1. In the picture below, energy is being converted to heat water. What type of energy transformations take place in this model?

Answer:

Commutator.

Explanation:

It is electrical switch and consist of cylinder. Split rings are two halves of a ring that act as a Commutator. When current in the coil is Zero and is about to change the direction.

Answer:

The internal energy of the gas remains constant,

Explanation:

The isothermal process can be defined as a thermodynamic process in which the temperature of the system remains constant.In isothemal process the process of transfer of heat energy from the surrounding to the system or to the system from the surrounding takes place at constant temperature.

in isothermal process internal energy of the system remains unchanged or constant.

For ideal gas internal energy of the system depends on the temperature.

For ideal gas when it is compressed the work is done by the surrounding on the gas is positive and heat added to the system is negative .When the heat energy is absent,both temperature and internal energy increases. As it is a isothermal process and temperature remains constant the gas must transfer the positive amount of heat to the system.

Hence internal energy of the system of gas remains constant.

Final answer:

During an isothermal compression of a gas, energy is transferred into the gas by heat to maintain its constant temperature, while work is done on the gas and its internal energy remains constant.

Explanation:

When a gas is compressed isothermally, specific events related to energy take place. Isothermal processes occur at a constant temperature, implying that the thermal energy of the gas does not change. If we refer to the first law of thermodynamics, stating that the change in internal energy (\(\Delta E_{int}\)) is equal to heat added to the system (Q) minus the work done by the system (W), we can understand these events. Since the temperature remains constant during an isothermal compression, the internal energy of an ideal gas, which depends only on temperature, also remains constant (\(\Delta E_{int} = 0\)). Work is done on the gas during compression, meaning that there is a transfer of energy into the gas by way of work (W > 0).

To maintain the constant temperature, heat must be removed from the gas equal to the work done on it (Q = W), so there is effective energy transfer into the gas by heat to maintain isothermality. Therefore, the correct statement among the given options is: Energy is transferred into the gas by heat.

Titan, Saturn's moon, has a thick atmosphere primarily composed of nitrogen. Its unique chemistry and thick atmosphere make it an interesting place for scientific study.

Titan, Saturn's moon, has a thick atmosphere made mostly of nitrogen. The atmosphere on Titan is thicker than Earth's and is primarily composed of nitrogen, with about 5% methane. The thick atmosphere and unique chemistry of Titan make it an interesting place for scientific study, as it may provide clues about the early chemistry of the solar system and the potential for life.

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A falling skydiver opens his parachute. A short time later, the weight of the skydiver-parachute system and the drag force exerted on the system are equal in magnitude. The following statements predicts the motion of the skydiver at this time

The skydiver is moving downward with constant speed.

Explanation:

Immediately on leaving the aircraft, the skydiver accelerates downwards due to the force of gravity. There is no air resistance acting in the upwards direction, and there is a resultant force acting downwards. The skydiver accelerates towards the ground.

The forces acting on a falling leaf are : gravity and air resistance.

The net force and the acceleration on the falling skydiver is upward.

An upward net force on a downward falling object would cause that object to slow down. The skydiver thus slows down.

As the speed decreases, the amount of air resistance also decreases until once more the skydiver reaches a terminal velocity.

A skydiver falling at a constant speed opens his parachute. When the skydiver is falling, the forces are unbalanced.

The drag force opposes the relative motion of an object in a fluid

The option that gives the statement that predicts the motion of the skydiver when the magnitude of the weight of the skydiver-parachute system and the drag force are equal is option b.

b. The skydiver is moving downward with constant speed

The reason the above option is correct is as follows:

The skydiver is falling through the air which is a fluid and is therefore subject to drag force, [tex]F_D[/tex], which is given as follows;

[tex]F_D = \dfrac{1}{2} \times \rho \times v^2 \times C_D \times A[/tex]

Where;

[tex]F_D[/tex] = The drag force

ρ = The fluid density

v = The relative object speed compared to the fluid

A = Cross sectional area

[tex]C_D[/tex] = The drag coefficient

Therefore, given that the surface area of the falling skydiver is largely increased when the parachute is opened and the drag force is a function of the square of the velocity, we have that the drag force rapidly equals the weight of the skydiver, such that the net force becomes zero, and therefore, the speed of the skydiver becomes constant

Therefore, the correct option is that the skydiver is moving downward with constant speed

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Explanation:

Given that,

Wavelength of light, [tex]\lambda=600\ nm=6\times 10^{-7}\ m[/tex]

Angle, [tex]\theta=25^{\circ}[/tex]

We need to find the slit spacing for diffraction. For a diffraction, the first order principal maximum is given by :

[tex]d\sin\theta=n\lambda[/tex]

n is 1 here

d is slit spacing

[tex]d=\dfrac{\lambda}{\sin\theta}\\\\d=\dfrac{6\times 10^{-7}}{\sin(25)}\\\\d=1.41\times 10^{-6}\ m\\\\d=1.41\ \mu m[/tex]

So, the slit spacing is [tex]1.41\ \mu m[/tex].

Answer:

300 minutes.

Explanation:

Given,

Speed of car 1 = 60 Km/h

Speed of car 2 = 75 Km/h

distance travel by car 2 to catch car 1 = 75 Km

time taken to catch the car = ?

Relative velocity between the car = 75 - 60 = 15 Km/h

[tex]time = \dfrac{distance}{speed}[/tex]

[tex]time = \dfrac{75}{15}[/tex]

t = 5 hrs

t = 5 x 60 = 300 minutes.

Two cars passing a point at different speeds can be calculated using time, distance, and speed formulas.

A car passes the Eiffel Tower driving at a constant rate of 60 km per hour. A second car, traveling at a constant rate of 75 km per hour, passes the Eiffel Tower later and catches the first car after traveling 75 km past the tower.

To calculate the time difference between the two cars passing the Eiffel Tower, we can use the formula: Time = Distance / Speed. The second car passes the tower 15 minutes after the first car.

Answer:

d) The image would be dimmer

Explanation:

When inverted image of an object is viewed on a screen from the side facing a converging lens, the

lens focuses the diverging, and blurred light rays from the distance object through refraction of the rays two times.Then the rays is converge by the double bending cl at a focal point behind the lens inorder for a sharper image to be be observed.

But when an opaque card is then introduced to cover only the upper half of the lens then the image becomes dimmer because of the reduction in the light intensity on the screen by 50%

When the opaque card should be introduced so here the image should be considered as the dimmer.

At the time when the inverted image should be viewed on the screen that faced the converging lens so here the lens should be focused on the diverging also there should be blurred light rays from the distance object via the refraction of the rays.

However, at the time there is the opaque card so here the image should be dimmer since there is the decrease in the light intensity by 50%/

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Answer:

a)   W = m g h , b)     W / U = 1

Explanation:

a) work is defined by

        W = F. dy

in this case the force of gravity goes down and the displacement of the particle is down, so the two are parallel and the scalar product is reduced to the algebraic product

        W = F y

strength is the weight of the body

         F = m g

        W = m g h

where h is the distance the body descends

b) the only force acting on the body is the weight of the work and we calculated them in part a, the potential energy is

     U = m g h

to compare the two magnitudes of let's find their relationship

    W / U = mgh / mgh

    W / U = 1

Answer: plate tectonics

Explanation: plate tectonics move constantly movement in narrow zones along plate boundaries causes earthquakes and plate tectonics come together and create pressure upon the magma deposit underground forcing it up into a volcanic eruption

Answer:

[tex]T(1)=21 [/tex]

Explanation:

The equation of the position in kinematics is given:

[tex]x(t)=x_{0}+v_{0}t+0.5at^{2}[/tex]

So the equation will be:

[tex]x(t)=20t+0.5*2*t^{2}[/tex]

[tex]x(t)=20t+t^{2}[/tex]    

Now, the Taylor polynomial equation is:

[tex]f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+...[/tex]

Using our position equation we can find f'(t)=v(t) and f''(x)=a(t). In our case a=0, so let's find each derivative.

[tex]f(t)=x(t)=20t+t^{2}[/tex]

[tex]f'(t)=\frac{dx(t)}{dt}=v(t)=20+2t[/tex]

[tex]f''(t)=\frac{dv(t)}{dt}=a(t)=2[/tex]

Using the Taylor polynomial with a = 0 and take just the second order of the derivative.

[tex]f(0)+\frac{f'(0)}{1!}(x)+\frac{f''(0)}{2!}(x)^{2}[/tex]

[tex]f(0)=x(0)=0[/tex]

[tex]f'(0)=v(0)=20[/tex]

[tex]f''(0)=a(0)=2[/tex]

[tex]T(t)=f(0)+\frac{f'(0)}{1!}(t)+\frac{f''(0)}{2!}(t)^{2}[/tex]

[tex]T(t)=\frac{20}{1!}(t)+\frac{2}{2!}(t)^{2}[/tex]

[tex]T(t)=20t+t^{2}[/tex]

Let's put t=1 so find the how far the car moves in the next second:

[tex]T(1)=20*1+1^{2}[/tex]

[tex]T(1)=21 [/tex]

Therefore, the position in the next second is 21 m.

We need to know if the acceleration remains at this value to use this polynomial in the next minute, so I suggest that it would be reasonable to use this method just under this condition.

I hope it helps you!

Answer:

Kinetic energy of the car = 4637.04 J

Explanation:

Mass of the car (m) =1200 kg

speed of the car (v) = 2.78 m/sec

Kinetic energy =[tex]\frac{1}{2}mv^{2}[/tex]

Putting the values of m and v,

Hence, Kinetic Energy of the car =[tex]\frac{1}{2}1200\cdot 2.78^{2}[/tex] =  4637.04 J

The kinetic energy of a 1,200kg car traveling at a speed of 2.78 m/s is calculated using the formula KE = ½ mv², yielding a kinetic energy of 4,634.88 joules.

To calculate the kinetic energy (KE) of a car with a mass of 1,200kg traveling at a speed of 2.78 m/s, we need to use the kinetic energy formula KE = ½ mv². Plugging in the given values, we have:

KE = ½ (1,200 kg) × (2.78 m/s)²

KE = ½ × 1,200 kg × 7.7284 m²/s²

KE = 4,634.88 J

Therefore, the kinetic energy of the car is calculated to be 4,634.88 joules.

Answer:

an electronic symbol is a pictogram used to represent various electrical and electronic devices or functions such as wires batteries resistors and transistors in a symptomatic diagram of an electrical or electronic circuit.

Explanation:

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Final answer:

Electrical circuit diagrams use specific standardized symbols to represent various components like wires, batteries, resistors, ground, and capacitors. Symbols for wire, resistor, and battery are straightforward, while capacitors have variations depending on type. ANSI and IEC provide different standard symbol guidelines.

Explanation:

Symbols for Components of Electrical Circuits

Electrical circuit diagrams use standardized symbols to represent different components. The wire is depicted as a thin black line and is assumed to be a perfect conductor in diagrams, although in reality it is not perfect but typically close enough for practical purposes. A battery or voltage source is represented by a set of parallel lines, with the longer line denoting the positive terminal and the shorter line the negative terminal. The resistor is illustrated with a zigzag symbol, which includes any resistance within the circuit connections. Ground is typically shown as a set of one to three horizontal lines, sometimes with one of the lines angled towards the bottom.

When it comes to alternating current (AC) sources, the symbol is a sine wave enclosed in a circle. This shows the oscillating nature of the AC voltage. Capacitors are represented with two parallel lines for a fixed capacitor; an electrolytic capacitor is shown with one curved line indicating the negative terminal and one straight line for the positive terminal. A variable-capacitance capacitor is represented with an arrow through the parallel lines, signifying adjustability.

Standards such as the American National Standard Institute (ANSI) and the International Electrotechnical Commission (IEC) provide guidelines for these symbols. While the ANSI standard is widely used for its recognizability, the IEC standard might be preferred for more complex circuits due to its cleaner presentation.

The electromagnetic waves from the lowest frequency to the highest frequency are gamma rays, X-ray, visible light, and infrared.

Electromagnetic waves are the waves which are formed when an electric field couples with a magnetic field. Magnetic and the electric fields of an electromagnetic wave are perpendicular to each other and to the direction of the wave propagation. Electromagnetic waves are used widely in the food processing for destroying microbes.

Therefore, the correct order of electromagnetic waves from the lowest frequency to the highest frequency are gamma rays, X-ray, visible light, and infrared.

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Answer:

a. 7.38 N b. 40.87 N c. 0.113 kg-m²

Explanation:

Answer:

In the lab activity, I used a virtual watershed to demonstrate how pollutants enter and flow through the watershed. I constructed food chains to show how biotic factors are related. I also predicted and modeled how human activity affected the biotic factors in an ecosystem. Based on my observations, I can conclude that biotic and abiotic factors are closely linked and that human activity that changes the environment, like pollution, can affect all the other parts of an ecosystem in a negative way.

Explanation:

Answer:

In the lab activity, I used a virtual watershed to demonstrate how pollutants enter and flow through the watershed. I constructed food chains to show how biotic factors are related. I also predicted and modeled how human activity affected the biotic factors in an ecosystem. Based on my observations, I can conclude that biotic and abiotic factors are closely linked and that human activity that changes the environment, like pollution, can affect all the other parts of an ecosystem in a negative way.

Answer:

The torque applied by the drill bit is T = 16.2 Nm and the cutting force of the drill bit is F = 33 N.

Explanation:

Given:-

- The diameter of the drill bit, d = 98 cm

- The power at which drill works, P = 5.85 hp

- The rotational speed of drill, N = 1900 rpm

Find:-

What Torque And Force Is Applied To The Drill Bit?

Solution:-

- The amount of torque (T) generated at the periphery of the cutting edges of the drilling bit when it is driven at a power of (P) horsepower at some rotational speed (N).

- The relation between these quantities is given:

                         T = 5252*P / N

                         T = 5252*5.85 / 1900

                         T = 16.171 Nm

- The force (F) applied at the periphery of the drill bit cutting edge at a distance of radius from the center of drill bit can be determined from the definition of Torque (T) being a cross product of the Force (F) and a moment arm (r):

                          T = F*r

Where,   r = d / 2

                          F = 2T / d

                          F = 2*16.171 / 0.98

                          F = 33 N

Answer: The torque applied by the drill bit is T = 16.2 Nm and the cutting force of the drill bit is F = 33 N.

Answer:

1) Torque = 21.934 Nm

2) force = 44.76 N

Explanation:

Power = 5.88 hp

1 hp = 746 W

Power = 5.88 x 746 = 4364.1 W

Angular speed in Rpm = 1900 rpm

But angular speed w = (2¶N)/60 rad/s

= (2 x 3.142 x 1900)/60 = 198.968 rad/s

From,

1) Power P = T x w

Where T = torque

T = P/W = 4364.1/198.968 = 21.934 Nm

2) diameter of drill = 98 cm

Radius = 98/2 = 49 cm = 49x10^-2 m

From torque T = Force x radius

Force = Torque /radius

F = 21.934/49x10^-2 = 44.76 N

Answer:

D) Einstein proved it is impossible to have a redshift greater than 1; these are all due to gravitational lensing tricking us

Explanation:

By Einstein's theory of relativity the maximum value of the red shift is 1. This correspond to a case in which the observed frequency equals the emitted frequency:

[tex]z=\frac{f_e-f_o}{f_o}[/tex]

in other cases the red shift is lower than one.

Hence, the gravitational lens change the measurements of the observed frequency and because of that we calculate a greater red shift.

Hence, the answer is:

D) Einstein proved it is impossible to have a redshift greater than 1; these are all due to gravitational lensing tricking us

hope this helps!!

Answer:

the magnitude of the torque about the point where the beam is bolted into place 12.32kNm

Explanation:

N/B The units given in the question are meant to be m, kg and kg respectively

Given data

Length of beam 3.7m

Mass of beam 520kg

Mass of construction worker 73kg

Torque is expressed as the moment or turning effect brought about by a couple

T= F*d

Where d=perpendicular distance

F= force

Converting The mass of the beam and the worker to force we have

520*10= 5200N

73*10=730N

Note that the force due to the mass of the beam is acting at the center of the beam

See the free body diagram attached

Taking moments about point O

Torque =Σ Fxd

Torque =5200x1.85+7x3.7

=9620+2701

=12321 Nm

=12.32kNm

Answer :

D. You should schedule an individual meeting with the system engineer to determine whether he has issues with the project that need to be resolved. Get his perspective on how the project is progressing and how he feels about his role.

Answer:

The magnitude of the net force is √2F.

Explanation:

Since the two particles have the same charge Q, they exert the same force on the test charge; both attractive or repulsive. So, the angle between the two forces is 90° in any case. Now, as we know the magnitude of these forces and that they form a 90° angle, we can use the Pythagorean Theorem to calculate the magnitude of the resultant net force:

[tex]F_N=\sqrt{F^{2}+F^{2}}\\\\F_N=\sqrt{2F^{2}}\\\\F_N=\sqrt{2}F[/tex]

Then, it means that the net force acting on the test charge has a magnitude of √2F.

The net force on the test charge q due to the two charges Q is calculated using vector addition and the Pythagorean theorem. Because the two forces act at a right angle, the total force is the vector sum and is equal to F√2.

The magnitude of the net force acting on the test charge due to both of these charges can be calculated using the principles of vector addition in Physics. When a charge q is placed at the corner of a square, the forces exerted by the two charges, Q and Q, at the opposite corners are equal in magnitude but act along different directions. Because these charges are placed at a right angle with respect to charge q, the total force is the vector sum of individual forces, which results in a diagonal force directed along the line joining the charge q and the empty corner of the square. Therefore, the magnitude of the net force is given by the Pythagorean theorem, √(F^2 + F^2), which simplifies to F√2.

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Answer:

The second trombones might produce frequency of 435 Hz or 441Hz.

Explanation:  

Beat frequency is defined as number of beats produced per second and numerically equal to the difference between the frequency of super imposing waves.The sound waves interfere constructively.

Beat is the phenomena of interference.

The frequency of the resultant wave is given by

F = f₁ - f₂ eqn 1

where f₁ , f₂ are frequency of waves.

The frequency of first trombones is 438Hz, 6 beats are heard every 2 second,

Beat frequency is F = [tex]\frac{6}{2}[/tex] = 3Hz.

Substituting in eqn 1 , we get two possible solution for frequency of second trombones.

f₂ =f₁ -F = 438 -3 =435Hz

f₂ = f₁ +F = 438+3 =441 Hz.

The second trombones might produce frequency of 435 Hz or 441Hz.

Answer:

The answer to your question is given below

Explanation:

Since both object A and B were dropped from the same height and the air resistance is negligible, both object A and B will get to the ground at the same time.

From the question, we were told that object A falls through a distance to dA at time t and object B falls through a distance of dB at time 2t.

Remember, both objects must get to the ground at the same time..!

Let the time taken for both objects to get to the ground be t.

Time A = Time B = t

But B falls through time 2t

Therefore,

Time A = Time B = 2t

Height = 1/2gt^2

For A:

Time = 2t

dA = 1/2 x g x (2t)^2

dA = 1/2g x 4t^2

For B

Time = t

dB = 1/2 x g x t^2

Equating dA and dB

dA = dB

1/2g x 4t^2 = 1/2 x g x t^2

Cancel out 1/2, g and t^2

4 = 1

4dA = dB

Divide both side by 4

dA = 1/4 dB

Answer:

Picture C

Explanation:

a.  The greatest angle will be bouncing, then embedded and finally the least is passing angle.

b. The  Momentum is greatest when the arrow bounces and least when the arrow passes.  This justify the obtained ranking of angles.

The given problem is based on the concept of momentum and angle made at various instance. Momentum is defined as an impact caused due to applied force. It is expressed as the product of mass and velocity of an object, with respect to the direction. Taking forward direction of arrow as positive then backward direction will be negative.

From the law of conservation of linear momentum, the sum of initial and final momentum should be equal.

Thus we can conclude that by ranking the angles, consider that the maximum angle that the pumpkin reaches depends on the speed of the pumpkin. The greatest angle will be bouncing, then embedded and finally the least is passing angle.

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Answer:

1.349 × 10¹² electrons

Explanation:

At equilibrium, the tension T in the string is resolved into horizontal and vertical components. Tcos17 being the vertical component and Tsin17 the horizontal component. The electrostatic force of repulsion at equilibrium, F = kq²/r² where q = excess charge on sphere and r = distance apart at equilibrium,F acts horizontally to the left and its weight, mg acts vertically downwards.

For equilibrium, sum of horizontal components = 0 and  sum of horizontal components = 0. So, Tsin17 - F = 0

Tsin17 = F ⇒ Tsin17 = kq²/r² (1)

Also Tcos17 - mg = 0  ⇒ Tcos17 = mg (2)

Dividing (1) by (2), we have

Tsin17/Tcos17 = kq²/r² ÷ mg

tan17 = kq²/mgr²

q = r√(mgtan17/k)

We find r using cosine rule r = √(500² + 500² -2 × 500²cos 2 × 17) since the string and the masses form an isosceles triangle at equilibrium.

r = √(2 × 500² -2 × 500²cos34) = 500√2(1 - cos 34) = 500√2 × 0.1710 =120.89 mm = 0.1209 m

substituting m = 9.60 g = 9.6 × 10⁻³ g, k = 9 × 10⁹ Nm²/C² and r into q, we have,

q = r√(mgtan17/k)

  = 0.1209 m√(9.6 × 10⁻³ g × 9.8 m/s² × tan17/9 × 10⁹ Nm²/C²)

  = 0.1209 m√(28.76 × 10⁻³ N/9 × 10⁹ Nm²/C²)

  =  0.1209 m√(3.195 × 10⁻¹² C²/m²)

  =  0.1209 m × 1.7877 × 10⁻⁶ C/m

  = 0.2161 × 10⁻⁶ C

  = 2.161 × 10⁻⁷ C

To find the number of surplus electrons, n on each sphere, we divide q by e the electron charge.

So, n = q/e = 2.161 × 10⁻⁷ C ÷ 1.602 × 10⁻¹⁹ C = 0.6825 × 10¹² = 1.349 × 10¹² electrons

The number of surplus electrons on each sphere : 3.27 * 10¹²

Given data :

Mass of metal spheres = 9.60 g

length of string = 500 mm

Angle made by each string with the vertical ( ∅ )  = 17°

we will apply the formula below

number of surplus electrons ( n  ) = [tex]\frac{q}{e}[/tex]  ----- ( 1 )

whereby :

Considering The forces acting on the metal spheres

Tan ∅ = [tex]\frac{q^{2} }{4\pi e_{o}mgr^{2} }[/tex]   ----- ( 2 )

First step : Determine the distance between the spheres

r = 2l * sin∅

 = 2 * 0.500 * sin17°

 = 0.2924 m

Next step : Determine the charge on each sphere

Back to equation ( 2 )

Tan 17° = [tex]\frac{(8.99*10^{9}) * q^{2} }{(9.60 * 10^{-3})*(9.81)*(0.2924 )^{2} }[/tex]

q² = [ ( 0.3057 ) * ( 9.60 * 10⁻³ ) * ( 9.81 ) * ( 0.2924 )² ] / ( 8.99 * 10⁹ )

    = 0.00246 / 8.99 * 10⁹

    = 2.7364 * 10⁻¹³

Charge on each sphere ( q ) = √ (  2.7364 * 10⁻¹³ ) = 5.23 * 10⁻⁷ C

Final step : The number of surplus electrons on each sphere

Back to equation ( 1 )

n = q / e

  = ( 5.23 * 10^-7 ) / ( 1.6 * 10^-19 )

  = 3.27 * 10¹²

Hence we can conclude that the number of surplus electrons on each sphere : 3.27 * 10¹².

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Explanation:

Given that,

(a) Frequency, [tex]f_1=4\times 10^{19}\ Hz[/tex]

All electromagnetic wave moves with the speed of light. It is given by :

[tex]c=f\lambda[/tex]

[tex]\lambda_1=\dfrac{c}{f_1}\\\\\lambda_1=\dfrac{3\times 10^8}{4\times 10^{19}}\\\\\lambda_1=7.5\times 10^{-12}[/tex]

(b) Frequency, [tex]f_2=5.5\times 10^{1=9}\ Hz[/tex]

All electromagnetic wave moves with the speed of light. It is given by :

[tex]c=f\lambda[/tex]

[tex]\lambda_2=\dfrac{c}{f_2}\\\\\lambda_2=\dfrac{3\times 10^8}{5.5\times 10^{9}}\\\\\lambda_2=0.054\ m[/tex]

Hence, this is the required solution.

Final answer:

The work-energy theorem states that the work done by all forces on a particle equals the change in the particle's kinetic energy, expressed as W = ΔKE = ½m( [tex]vf^2 - vi^2[/tex]). The most work is done between the points where kinetic energy changes the most.

Explanation:

The work-energy theorem states that the work done by all forces acting on a particle is equal to the change in the particle's kinetic energy. In formula form, this can be expressed as W = ΔKE, where W is the work done by the forces, and ΔKE is the change in kinetic energy, often written as ΔKE = KEf - KEi, with KEf denoting the final kinetic energy and KEi the initial kinetic energy. For a roller coaster car, the point where it does the most work is where the change in kinetic energy is greatest, typically between the top of the highest hill (where kinetic energy is lowest) and the bottom of the following descent (where kinetic energy is highest).

The formula representing the work-energy theorem is W = ΔKE = ½m( [tex]vf^2 - vi^2[/tex]), where m represents the mass of the object, vf the final velocity, and vi the initial velocity.