A 900 kg roller coaster car starts from rest at point A. rolls down the track, goes
around a loop (points B and C) and then flies off the inclined part of the track (point D),
Figure 2.
The dimensions are: H =80 m.
r= 15m, h=10m and theta =9.30°

Calculate the
(a) gravitational potential energy at point A.

(b) velocity at point C, if the work done to move the roller coaster from point B to C is 264870 J.

c) distance of the car land (in the horizontal direction) from point D if given the
velocity at point D is 37.06 m/s

I​

Answer: it is easier to read in bright light than dim light.

Explanation:

The ray of light is the direction that is used by light in travelling through a medium. Rays that pass through a lens very close to the principle axis are more sharply focused than those that are very far from the axis.

Because of the fact that the rays are close to the principle axis, the spherical aberration helps us to understand the reason why it is easier for people to read in bright light than readin iin dim light.

Answer:

Nails are made of iron. Iron consists of 26 protons and 26 electrons. protons are positively charged and electrons are negatively charged, so this force of attraction keeps the electrons together.

If electrons repel from each other, the positively charge protons and nucleus allow them to move in a definite orbit and prevent them flying out of the nail.

Answer:

The  time constant is  [tex]\tau = 1.265 s[/tex]

Explanation:

From the question we are told that

     the time take to charge is  [tex]t = 2.4 \ s[/tex]

The mathematically representation for voltage potential of a capacitor at different time is

        [tex]V = V_o - e^{-\frac{t}{\tau} }[/tex]

Where  [tex]\tau[/tex]  is the time constant  

           [tex]V_o[/tex] is the potential of the capacitor when it is full

     So  the capacitor potential will be  100%  when it is full thus  [tex]V_o =[/tex]100%  =  1  

and from the  question we are told that the  at the given time the potential of the capacitor is 85% = 0.85 of its final potential so

      V  = 0.85

Hence

     [tex]0.85 = 1 - e^{-\frac{2.4}{\tau } }[/tex]

       [tex]- {\frac{2.4}{\tau } } = ln0.15[/tex]

        [tex]\tau = 1.265 s[/tex]

Answer:

The wave takes 0.132 seconds to travel from one end of the string to the other.

Explanation:

The velocity of a transversal wave ([tex]v[/tex]) travelling through a string pulled on both ends is determined by this formula:

[tex]v = \sqrt{\frac{T\cdot L}{m} }[/tex]

Where:

[tex]T[/tex] - Tension, measured in newtons.

[tex]L[/tex] - Length of the string, measured in meters.

[tex]m[/tex] - Mass of the string, measured in meters.

Given that [tex]T = 15\,N[/tex], [tex]L = 7.6\,m[/tex] and [tex]m = 0.034\,kg[/tex], the velocity of the tranversal wave is:

[tex]v = \sqrt{\frac{(15\,N)\cdot (7.6\,m)}{0.034\,kg} }[/tex]

[tex]v\approx 57.522\,\frac{m}{s}[/tex]

Since speed of transversal waves through material are constant, the time required ([tex]\Delta t[/tex]) to travel from one end of the string to the other is described by the following kinematic equation:

[tex]\Delta t = \frac{L}{v}[/tex]

If [tex]L = 7.6\,m[/tex] and [tex]v\approx 57.522\,\frac{m}{s}[/tex], then:

[tex]\Delta t = \frac{7.6\,m}{57.522\,\frac{m}{s} }[/tex]

[tex]\Delta t = 0.132\,s[/tex]

The wave takes 0.132 seconds to travel from one end of the string to the other.

Answer:

Explanation:

Kinetic energy is expressed as KE = 1/2 mv² where;

m is the mass of the body

v is the velocity of the body.

For the heavier shell;

m = 5kg

KE gained = 100J

Substituting this values into the formula above to get the velocity v;

100 = 1/2 * 5 * v²

5v² = 200

v² = 200/5

v² = 40

v = √40

v = 6.32 m/s

Note that after the explosion, both body fragments will possess the same velocity.

For the lighter shell;

mass = 2.0kg and v = 6.32m/s

KE of the lighter shell = 1/2 * 2 * 6.32²

KE of the lighter shell = 6.32²

KE of the lighter shell= 39.94m/s

Hence, the lighter one gains a kinetic energy of 39.94m/s.

The gain in the kinetic energy of the smaller fragment is 249.64 J.

The given parameters;

The velocity of the heavier fragment is calculated as follows;

[tex]K.E = \frac{1}{2} mv^2\\\\mv^2 = 2K.E\\\\v^2 = \frac{2K.E}{m} \\\\v= \sqrt{\frac{2K.E}{m} } \\\\v = \sqrt{\frac{2 \times 100}{5} }\\\\v = 6.32 \ m/s[/tex]

Apply the principle of conservation of linear momentum to determine the velocity of the smaller fragment as;

[tex]m_1 u_1 + m_2 u_2 = v(m_1 + m_2)\\\\-6.32(5) \ + 2u_2 = 0(7)\\\\-31.6 + 2u_2 = 0\\\\2u_2 = 31.6\\\\u_2 = \frac{31.6}{2} \\\\u_2 = 15.8 \ m/s[/tex]

The gain in the kinetic energy of the smaller fragment is calculated as follows;

[tex]K.E_2 = \frac{1}{2} mu_2^2\\\\K.E_2 = \frac{1}{2} \times 2 \times (15.8)^2\\\\K.E_2 = 249.64 \ J[/tex]

Thus, the gain in the kinetic energy of the smaller fragment is 249.64 J.

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Answer:

The voltage across the capacitor will remain constant

The capacitance of the capacitor will increase

The electric field between the plates will remain constant

The charge on the plates will increase

The energy stored in the capacitor will increase

Explanation:

First of all, if a capacitor is connected to a voltage source, the voltage or potential difference across the capacitor will remain constant. The electric field across the capacitor will stay constant since the voltage is constant, because the electric field is proportional to the voltage applied. Inserting a dielectric material into the capacitor increases the charge on the capacitor.

The charge on the capacitor is equal to

Q = CV

Since the voltage is constant, and the charge increases, the capacitance will also increase.

The energy in a capacitor is given as

E = [tex]\frac{1}{2}CV^{2}[/tex]

since the capacitance has increased, the energy stored will also increase.

Answer:

a) 0.72 V

b) 19.2 mA

c) 2.304 Watts

Explanation:

A transformer is used to step-up or step-down voltage and current. It uses the principle of electromagnetic induction. When the primary coil is greater than the secondary coil, the it is a step-down transformer, and when the primary coil is less than the secondary coil, the it is a step-up transformer.

number of primary turns = [tex]N_{p}[/tex] = 500 turns

input voltage = [tex]V_{p}[/tex] = 120 V

number of secondary turns = [tex]N_{s}[/tex] = 3 turns

output voltage = [tex]V_{s}[/tex] = ?

using the equation for a transformer

[tex]\frac{V_{s} }{V_{p} } = \frac{N_{s} }{N_{p} }[/tex]

substituting values, we have

[tex]\frac{V_{s} }{120 } = \frac{3 }{500} }[/tex]

[tex]500V_{p} = 120*3\\500V_{p} = 360[/tex]

[tex]V_{p}[/tex] = 360/500 = 0.72 V

b) by law of energy conservation,

[tex]I_{P}V_{p} = I_{s}V_{s}[/tex]

where

[tex]I_{p}[/tex] = input current = ?

[tex]I_{s}[/tex] = output voltage = 3.2 A

[tex]V_{s}[/tex] = output voltage = 0.72 V

[tex]V_{p}[/tex] = input voltage = 120 V

substituting values, we have

120[tex]I_{p}[/tex] = 3.2 x 0.72

120[tex]I_{p}[/tex] = 2.304

[tex]I_{p}[/tex]  = 2.304/120 = 0.0192 A

= 19.2 mA

c) power input = [tex]I_{p} V_{p}[/tex]

==> 0.0192 x 120 = 2.304 Watts

Answer:

Acidosis is defined as the formation of excessive acid in the body due to kidney disease or kidney failure.

In order to compensate acidosis, the kidneys will reabsorb more HCO3 from the tubular fluid through tubular cells and collecting duct cell will secret more H+ and ammoniagenesis, which form more NH3 buffer.

Answer:

a

  [tex]\phi = 1.78 *10^{-7} \ Weber[/tex]

b

 [tex]L = 1.183 *10^{-7} \ H[/tex]

Explanation:

From the question we are told that

   The radius is  [tex]r = 6 \ cm = \frac{6}{100} = 0.06 \ m[/tex]

   The current it carries is  [tex]I = 1.50 \ A[/tex]

The  magnetic flux of the coil is mathematically represented as

       [tex]\phi = B * A[/tex]

Where  B is the  magnetic field which is mathematically represented as

         [tex]B = \frac{\mu_o * I}{2 * r}[/tex]

Where  [tex]\mu_o[/tex] is the magnetic field with a constant value  [tex]\mu_o = 4\pi * 10^{-7} N/A^2[/tex]

substituting  value

          [tex]B = \frac{4\pi * 10^{-7} * 1.50 }{2 * 0.06}[/tex]

          [tex]B = 1.571 *10^{-5} \ T[/tex]

The area A is mathematically evaluated as

       [tex]A = \pi r ^2[/tex]

substituting values

       [tex]A = 3.142 * (0.06)^2[/tex]

       [tex]A = 0.0113 m^2[/tex]

the magnetic flux is mathematically evaluated as    

        [tex]\phi = 1.571 *10^{-5} * 0.0113[/tex]

         [tex]\phi = 1.78 *10^{-7} \ Weber[/tex]

The self-inductance is evaluated as

       [tex]L = \frac{\phi }{I}[/tex]

substituting values

        [tex]L = \frac{1.78 *10^{-7} }{1.50 }[/tex]

         [tex]L = 1.183 *10^{-7} \ H[/tex]

Answer:

the phase relationship between two waves.

Explanation:

Coherence describes all properties of the correlation between physical quantities between waves. It is an ideal property of waves that determines their interference. In a situation in which there is a correlation or phase relationship between two waves. If the properties of one of the waves can be measure directly, then, some of the properties of the other wave can be calculated.

Answer:

The induced voltage is  [tex]\epsilon = 4.53 \ V[/tex]

Explanation:

From the question we are told that

    The  number of turns is  [tex]N = 1300 \ turns[/tex]

     The diameter is  [tex]d = 2.2 \ cm =0.022 \ m[/tex]

     The  initial magnetic field is  [tex]B_i = 0.11 \ T[/tex]

     The final magnetic field is  [tex]B_f = 0 \ T[/tex]

    The time taken is  [tex]t = 12 \ ms = 12*10^{-3} \ s[/tex]

The radius is mathematically evaluated as

         [tex]r = \frac{d}{2}[/tex]

substituting values

         [tex]r = \frac{0.022}{2}[/tex]

         [tex]r = 0.011 \ m[/tex]

Generally the induced emf  is mathematically represented as

      [tex]\epsilon = - N * \frac{d\phi}{dt}[/tex]

Where  [tex]d\phi[/tex] is the change in magnetic flux of the wire which is mathematically represented as

      [tex]d \phi = dB* A * cos \theta[/tex]

=>  [tex]d \phi = (B_f - B_i )* A * cos \theta[/tex]

Here  [tex]\theta = 0[/tex]

since the axis of the coil is parallel to the field

    Where A  is the cross-sectional area of the coil which is mathematically represented as

      [tex]A = \pi * r^2[/tex]

       [tex]A = 3.142 * 0.011^2[/tex]

      [tex]A = 3.80*10^{-4} \ m^2[/tex]

So the induced emf

        [tex]\epsilon = - 1300 * \frac{(0- 0.11) * 3.80*10^{-4}}{12*10^{-3}}[/tex]   Here we substituted the values of  [tex]d \phi[/tex]

       [tex]\epsilon = 4.53 \ V[/tex]

The emf induced in the coil at the given magnetic field strength is 4.53 V.

The given parameters;

The area of the coil is calculated as follows;

[tex]A = \pi r^2 = \frac{\pi d^2}{4} \\\\A = \frac{\pi \times 0.022^2}{4} = 0.00038 \ m^2[/tex]

The emf induced in the coil is calculated as follows;

[tex]emf = -N\frac{d\phi}{dt} \\\\emf = N (\frac{\phi_1 - \phi_2}{t} )\\\\emf = N(\frac{AB_1 - AB_2}{t} )\\\\emf = NA(\frac{B_1 - B_2}{t} )\\\\emf = 1300 \times 0.00038 (\frac{0.11 - 0}{12 \times 10^{-3}} )\\\\emf = 4.53 \ V[/tex]

Thus, the emf induced in the coil at the given magnetic field strength is 4.53 V.

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Answer:

- 3.3571

Explanation:

the negative sign means there were moving from right to left

The total momentum of a colliding system is constant always. Therefore, by using this concept, the final velocity of the coupled mass will be - 3.3 m/s.

Momentum of an object is the ability of an object to bring the force applied to make a maximum displacement. It is the product of its mass and velocity. Momentum is a vector quantity. It have both magnitude and direction.

The momentum in a collision is conserved. Thus total initial momentum is equal to the final momentum. Let m1 and m2 be the colliding masses and the u and v be the initial and final velocity.

Then, m1 u1 + m2u2 = (m1 + m2)v for a coupled mass after collision.

then final velocity v =  ( m1 u1 + m2u2 )/ (m1 + m2)

Apply the given values in the equation as follows:

v = (15 kg ×1.5 m/s + 6 kg× (-15.5 m/s)) (15 +6) = -3.3 m/s

Therefore, the final velocity of the coupled masses after collision is  -3.3 m/s.

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Answer:

Explanation:

E= −L ΔI / Δt.

L = E Δt / ΔI

Hence the unit of inductance may be V s A⁻¹

or volt s per ampere .

In the given case

change in current ΔI = - 2.5 A

change in time = .015 s

L = .56 H

E = − L ΔI / Δt.

= .56 x 2.5 / .015

= 93.33 V .

Answer:

Explanation:

moment of inertia of satellite I = 1/2 m R²

m is mass and R is radius of the disc

I = 0.5 x 8.5 x 0.8²

= 2.72 kg m²

angular acceleration α = change in angular velocity / time

α = (14.5 - 0) / 30

α = .48333

Let force of each rocket = F

torque created by one rocket = F x R

= F x .8

Torque created by 4 rockets = 4 x .8 F = 3.2 F

3.2 F = I x α

3.2 F =  2.72 x   .48333

F = 0 .41 N

Answer:

Explanation:

The sum of force along both components x and y is expressed as;

[tex]\sum Fx = ma_x \ and \ \sum Fy = ma_y[/tex]

The magnitude of the net force which is also known as the resultant will be expressed as [tex]R =\sqrt{(\sum Fx)^2 + (\sum Fx )^2}[/tex]

To get the resultant, we need to get the sum of the forces along each components. But first lets get the acceleration along the components first.

Given the position of the object along the x-component to be x = 6t² − 4;

[tex]a_x = \frac{d^2 x }{dt^2}[/tex]

[tex]a_x = \frac{d}{dt}(\frac{dx}{dt} )\\ \\a_x = \frac{d}{dt}(6t^{2}-4 )\\\\a_x = \frac{d}{dt}(12t )\\\\a_x = 12m/s^{2}[/tex]

Similarly,

[tex]a_y = \frac{d}{dt}(\frac{dy}{dt} )\\ \\a_y = \frac{d}{dt}(5t^{3} +6 )\\\\a_y = \frac{d}{dt}(15t^{2} )\\\\a_y = 30t\\a_y \ at \ t= 2.15s; a_y = 30(2.15)\\a_y = 64.5m/s^2[/tex]

[tex]\sum F_x = 3.15 * 12 = 37.8N\\\sum F_y = 3.15 * 64.5 = 203.18N[/tex]

[tex]R = \sqrt{37.8^2+203.18^2}\\ \\R = \sqrt{1428.84+41,282.11}\\ \\R = \sqrt{42.710.95}\\ \\R = 206.67N[/tex]

Hence, the magnitude of the net force acting on this object at t = 2.15 s is approximately 206.67N

Answer:

v=54-6t^2, 54-6 (9)=0

Answer:

vi = 19.6 m/s

Explanation:

Given:

final velocity vf = 0

gravity a = -9.8

time t = 1

Initial velocity vi = vf - at

vi = 0 + 9.8 (1.0)

vi = 9.8 m/s

the y component of velocity is the initial velocity.

therefore v sin 30 = 9.8

vi/2 = 9.8

vi = 19.6 m/s

A tennis ball is thrown from ground level with velocity v0 directed 30 degrees above the horizontal. If it takes the ball 1.0s to reach the top of its trajectory, the magnitude of the initial velocity vi = 19.62 m/s.

Given data to find the initial velocity,

final velocity vf = 0

gravity a = -9.8

time t = 1

The motion of the tennis ball on the vertical axis is an uniformly accelerated motion, with deceleration of  (gravitational acceleration).

The component of the velocity on the y-axis is given by the following law:

Initial velocity vi = vf - at

At the time t=0.5 s, the ball reaches its maximum height, and when this happens, the vertical velocity is zero (because it is a parabolic motion), Substituting into the previous equation, we find the initial value of the vertical component of the velocity:

vi = 0 + 9.8 (1.0)

vi = 9.8 m/s

the y component of velocity is the initial velocity.

However, this is not the final answer. In fact, the ball starts its trajectory with an angle of 30°. This means that the vertical component of the initial velocity is,

therefore, v sin 30° = 9.8

We found before the value of y component, so we can substitute to find the initial speed of the ball:

vi/2 = 9.8

vi = 19.6 m/s

Thus, the initial velocity can be found as 19.62 m/s.

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Answer:

The fraction of light that escapes the water surface as the water moves deeper into the water will decrease.

Explanation:

The speed of light in water is small compared to the speed of light in air, and a larger part of the light energy is absorbed in water than in air. When the bulb is immersed in water, some of the light energy is absorbed by the mass of water. When the light bulb is further moved deeper into the water, the fraction of light that escapes decreases, because more mass of water is made available to absorb more of the light energy from the bulb.

Answer:

The rate at which the magnetic field changes is  [tex]\frac{\Delta B }{\Delta t } = - 5.33*10^{-3} \ T/ s[/tex]

Explanation:

From the question we are told that

   The  electric field strength is [tex]E = 4.0 mV/m = 4.0 *10^{-3} V/m[/tex]

   The  radius of the  circular region where the electric field is induced is

   [tex]d = 1.5 \ m[/tex]

Generally the induced electric field is mathematically represented as

     [tex]E = - \frac{r}{2} * \frac{\Delta B }{\Delta t }[/tex]

The  negative sign show that the induced electric field is acting in opposite direction to the change in magnetic field

Where  [tex]\frac{\Delta B }{\Delta t }[/tex] is the change in magnetic field

So  

       [tex]\frac{\Delta B }{\Delta t } = - \frac{2 * E }{r}[/tex]

substituting values

       [tex]\frac{\Delta B }{\Delta t } = - \frac{2 * 4.0 *10^{-3}}{ 1.5 }[/tex]

       [tex]\frac{\Delta B }{\Delta t } = - 5.33*10^{-3} \ T/ s[/tex]

The arrows point away from a positive charge and towards a negative charge.

The electric field is a vector field, which means that it has both a magnitude and a direction. The magnitude of the electric field is a measure of how strong the field is, and the direction of the electric field is a measure of the direction in which the force would be exerted on a positive charge.

The brightness of the arrow in an electric field simulation is a representation of the magnitude of the electric field. The brighter the arrow, the stronger the electric field. When you check the "Direction only" box, the arrows will only show the direction of the electric field. This is because the "Direction only" box only shows the direction of the vector field, not the magnitude.

When you uncheck the box, the arrows will show both the direction and the strength of the electric field. This is because the "Direction only" box is unchecked, so the arrows will show the full vector field. The arrows point away from a positive charge and towards a negative charge because positive charges repel each other and negative charges attract each other.

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Answer:

Explanation:

Given two vectors as follows

E₁ = 13.5 i -12 j

E₂ = -7.4 i - 4.7 j

Resultant E = E₁ + E₂

= 13.5 i -12 j -7.4 i - 4.7 j

E = 6.1 i - 16.7 j

a ) X component of resultant = 6.1 N

b ) y component of resultant = -16.7 N

Magnitude of resultant = √ ( 6.1² + 16.7² )

= 17.75 N

d ) If θ be the required angle

tanθ = 16.7 / 6.1 = 2.73

θ = 70° .

counterclockwise = 360 - 70 = 290°

By working with the vector forces, we will get:

Remember that we can directly add vector forces, so if our two forces are:

F₁ = <13.5 N, -7.5 N>

F₂ = < -12 N, -4.70 N>

Then the resultant force is:

F = F₁ + F₂ = <13.5 N + (-12 N), -7.5 N + ( -4.70 N) >

F = < 1.5 N, -12.2 N>

so we have:

a) The x-component is 1.5 N

b) The y-component is -12.2 N

c) The magnitude will be:

|F| = √( (1.5 N)^2 + (-12.2 N)^2) = 12.29 N

d) The direction of a vector <x, y> measured counterclockwise from the positive x-axis is given by:

θ = Atan(y/x)

Where Atan is the inverse tangent function, then here we have:

θ = Atan(-12.2 N/1.5 N) = 277.01°

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Answer:

0

Explanation:

On a graph, if there is plot of time vs velocity, then area of velocity plot gives the displacement.

also we can see area of plot is velocity* time which is equal to formula of displacement.

area for this plot is velocity * time

Thus,

from

t =0 to t = 0.2

v = 20

t = 0.2 - 0 = 0.2

thus, displacement till 0.2 seconds = 20*0.2 = 4 Km

____________________________________________________

from

t =0 to t = 0.4

v = 0

t = 0.4 - 0.2 = 0.2

thus, displacement from  0.2 seconds to 0.4 seconds = 0*0.2 = 0 Km

____________________________________________________

from

t =0.4 to t = 0.6

v = -20

t = 0.6 - 0.4 = 0.2

thus, displacement from  0.4 seconds to 0.6 seconds = -20*0.2 = -4 Km

Thus, total displacement = 4+0 -4 = 0

Thus, net displacement made by car is 0.

Answer:

Since their wings and body develop the drag. When there is warm air then they expand their wings. Since,soaring birds and glider pilots have no engine, they always maintain their high speed to lift their weight in air for hours without expending power by convection

Explanation:

Answer:

(a) 4.334 × 10¹¹ joules are required to propel the rocket an unlimited distance away from Earth's surface, (b) The rocket has travelled 3999.865 miles from the Earth's surface with the half of the total work.

Explanation:

The complete statement is: "Use the weight of the rocket to answer the question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) 7 metric ton rocket (a) How much work is required to propel the rocket an unlimited distance away from Earth's surface, (b) How far has the rocket traveled when half the total work has occurred?"

(a) The work required to propel the rocket is given by the change in gravitational potential energy, whose expression derives is described below:

[tex]U_{g, f} - U_{g, o} = -G\cdot M\cdot m \cdot \left[\frac{1}{r_{f}}-\frac{1}{r_{o}} \right][/tex]

Where:

[tex]U_{g,o}[/tex], [tex]U_{g,f}[/tex] - Initial and final gravitational potential energies, measured in joules.

[tex]m[/tex], [tex]M[/tex] - Masses of the rocket and planet Earth, measured in kilograms.

[tex]G[/tex] - Universal gravitation constant, measured in newton-square meters per square kilogram.

[tex]r_{o}[/tex], [tex]r_{f}[/tex] - Initial and final distances of the rocket with respect to the center of the Earth, measured in meters.

The initial distance and rocket mass are converted to meters and kilograms, respectively:

[tex]r_{o} = (4000\,mi)\cdot \left(1609.34\,\frac{m}{mi} \right)[/tex]

[tex]r_{o} = 6,437,360\,m[/tex]

[tex]m = (7\,ton)\cdot \left(1000\,\frac{kg}{ton} \right)[/tex]

[tex]m = 7000\,kg[/tex]

Given that [tex]m = 7000\,kg[/tex], [tex]M = 5.972\times 10^{24}\,kg[/tex], [tex]G = 6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}}[/tex], [tex]r_{o} = 6,437,360\,m[/tex] and [tex]r_{f} \rightarrow +\infty[/tex], the work equation is reduced to this form:

[tex]U_{g,f} - U_{g,o} = \frac{G\cdot m \cdot M}{r_{o}}[/tex]

[tex]U_{g,f} - U_{g,o} = \frac{\left(6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}} \right)\cdot (7000\,kg)\cdot (5.972\times 10^{24}\,kg)}{6,437,360\,m}[/tex]

[tex]U_{g,f} - U_{g,o} = 4.334\times 10^{11}\,J[/tex]

4.334 × 10¹¹ joules are required to propel the rocket an unlimited distance away from Earth's surface.

(b) The needed change in gravitational potential energy is:

[tex]U_{g,f} - U_{g,o} = 2.167\times 10^{11}\,J[/tex]

The expression for the change in gravitational potential energy is now modified by clearing the final distance with respect to the center of Earth:

[tex]U_{g, f} - U_{g, o} = -G\cdot M\cdot m \cdot \left[\frac{1}{r_{f}}-\frac{1}{r_{o}} \right][/tex]

[tex]\frac{U_{g,o}-U_{g,f}}{G\cdot M \cdot m} = \frac{1}{r_{f}} - \frac{1}{r_{o}}[/tex]

[tex]\frac{1}{r_{f}} = \frac{1}{r_{o}} + \frac{U_{g,o}-U_{g,f}}{G\cdot M\cdot m}[/tex]

[tex]r_{f} = \left(\frac{1}{r_{o}} + \frac{U_{g,o}-U_{g,f}}{G\cdot M\cdot m} \right)^{-1}[/tex]

If [tex]m = 7000\,kg[/tex], [tex]M = 5.972\times 10^{24}\,kg[/tex], [tex]G = 6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}}[/tex], [tex]r_{o} = 6,437,360\,m[/tex]  and [tex]U_{g,f} - U_{g,o} = 2.167\times 10^{11}\,J[/tex], then:

[tex]r_{f} = \left[\frac{1}{6,437,360\,m}-\frac{2.167\times 10^{11}\,J}{\left(6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}} \right)\cdot (7000\,kg)\cdot (5.972\times 10^{24}\,kg)} \right]^{-1}[/tex]

[tex]r_{f} \approx 12,874,502.49\,m[/tex]

The final distance with respect to the center of the Earth in miles is:

[tex]r_{f} = (12,874,502.49\,m)\cdot \left(\frac{1}{1609.34}\,\frac{mi}{m} \right)[/tex]

[tex]r_{f} = 7999.865\,mi[/tex]

The distance travelled by the rocket is: ([tex]r_{f} = 7999.865\,mi[/tex], [tex]r_{o} = 4000\,mi[/tex])

[tex]\Delta r = r_{f}-r_{o}[/tex]

[tex]\Delta r = 7999.865\,mi - 4000\,mi[/tex]

[tex]\Delta r = 3999.865\,mi[/tex]

The rocket has travelled 3999.865 miles from the Earth's surface with the half of the total work.

Answer:

Final velocity (v) = 0.509 m/s (Approx)

Explanation:

Ant use impulse power

Given:

Mass of ant = 12 mg = 12 × 10⁻⁶ kg

Average force = 47 mN = 47 × 10⁻³ N

Initial velocity(u) = 0

Time taken = 0.13 ms = 0.13 × 10⁻³ s

Find:

Final velocity (v)

Computation:

Force × Time =  change in momentum

(47 × 10⁻³ N)(0.13 × 10⁻³ s) = mv - mu

(47 × 10⁻³ N)(0.13 × 10⁻³ s) = m(v - u)

6.11 × 10⁻⁶ = 12 × 10⁻⁶(v - 0)

6.11 = 12 v

Final velocity (v) = 0.509 m/s (Approx)

Answer:Centripetal force that acts an object keep it along a moving circular path.

Explanation:Centripetal force along a path circular of radius(r) with velocity(V) acceleration the center of the path.

a=v/r

object will along moving continue a straight path unless by the external force.External force is the centripetal force.

Centripetal force is to moving in horizontal circle,Centripetal force is not a fundamental force.Gravitational force satellite and orbit of centripetal force.

Centripetal acceleration and centripetal force are used to calculate the motion of objects in circular motion. The main answer to the question is given below:The centripetal force is given by:F = mv²/rwhere m is the mass of the object, v is the speed of the object and r is the radius of the circle. The unit of centripetal force is Newtons (N).The centripetal acceleration is given by:a = v²/rThe unit of centripetal acceleration is meters per second squared

(m/s²).Explanation:When an object moves in a circular motion, there is a force that acts upon it. This force is called the centripetal force. This force always points towards the center of the circle. It is responsible for keeping the object moving in a circular motion.The centripetal force is related to the centripetal acceleration.

The centripetal acceleration is the acceleration of an object moving in a circle. It is always directed towards the center of the circle.The magnitude of the centripetal force is given by:F = mv²/rwhere F is the force, m is the mass of the object, v is the speed of the object and r is the radius of the circle.The magnitude of the centripetal acceleration is given by:a = v²/rwhere a is the acceleration, v is the speed of the object and r is the radius of the circle.

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Answer:

Plss see attached file

Explanation:

Answer:

a

The  speed of  wave is   [tex]v_1 = 129.1 \ m/s[/tex]

b

The new speed of the two waves is [tex]v = 129.1 \ m/s[/tex]

Explanation:

From the question we are told that

    The mass of the string is  [tex]m = 60 \ g = 60 *10^{-3} \ kg[/tex]

    The length is  [tex]l = 2.0 \ m[/tex]

    The tension is  [tex]T = 500 \ N[/tex]

Now the velocity of the first wave is mathematically represented as

     [tex]v_1 = \sqrt{ \frac{T}{\mu} }[/tex]

Where  [tex]\mu[/tex] is the linear density which is mathematically represented as

      [tex]\mu = \frac{m}{l}[/tex]

substituting values    

     [tex]\mu = \frac{ 60 *10^{-3}}{2.0 }[/tex]

     [tex]\mu = 0.03\ kg/m[/tex]

So

   [tex]v_1 = \sqrt{ \frac{500}{0.03} }[/tex]

   [tex]v_1 = 129.1 \ m/s[/tex]

Now given that the Tension, mass and length are constant the velocity of the second wave will same as that of first wave (reference PHYS 1100 )

Answer:

a) The speed of the electron as it reaches the inner conductor is closest to:

v = 1.45 × 10⁷m/s

b) The electric field magnitude between the cylinders is

E = 10,000V/m

Explanation:

given

inner radius of the cylinder r₁ = 20mm = 0.02m

outter radius of the cylinder r₂ = 80mm = 0.08m

potential difference V= 600V

mass of electron = 9.1×10⁻³¹kg

charge on electron = 1.6×10⁻¹⁹C

calculating the work done in bringing electron at inner conductor is

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

note:

[tex]V = \frac{W}{q}[/tex]

∴W = (ΔV)q

(ΔV)q = [tex]\frac{1}{2}mv^{2}[/tex]

(600)1.6×10⁻¹⁹ = ¹/₂ × 9.1×10⁻³¹ × v²

v² ≈ 2.11 × 10¹⁴

v = 1.45 × 10⁷m/s

According to the energy conservation law, the total energy of an isolated system is always constant.  

The energy of an isolated system can neither be created nor be destroyed, it can only convert one form to another form.

∴ the maximum electric field

E = ΔV/d

E = 600/d

where d is the distance between the two points

where d = 0.06m

E = 600/0.06

E = 10,000V/m

Note: the electric field due to the potential difference between to points depends upon the potential difference V and the distance between both points d.

a) The speed of the electron as it reaches the inner conductor is closest to: v = 1.45 × 10⁷m/s

b) The electric field magnitude between the cylinders is, E = 10,000V/m

Given:

Inner radius of the cylinder r₁ = 20mm = 0.02m

Outer radius of the cylinder r₂ = 80mm = 0.08m

Potential difference V= 600V

Mass of electron = [tex]9.1*10^{-31}kg[/tex]

Charge on electron = 1.6×10⁻¹⁹C

A)

Calculation for Work Done:

[tex]W=1/2mv^2[/tex]............(1)

Also.

[tex]V=\frac{W}{q}[/tex]

Thus, [tex]W=\triangle V*q[/tex]...........(2)

On equating 1 and 2:

[tex]\triangle V*q=1/2mv^2\\\\(600)1.6*10^{-19} = 1/2 * 9.1*10^{-31}* v^2\\\\v^2 =2.11 * 10^{14}\\\\v = 1.45 * 1067m/s[/tex]

B)

The energy of an isolated system can neither be created nor be destroyed, it can only convert one form to another form.

Thus, the maximum electric field

[tex]E = \triangle V/d\\\\E = 600/d[/tex]

where d is the distance between the two points

d = 0.06m

[tex]E = 600/0.06\\\\E = 10,000V/m[/tex]

Thus, the maximum electric field is 10,000V/m.

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Answer:

The meaning of the integral (120, 60)∫ |v(t)| dt is simply the distance covered by the particle from time t = 60 seconds to time t = 120 seconds

Explanation:

We are told that the particle moves back and forth along a straight line with velocity v(t).

Now, velocity is the rate of change of distance with time. Thus, the integral of velocity of a particle with respect to time will simply be the distance covered by the particle.

Thus, the meaning of the integral (120, 60)∫ |v(t)| dt is simply the distance covered by the particle from time t = 60 seconds to time t = 120 seconds