. Option a is correct.

The statement consistent with the first law of thermodynamics is (a) The internal energy of the gas will increase. This is because the work done on the gas is being converted into an increase in internal energy.

Option (b) (c) The internal energy of the gas will decrease, or will not change is incorrect, because work is being done on the gas which will increase its internal energy.

Option (d) does not address the fact that work is being done on the gas, which is also contributing to changes in its internal energy.

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To calculate the velocity of a spherical raindrop falling from 4.4 km without air drag, we can use the equations of motion:

v² = u² + 2as

v² = 0 + 2(-9.81 m/s²)(4.4 km) = -2(9.81 m/s²)(4400 m) = -86,140 m²/s²

v = sqrt(86,140 m²/s²) = 293.9 m/s

Therefore, the velocity of the raindrop falling from 4.4 km without air drag is approximately 293.9 m/s.

b. To calculate the velocity of the raindrop falling with air drag, we can use the following equation:

F_drag = 1/2 * rho * v^2 * A * C_d

F_drag = m * g

m * g = 1/2 * rho * v^2 * A * C_d

Solving for v, we get:

v = sqrt((2 * m * g) / (rho * A * C_d))

Substituting the values, we get:

v = sqrt((2 * (4/3) * pi * (3.8/2)^3 * 1000 kg/m³ * 9.81 m/s²) / (1.16 kg/m³ * pi * (3.8/2)^2 * 1.0))

Simplifying, we get:

v = 9.7 m/s

Therefore, the velocity of the raindrop falling from 4.4 km with air drag is approximately 9.7 m/s.

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When the liquid cools down, it loses heat energy.

As the liquid cools, it loses heat energy. As a result, its particles slow down in movement and come closer to one another. Attractive forces begin to hold particles and the crystals of a solid form.

If water is cooled, it can change into ice. If ice is warmed, it can change into a liquid state. Heating a substance makes the molecules move very fast whereas cooling a substance makes the molecules move very slowly.

Heating a liquid increases the speed of the molecules present in it. An increase in the molecule's speed competes with the attraction between molecules and results in the molecules moving apart whereas Cooling a liquid decreases the movement of the molecules.

So we can conclude that the liquid cools down when it loses heat energy.

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Part A, the minimum stopping distance for the car traveling at 46 m/s is 116 m. For the minimum stopping distance to find the minimum stopping distance for the car traveling at 46 m/s:

dstop = vo Treact + (vo^2)/(2abrake)

The minimum stopping distance of a car traveling at speed vo with a reaction time Treact and constant maximum braking acceleration abrake can be found by considering the distance traveled during the driver's reaction time and the distance traveled while the car is slowing down due to braking. The total stopping distance is given by:

dstop = dreact + dbrake

where dreact is the distance traveled during the driver's reaction time and dbrake is the distance traveled while the car is slowing down due to braking.

The distance traveled during the driver's reaction time is given by:

dreact = vo Treact

The distance traveled while the car is slowing down due to braking can be found using the kinematic equation:

v^2 = vo^2 + 2a(x - xo)

where v is the final velocity, xo is the initial position, x is the final position, and a is the acceleration. In this case, we want to find the distance traveled while the car is coming to a stop, so v = 0, xo = 0, and x = dbrake. Solving for dbrake, we get:

dbrake = (vo^2)/(2abrake)

Substituting this expression into the total stopping distance equation, we get:

dstop = vo Treact + (vo^2)/(2abrake)

This is the expression for the minimum stopping distance of a car traveling at speed vo with a reaction time Treact and constant maximum braking acceleration abrake.

Part B:

We can use the expression for the minimum stopping distance derived in Part A to find the minimum stopping distance of the car traveling at 46 m/s. We are given that the car traveling at 30 m/s can stop in a distance of 70 m, including the distance traveled during the driver's reaction time of 0.50 s. Using this information, we can find the value of abrake as follows:

dstop = vo Treact + (vo^2)/(2abrake)

70 m = (30 m/s)(0.50 s) + (30 m/s)^2/(2abrake)

Solving for abrake, we get:

abrake = (30 m/s)^2/(2(70 m - 15 m)) = 4.08 m/s^2

Note that we subtracted the distance traveled during the driver's reaction time of 0.50 s, which is 15 m, from the total stopping distance of 70 m to get the distance traveled while braking.

Now that we know the value of abrake, we can use the same expression for the minimum stopping distance to find the minimum stopping distance for the car traveling at 46 m/s:

dstop = vo Treact + (vo^2)/(2abrake)

dstop = (46 m/s)(0.50 s) + (46 m/s)^2/(2(4.08 m/s^2)) = 116 m

Therefore, the minimum stopping distance for the car traveling at 46 m/s is 116 m

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A solar system's planets, sun, and other things are created in the solar nebula, a huge disc-shaped mass of gas and dust. Nebula is a Latin term that means "cloud." The

Describe solar energy.

Visit the Article History. Solar energy is another name for it. Solar energy is the name given to the Sun's radiation that can ignite chemical reactions, produce heat, or create electricity. The total solar energy incidence on Earth is far greater than the global energy needs at the moment and in the future. This is extremely dispersed if properly harnessed.

What advantages does solar energy offer?

When combined with storage, solar energy may supply backup power for nights and outages, lower electricity costs, help build a more robust electrical grid, promote economic growth, create jobs, and operate at equivalent effectiveness on both small and big sizes. There are numerous types and sizes of solar energy systems.

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It will take about 233.34 seconds for the hot dog cooker to raise the temperature of the hot dog from 20∘C to 85∘C.

Specific heat is the amount of heat energy required to raise the temperature of a substance by one degree Celsius or one Kelvin per unit mass.

Here,
To solve this problem, we can use the formula for the amount of heat required to change the temperature of an object,

Q = mcΔT

First, we need to calculate the amount of heat required to raise the temperature of the hot dog from 20∘C to 85∘C:

Q = (0.07 kg)(2500 J/kg⋅K)(85∘C - 20∘C)

Q = 1058.5 J

Next, we can use the formula for electrical power,

P = IV

We can rearrange this formula to solve for the current:

I = P/V

The power required to heat the hot dog can be calculated using the formula for electrical power:

P = V²/R

Substituting the given values, we get:

P = (120 V)²/160 Ω

P = 90 W

I = 90 W/120 V

I = 0.75 A

Finally, we can use the formula for the amount of time required to transfer a certain amount of heat:

t = Q/(IΔT)

Substituting the values we calculated, we get:

t = 1058.5 J/(0.75 A)(65∘C)

t = 233.34 s

Therefore, it will take about 233.34 seconds for the hot dog cooker to raise the temperature of the hot dog from 20∘C to 85∘C.

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For solution of a) of the particular question where a thin rod and uniform change lies along the x axis=

λ = linear charge density

Consider a small length "dx" at distance "x" from the origin

dq = small charge on the small length = λ dx

r = distance of small length from point P = sqrt(x2 + d2)

small electric field at P due to small length is given as

dE = k dq/r2

dE = k λ dx /(sqrt(x2 + d2))2

dE = k λ dx /(x2 + d2)

From the diagram , we see that "dE Sinθ" are equal and opposite, hence x-components cancel out.

Net electric field at P is given as

E = ∫ 2 dE Cosθ

E = ∫ 2 (k λ dx /(x2 + d2)) (d/sqrt(x2 + d2))

E = ∫ 2 (k λ d dx /(x2 + d2)3/2)

E = ∫ _{0}^{l/2} 2 (k λ d dx /(x2 + d2)3/2)

E = (2 k λ d) ∫_{0}^{l/2}dx /(x2 + d2)3/2

E = (2 k λ d) ((l/2)/ (d2 sqrt(d2 + (l/2)2))

E = (2 k λ d) (Sinθ _{o}/ d2 )                                    

Since

Sinθ _{o} = (l/2) /sqrt(d2 + (l/2)2)

E = 2 k λ Sinθ _{o}/ d

For solution of b)

for infinite length , θ _{o}= 90

E = 2 k λ Sin90/ d

E = 2 k λ / d

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Electric force by one sphere on other is [tex]-4.80 * 10^(-3) N[/tex] and after they have come to equilibrium is [tex]1.08 * (10^-3) N[/tex]

(a) The electric force exerted by one sphere on the other can be calculated using Coulomb's law:

F = k * (q1 * q2) / r^2

where F is the force, k is Coulomb's constant (9.0 x 10^9 Nm^2/C^2), q1 and q2 are the charges on the two spheres (12.0nC and -18.0nC), and r is the distance between their centers (0.300m).

Plugging in the values, we get:

F = 9.0 x 10^9 * (12.0 x 10^-9) * (-18.0 x 10^-9) / (0.300)^2 = -4.80 x 10^-3 N

The negative sign indicates that the force is attractive.

(b) When the spheres are connected by a conducting wire, they will share charge until they reach equilibrium. At equilibrium, the net force on each sphere will be zero. The electric force each sphere exerts on the other will be the same in magnitude, but opposite in direction.

To find the magnitude of the force, we can use the fact that the potential of each sphere will be the same at equilibrium. The potential can be calculated using:

V = k * q / r

where V is the potential, k is Coulomb's constant, q is the charge on the sphere, and r is the distance from the sphere's center.

At equilibrium, the potential of each sphere will be the same, so:

k * q1 / r1 = k * q2 / r2

Solving for q1 and q2, we get:

q1 = -q2 * r1 / r2

Plugging in the values, we get:

q1 =[tex]-(-18.0 * 10^(-9)) * 0.150m / 0.150m = 18.0 * 10^(-9) C[/tex]

q2 = -q1 = [tex]-18.0 * 10^-(9) C[/tex]

The electric force each sphere exerts on the other can be calculated using Coulomb's law with the new charges:

F = k * (q1 * q2) / r^2

Plugging in the values, we get:

F =[tex]9.0 * 10^(-9) * (18.0 * 10^(-9)^2 / (0.300)^2 = 1.08 * 10^(-3) N[/tex]

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The Sun appears smaller than the Earth from here on Earth, but that is only because the Earth is considerably closer to you than the Sun is. Jupiter due to its rapid revolution, which increases its diameter in the midsection.

Our solar system consists of the star, Sun, planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, and small planets such as Pluto.

While the Sun is 150 million kilometers away from where you are, you are on the surface of the Earth.

The planet is an oblate spheroid due to its rapid revolution, which increases its diameter in the midsection.

Therefore, Jupiter due to its rapid revolution increases its diameter in the midsection making it bigger as compared to the world, so it is believed small compared to the Sun and Jupiter.

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1.12 MPa is the shear stress in the glued joint.

If you think of an object as a 2D object, its cross section area is one of its areas. Consider a perfectly spherical ball as an illustration. A circle with a radius equal to the ball can be seen if you view the ball as a 2D object. Consider a cone for another illustration.

The axial or normal stress, σ, in the glued joint can be calculated as:

σ = P/A

A = 2 × b × h

A = 2 × 75 mm × 150 mm = 22,500 mm²

Substituting the values of P and A, we get:

σ = 18 kN / 22,500 mm² = 0.8 MPa

So the normal stress in the glued joint is 0.8 MPa.

The shear stress, τ, in the glued joint can be calculated as:

τ = VQ/It

V = P/2 = 9 kN

The first moment of area, Q, can be calculated as:

Q = b×h2/4

When we change the values of b and h, we obtain:

Q = 75 mm × (150 mm)2 / 4 = 1,406,250 mm³

Calculating the second instant of area, I, is as follows:

I = b×h3/12

The result of substituting the values of b and h is:

I = 75 mm × (150 mm)3 / 12 = 1,054,687.5 mm⁴

Substituting the values of V, Q, I, and t, we get:

τ = 9 kN × 1,406,250 mm³/ (1,054,687.5 mm⁴ × 75 mm) = 1.12 MPa

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When a charge Q is given to a thin metallic plate then the charge gets equally distributed between the two plates. Thus we have an electric field produced by one side of the plate is

E=σ2ϵo

where σ=QA and A is the area.

A)

Electric field, E = Q/(A e0)

E = (5.952 x 10^-6)/(pi x 0.0522^2 x 8.85 x 10^-12)

E = 7.86 x 10^7 N/C

B)

Stays the same

An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles.

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The correct answer is e) both b and c.

The quantities that will always be larger for waves on A than for waves on B are the frequency of the first harmonic and the wave velocity.

The frequency of the first harmonic is determined by the tension in the wire. Since A is stretched more tightly than B, the frequency of the first harmonic will be larger for waves on A than for waves on B.

The wave velocity is also determined by the tension in the wire. A higher tension results in a higher wave velocity. Therefore, the wave velocity will also be larger for waves on A than for waves on B.

The amplitude and wavelength of the first harmonic are not affected by the tension in the wire, so they will not be larger for waves on A than for waves on B.

In conclusion, the frequency of the first harmonic and the wave velocity will always be larger for waves on A than for waves on B.

Therefore, the correct answer for quantities that will be larger for waves on A than for waves on B is option e) both b and c.

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Answer:

KE = 78 J

Explanation:

Answer:

Explanation:

Kinetic energy = 1/2*m*v^2

The kinetic energy is 4400 joules. The mass is 29 kg. What is the speed

4400 = 1/2*29*v^2

v^2 = 303.44

v=17.42m/s^2

The mass is 18 kg. The velocity is 4.7 m/s. What is the kinetic energy?

KE= 1/2*18*4.7*4.7=198.81J

The organic compounds in the homologous series have similar chemical properties. The simplest example of homologous series is the first four hydrocarbons; methane, ethane, propane and butane.

The homologous series is known as the group of organic compounds that differ from each other by a methylene group. They are series of compounds with the same functional group and similar chemical properties.

The compounds of carbon in homologous series have different number of carbon atoms. But they contain the same functional group. Alkanes, alkenes and alkynes form the homologous series.

Thus all the alkanes, alkenes and alkynes form homologous series.

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The maximum range of the water jet is[tex]Rmax = h(sqrt(2)),[/tex] and it is achieved when the height of the hole is [tex]yo = h/2.[/tex]

We can use the equations of motion for an object under constant acceleration to derive the formula for the horizontal distance R that a jet of water will travel before striking the ground. The acceleration of the water jet is due to gravity, and it is constant and equal to the acceleration due to gravity, g.

Let t be the time it takes for the water jet to hit the ground after leaving the hole. We can use the equation of motion for the vertical direction:

[tex]y = yo + voy t - (1/2)gt^2[/tex]

where y is the vertical displacement of the water jet, yo is the initial vertical displacement (the height of the hole from the bottom of the tank), voy is the initial vertical velocity (which is zero), and g is the acceleration due to gravity.

Solving for t, we get:

[tex]t = sqrt((2(yo - y))/g)[/tex]

Now we can use the equation of motion for the horizontal direction:

[tex]x = v_{0} x t[/tex]

where x is the horizontal displacement (which is R), and vox is the initial horizontal velocity (which is constant and equal to the velocity of the water jet as it emerges from the hole).

We can express [tex]v_{0} x[/tex] in terms of the vertical displacement y and the time t:

[tex]v_{0} x = R/t = R(sqrt(g/(2(yo - y))))[/tex]

Substituting for t and simplifying, we get:

[tex]v_{0} x = R(sqrt(2g/h))[/tex]

Now we can express the range R in terms of the tank height h and the height of the hole yo:

[tex]R = (v_{0} x^2/h) = 2h(sqrt(yo/h))[/tex]

To derive the formula for an identical tank on the moon where the acceleration due to gravity is Jm = 1/4 of the acceleration due to gravity on Earth (g), we can substitute g/4 for g in the equation for the horizontal velocity vox. This gives:

[tex]vox = R(sqrt(g/(8h)))[/tex]

Substituting into the equation for R, we get:

[tex]R = (vox^2/h) = 8h(sqrt(yo/h))[/tex]

To show that the maximum range of the water jet is achieved when yo = h/2, we can differentiate R with respect to yo and set the result equal to zero:

[tex]dR/dyo = (4/h)(sqrt(yo/h))(h/2 - yo)[/tex]

Setting this equal to zero and solving for yo, we get:

[tex]yo = h/2[/tex]

To find the maximum range Rmax, we substitute yo = h/2 into the equation for R:

[tex]Rmax = 2h(sqrt(h/2h)) = h(sqrt(2))[/tex]

Therefore, the maximum range of the water jet is [tex]Rmax = h(sqrt(2))[/tex], and it is achieved when the height of the hole is yo = h/2.

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