determine the moment of inertia ixx of the sphere in appendix a ex- pressed in terms of its total mass, m. the material has a constant density rho and a uniform mass distribution.

Answers

Answer 1

The moment of inertia (Ixx) of the sphere in Appendix A expressed in terms of its total mass (m), with constant density (rho) and uniform mass distribution, is

Ixx = (8/15) * pi * r^5 * rho

To determine the moment of inertia (Ixx) of the sphere in Appendix A expressed in terms of its total mass (m), we will consider the sphere's constant density (rho) and uniform mass distribution. Here's a step-by-step explanation:

1. First, let's find the mass of the sphere. Mass (m) can be calculated using the formula:
m = (4/3) * pi * r^3 * rho
where r is the radius of the sphere.

2. Now, let's calculate the moment of inertia (Ixx) of the sphere. For a solid sphere, the moment of inertia along any axis passing through its center can be expressed as:
Ixx = (2/5) * m * r^2

3. We want to express Ixx in terms of the total mass (m). To do this, we can substitute the mass equation from step 1 into the moment of inertia equation from step 2:
Ixx = (2/5) * [(4/3) * pi * r^3 * rho] * r^2

4. Finally, simplify the equation:
Ixx = (8/15) * pi * r^5 * rho

So, the moment of inertia (Ixx) of the sphere in Appendix A expressed in terms of its total mass (m), with constant density (rho) and uniform mass distribution, is:
Ixx = (8/15) * pi * r^5 * rho

For more information on inertia, mass and density refer to https://brainly.com/question/1140505

#SPJ11

Related Questions

Which of the following is not a property of light?
a. light is a form of matter less dense than air
b. light travels in straight lines
c. light has different colors
d. light has different intensities, and cam be bright or dim

Answers

Answer: Light is not a form of matter, so a is not a property of light.

Explanation:

Name Period_Date Rotation: Worksheet 9 Angular Momentum The following masses are swung in horizontal circles at the end of a thin string at constant speed. a. A 2.0 kg mass moving at 2.0 on the end of a 2.0 m long thin string. b. A 3.0 kg mass moving at 1.om, on the end of a 2.0 m long thin string. c. A 1.0 kg mass moving at 3.0 on the end of a 2.0 m long thin string. d. A 2.0 kg mass moving at 1.0 on the end of a 4.0 m long thin string. e. A 2.0 kg mass moving at 2.0 on the end of a 4.0 m long thin string

Answers

The length of string is 16.0 kg·m²/s

What is Mass?

Mass is a fundamental property of matter that quantifies the amount of substance or material present in an object. It is a scalar quantity, meaning it only has magnitude and no direction. Mass is commonly measured in units such as kilograms (kg), grams (g), or other appropriate units depending on the context.

a. Mass: 2.0 kg

Velocity: 2.0 m/s

Length of string: 2.0 m

Angular momentum: 8.0 kg·m²/s

b. Mass: 3.0 kg

Velocity: 1.0 m/s

Length of string: 2.0 m

Angular momentum: 6.0 kg·m²/s

c. Mass: 1.0 kg

Velocity: 3.0 m/s

Length of string: 2.0 m

Angular momentum: 6.0 kg·m²/s

d. Mass: 2.0 kg

Velocity: 1.0 m/s

Length of string: 4.0 m

Angular momentum: 2.0 kg·m²/s

e. Mass: 2.0 kg

Velocity: 2.0 m/s

Length of string: 4.0 m

Angular momentum: 8.0 kg·m²/s

Learn more about Mass from the given link

https://brainly.com/question/86444

#SPJ1

water flows in a 10-cm diameter pipe at a velocity of 0.75 m/s. the mass flow rate of water in the pipe is:

Answers

The mass flow rate of water in the pipe is approximately 0.58875 kg/s using the formula of mass flow rate.

To find the mass flow rate of water in the pipe, we'll use the formula:
Mass flow rate = Area of the pipe × Velocity × Density of water

Step 1: Calculate the area of the pipe.
[tex]Area = \pi * (Diameter / 2)^2[/tex]

Diameter = 10 cm = 0.1 m (convert cm to m by dividing by 100)

[tex]Area = \pi * (0.1 / 2)^2 = \pi × (0.005)^2 = 0.000785 m^2[/tex]

Step 2: Use the given velocity.
Velocity = 0.75 m/s

Step 3: Determine the density of water.
The density of water is approximately 1000 kg/m³.

Step 4: Calculate the mass flow rate.
Mass flow rate = Area × Velocity × Density
[tex]Mass flow rate = 0.000785 m^2 * 0.75 m/s * 1000 kg/m^3 = 0.58875 kg/s[/tex]

Learn more about mass flow rate here:

https://brainly.com/question/23159800

#SPJ11

how would you shape a given length of wire to give it the greatest self-inductance? the least?

Answers

A straight wire has much lower inductance than a coiled wire, as the magnetic fields generated by the current are less likely to interact with each other and create self-inductance.

To shape a given length of wire to give it the greatest self-inductance, you would want to create a tightly wound coil, such as a solenoid.

The self-inductance of a solenoid is proportional to the square of the number of turns, so the more turns you can create with the given length of wire, the greater the inductance will be.

Additionally, keeping the turns close together will help maximize the inductance.

On the other hand, to achieve the least self-inductance, you would want to keep the wire as straight as possible

For more information on self-inductance and its effect on wire refer to https://brainly.in/question/14293968

#SPJ11

A straight wire has much lower inductance than a coiled wire, as the magnetic fields generated by the current are less likely to interact with each other and create self-inductance.

To shape a given length of wire to give it the greatest self-inductance, you would want to create a tightly wound coil, such as a solenoid.

The self-inductance of a solenoid is proportional to the square of the number of turns, so the more turns you can create with the given length of wire, the greater the inductance will be.

Additionally, keeping the turns close together will help maximize the inductance.

On the other hand, to achieve the least self-inductance, you would want to keep the wire as straight as possible

For more information on self-inductance and its effect on wire refer to https://brainly.in/question/14293968

#SPJ11

For ease of installation, a cabin that is used occasionally is supplied with baseboard electric heating. These are 30 amp circuits powered with 220 volts. To supply 70,000Btu/h (about 20kW)1. How many circuits are needed, and2. What is the resistance of each baseboard "strip" in a single circuit?

Answers

To supply 70,000 BTU/h (about 20 kW) of baseboard electric heating to the cabin, you would need 4 circuits, and the resistance of each baseboard strip in a single circuit would be approximately 7.3 ohms.

To determine how many circuits are needed and the resistance of each baseboard strip in a single circuit for a cabin with 70,000 BTU/h (about 20 kW) of baseboard electric heating, we'll need to follow these steps:

1. Convert the desired heating capacity to watts:
70,000 BTU/h * (1 kW / 3412.14 BTU/h) ≈ 20,500 W

2. Calculate the power per circuit:
Power per circuit = Voltage x Current = 220 V x 30 A = 6,600 W

3. Determine the number of circuits needed:
Number of circuits = Total Power / Power per circuit = 20,500 W / 6,600 W ≈ 3.1
Since you can't have a fraction of a circuit, you'll need 4 circuits to supply the required power.

4. Calculate the total resistance for each circuit:
Resistance (R) = Voltage² / Power = (220 V)² / 6,600 W ≈ 7.3 ohms.

Learn more about resistance:

https://brainly.com/question/17563681

#SPJ11

the current is from left to right in the conductor showmn. the magnetic field is onto the page and point is at higher potential than point t. the charge carriers are
a. positive b. negative c. neutral d. absent e. moving near the speed of light

Answers

The charge carriers in the conductor are likely to be negative, and they are not absent, but their speed is not near the speed of light due to the effects of collisions in the conductor.

Based on the given information, we know that the conductor is experiencing a magnetic force due to the magnetic field pointing onto the page. Additionally, we know that the point labeled as "point" is at a higher potential than the point labeled as "t".

The direction of the current flowing from left to right in the conductor is indicative of the direction in which the charge carriers are moving. Given this information, we can determine that the charge carriers in the conductor must be negative because electrons are negatively charged and move opposite to the direction of conventional current flow.

Furthermore, the fact that the charge carriers are moving in a conductor does not necessarily imply that they are moving near the speed of light. The speed at which electrons move in a conductor is known as the drift velocity and is typically much slower than the speed of light due to collisions with other particles in the conductor.

For more such questions on Charge.

https://brainly.com/question/27986842#

#SPJ11

suppose an ideal gas undergoes isobaric (constant pressure) compression)
which expression about the entropy of the environment and the gas is correct?
∆S > 0 ∆S + ∆S > 0 ∆ Seny + ∆S = 0

Answers

The correct expression about the entropy of the environment and the gas for an isobaric compression of an ideal gas is ∆S + ∆Senv > 0, where ∆S is the change in entropy of the gas and ∆Senv is the change in entropy of the environment.

During an isobaric compression, work is done on the gas to decrease its volume while the pressure remains constant. This leads to an increase in the temperature of the gas, which in turn leads to an increase in its entropy. However, the compression process also results in an increase in the entropy of the environment due to the release of heat.

Thus, the total change in entropy of the system (gas) and the environment is positive, which is expressed as ∆S + ∆Senv > 0. The other options given (∆S > 0 and ∆S + ∆Seny = 0) are not correct for an isobaric compression process.

for a particular process at 300 k, δg δ g = -10.0 kj and δh δ h = -7.0 kj. if the process is carried out reversibly, the amount of useful work (in kj) that .

Answers

The amount of useful work that can be obtained for this particular process at 300 K is -3.0 kJ.

For a reversible process, the amount of useful work (in kJ) that can be obtained is given by the equation:

w = -δg = δh - Tδs

where δs is the change in entropy of the system.

Since the process is carried out reversibly, δs can be calculated using the equation:

δs = δqrev / T

where δqrev is the heat absorbed by the system during a reversible process.

Since δh = -7.0 kJ and δg = -10.0 kJ, we know that the process is exothermic (δh < 0) and spontaneous (δg < 0). Therefore, δqrev must be negative, indicating that heat is released from the system.

We can calculate δqrev using the equation:

δqrev = -Tδs = -T(δh / T) = -δh = 7.0 kJ

Substituting this value into the equation for work, we get:

w = -δg = δh - Tδs = -10.0 kJ - (-7.0 kJ) = -3.0 kJ

Therefore, the amount of useful work for this particular process at 300 K is -10.0 kJ.

Learn more about work here: brainly.com/question/31050706

#SPJ11

what is the sensitivity (in µa) of the galvanometer (that is, what current gives a full-scale deflection) inside a voltmeter that has a 1.00 mω resistance on its 30.5 v scale?

Answers

The sensitivity of the galvanometer inside the voltmeter is 30,500,000 µA.

Hi! To find the sensitivity (in µA) of the galvanometer inside a voltmeter with a 1.00 mΩ resistance on its 30.5 V scale, you can follow these steps:

1. First, note the full-scale voltage, V = 30.5 V, and the internal resistance of the voltmeter, R = 1.00 mΩ.
2. Use Ohm's law to calculate the current for full-scale deflection, I = V/R.
3. Convert the calculated current to microamperes (µA).

Now, let's calculate the sensitivity of the galvanometer:

1. V = 30.5 V, R = 1.00 mΩ = 0.001 Ω (since 1 mΩ = 0.001 Ω).
2. I = V/R = 30.5 V / 0.001 Ω = 30500 A.
3. Convert the current to µA: 1 A = 1,000,000 µA, so 30500 A = 30,500,000 µA.

So, the sensitivity of the galvanometer inside the voltmeter is 30,500,000 µA.

To know more about galvanometer refer here:

https://brainly.com/question/17247518

#SPJ11

When helium capture occurs with a carbon 12 nucleus, what results?
A) Nitrogen 14
B) Oxygen 16
C) Neon 20
D) Silicon 28
E) Nickel 56

Answers

When helium capture occurs with a carbon 12 nucleus, it results in Nitrogen 14.

Helium capture, also known as alpha capture, is a type of nuclear reaction in which a helium nucleus (consisting of two protons and two neutrons, denoted as an alpha particle) is captured by a target nucleus.

When a helium capture occurs with a carbon 12 nucleus (which has 6 protons and 6 neutrons), the resulting nucleus will have 8 protons and 8 neutrons. This results in the formation of a nitrogen 14 nucleus, which has 7 protons and 7 neutrons, denoted as 14N or Nitrogen-14.

Learn more about carbon 12 at

https://brainly.com/question/31324610

#SPJ11

I need help with my physics homework
When a thin stick of mass M and length L is pivoted about one end, its moment of inertia is I=(1/3)ML^2. When the stick is pivoted about its midpoint, its moment of inertia is

A.) (1/12)ML^2
B.) (1/6)ML^2
C.) (1/3)ML^2
D.) (7/12)ML^2
E.) ML^2

Answers

The moment of inertia of the stick when it is pivoted about its midpoint is (1/6)ML². The answer choice is (B).

When the stick is pivoted about its midpoint, we can split it into two equal pieces of mass M/2 and length L/2, each with a moment of inertia of:

I₁ = (1/3)(M/2)(L/2)² = (1/12)ML²

The parallel axis theorem tells us that the moment of inertia of the whole stick about its midpoint is equal to the sum of the moments of inertia of the two pieces plus the moment of inertia of the stick as if it were a point mass at its center of mass:

I₂ = 2I₁ + (1/12)M(L/2)²

I₂ = (2/12)ML² + (1/48)ML²

I₂ = (1/6)ML²

Option B is correct.

To know more about inertia, here

brainly.com/question/3268780

#SPJ1

show that the transition matrix is regular and find its steady-state vector.

Answers

To show that a transition matrix is regular by raising the matrix to different powers until you find a power with all positive elements and its steady-state vector is equation πP = π while ensuring the elements of π sum to 1.

A transition matrix is regular if some power of the matrix has only positive elements, a steady-state vector is a probability vector that remains unchanged after being multiplied by the transition matrix. To demonstrate that the transition matrix is regular, raise the matrix to different powers until you find a power with all positive elements. If such a power exists, the matrix is considered regular.

Next, to find the steady-state vector, solve the following equation: πP = π, where π is the steady-state vector, and P is the transition matrix. Additionally, ensure the elements of π sum to 1, representing the total probability, you can solve this system of linear equations using methods like Gaussian elimination, matrix inversion, or iterative techniques. In summary, to show that a transition matrix is regular, find a power of the matrix with all positive elements, then, to find the steady-state vector, solve the equation πP = π while ensuring the elements of π sum to 1.

Learn more about transition matrix at:

https://brainly.com/question/28026040

#SPJ11

What eclipse occurs when the Moon is in between the Sun and the Earth and the Moon partially or completely blocks out the Sun?

Answers

The eclipse that occurs when the Moon is in between the Sun and the Earth and partially or completely blocks out the Sun is known as a solar eclipse.

During a solar eclipse, the Moon casts a shadow on the Earth's surface, creating a path of totality where the Sun is completely blocked out, and a partial eclipse where only part of the Sun is covered. Solar eclipses occur only during a new moon when the Moon passes between the Sun and the Earth.

During a solar eclipse, the Moon's shadow falls on the Earth's surface, causing the Sun to appear as if it is being covered or "eclipsed" by the Moon. Depending on the alignment of the Sun, Moon, and Earth, a solar eclipse can be either partial or total.

Therefore, A total solar eclipse occurs when the Moon completely covers the Sun, and only the Sun's corona (outer atmosphere) is visible as a glowing ring around the Moon.

To know more about solar eclipse refer here :

https://brainly.com/question/1077760

#SPJ11

An automobile dealer calculates the proportion of new cars sold that have been returned a various numbers of times for the correction of defects during the warranty period. The results are shown in the following table. Number of returns 0 Proportions 0.28 0.36 0.23 0.09 0.04 (a) Graph the probability distribution function. (b) Calculate and graph the cumulative probability distribution. (b) Calculate and graph the cumulative probability distribution. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period. (d) Find the variance of the number of returns of an automobile for correc- tions for defects during the warranty period.

Answers

(a) Bar graph with x-axis showing the number of returns and y-axis showing the proportion of new cars sold with that number of returns.

(b) Line graph with x-axis showing the number of returns and y-axis showing the cumulative proportion of new cars sold with up to that number of returns.

(d) Variance = 1.715 returns^2, calculated as [(0-0.95)^20.28]+[(1-0.95)^20.36]+[(2-0.95)^20.23]+[(3-0.95)^20.09]+[(4-0.95)^20.04].

In part (a), we represent the probability distribution function using a bar graph. The x-axis shows the number of returns, and the y-axis shows the proportion of new cars sold with that number of returns. In part (b), we plot the cumulative probability distribution using a line graph. The x-axis shows the number of returns, and the y-axis shows the cumulative proportion of new cars sold with up to that number of returns.

learn more about returns here:

https://brainly.com/question/29569139

#SPJ11

You will need to design a cascode amplifier, which should be built and tested to meet the following requirements:1. Magnitude of the voltage gain = 12*SQRT(Z+35):± 10%, where Z is the sum of the last 3 digits of yourstudent number.2. The load resistance RL = 6*(Z+40)2 Ω, rounded up to the nearest standard value stocked in the lab, i.e.,decade multiples of 1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, and 8.2 kΩ. As an example, if yourRL is 67.4 kΩ, you will round this to 68 kΩ.3. The high frequency cutoff fH is to be maximized. It must exceed 1 MHz.4. The output voltage should be able to get to 2 V peak-peak without appreciable distortion[1]. To ensurethis, the AC base-emitter voltage must be kept under 10 mV peak-peak for such an output.5. No DC current may flow in RL and no DC current may flow into or out of the signal generator.6. The low frequency fL must be less than 200 Hz.7. The input and output impedances are left to the discretion of the designer, but their magnitudes at 1kHz are to be determined by calculation and then measured.8. Total circuit power is not to exceed 50 mW.9. The transistors are all to be 2N3904.10. Collector currents in the transistors are to be 1.0 mA ±10%.11. Power-supply voltages are to be limited to +5 volts and/or +15 volts and/or -15 volts.12. No adjustable components, e.g. a trimmer potentiometer, will be allowed.13. Available capacitors are limited to: 1 x 100 µF, 1 x 33 µF, 1 x 10 µF, 2 x 1 µF, 1 x 0.1 µF [2].14. The choice of all other components is left up to the designer.

Answers

To design a cascode amplifier meeting the specified requirements, follow these steps:

1. Calculate the voltage gain (Av) using the formula Av = 12 * SQRT(Z + 35), where Z is the sum of the last 3 digits of your student number.
2. Determine the load resistance (RL) using the formula RL = 6 * (Z + 40)² Ω, and round up to the nearest standard value stocked in the lab.
3. Use a high-pass filter at the input to achieve the low frequency cutoff (fL) of less than 200 Hz.
4. Design the cascode amplifier using 2N3904 transistors with collector currents set to 1.0 mA ±10%. The power supply voltages should be limited to +5V, +15V, or -15V.
5. Maximize the high frequency cutoff (fH) to exceed 1 MHz by carefully selecting component values and minimizing parasitic capacitances.
6. Ensure the output voltage can reach 2 V peak-peak without distortion by keeping the AC base-emitter voltage below 10 mV peak-peak.
7. Prevent DC current from flowing in RL and the signal generator by using coupling capacitors.
8. Calculate and measure the input and output impedances at 1 kHz.
9. Limit the total circuit power to 50 mW.
10. Use the available capacitors (1 x 100 µF, 1 x 33 µF, 1 x 10 µF, 2 x 1 µF, 1 x 0.1 µF) in the design.
11. Choose all other components as needed to achieve the desired performance while adhering to the constraints.

By following these steps, you will design a cascode amplifier that meets the given requirements. Remember, no adjustable components are allowed, and all transistors must be 2N3904.

For more such questions on cascode amplifier, visit:

brainly.com/question/14869225

#SPJ11

A typical power for a laser used in physics labs is 0.75 mW. This laser would produce a beam that is about 1 mm in diameter I found the average intensity: to be 955 W/m2 I found the average energy density of this beam: 3.183*10^-6 J/m^3 I need help with this question. Lets say the laser beam is reflected completely off a mirror. What is the maximum force the beam can exert on the mirror? I did 955 x 2x (4pie x 10^-7)/(3*10^8) but says my answer is wrong, is this the right equation???

Answers

The maximum force of a typical power for a laser used in physics labs is 0.75 mW and would produce a beam that is about 1 mm in diameter can exert on the mirror is 5 x 10⁻⁹ N.

In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.

To find the maximum force the laser beam can exert on the mirror when it is reflected completely, you should use the following formula:

Force (F) = (2 × Power (P)) / Speed of light (c)

where Power (P) = 0.75 mW (convert to Watts: 0.75 x 10⁻³ W), and Speed of light (c) = 3 x 10⁸ m/s.

F = (2 × (0.75 x 10⁻³ W)) / (3 x 10⁸ m/s)

Thus, the maximum force of the beam can exert on the mirror is approximately 5 x 10⁻⁹ N.

Learn more about maximum force: https://brainly.com/question/14434333

#SPJ11

A closed curve encircles several conductors. The line integral around this curve is ∮B⃗ ⋅dl⃗ = 4.25×10^-4 T⋅m .
A) What is the net current in the conductors?
B) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral?

Answers

a) the net current in the conductors is 3.38 A. b) the line integral in the opposite direction will be: ∮B⃗ ⋅dl⃗ = -4.25×10^-4 T⋅m

To solve this problem, we can use Ampere's Law, which relates the line integral of the magnetic field around a closed loop to the net current passing through the loop.

A) The equation for Ampere's Law is: ∮B⃗ ⋅dl⃗ = μ0I, where μ0 is the permeability of free space and I is the net current passing through the loop. Solving for I, we get:

I = ∮B⃗ ⋅dl⃗ / μ0

Substituting the given values, we get:

I = (4.25×10^-4 T⋅m) / (4π×10^-7 T⋅m/A)

I = 3.38 A

Therefore, the net current in the conductors is 3.38 A.

B) If we integrate around the curve in the opposite direction, the value of the line integral will be negative, since the direction of the magnetic field will be opposite. Specifically, we can use the fact that reversing the direction of the line integral is equivalent to reversing the direction of the loop, which changes the sign of the enclosed current.

Therefore, the line integral in the opposite direction will be: ∮B⃗ ⋅dl⃗ = -4.25×10^-4 T⋅m

To learn more about conductors click here

brainly.com/question/8426444

#SPJ11

Johnny, of mass 65 kg, and Lucy, of mass 45 kg, are facing each other on roller blades. The coefficient of kinetic friction between the roller blades and concrete surface is 0.20. When Johnny pushes Lucy from rest he applies a force for 1.0 s. Lucy then slows down to a stop in another 8.0 s. Calculate:
a. The applied force exerted by Johnny on Lucy.
b. How long it takes Johnny to come to rest.

I tried calculated the force exerted but I would need acceleration which I don't have...any tips on how to solve this one??? help is appreciated!!

Answers

Answer:

John applied a force of approximately [tex]795\; {\rm N}[/tex] (on average, rounded) on Lucy.

John slows down to a stop after approximately another [tex]5.37\; {\rm s}[/tex].

(Assuming that [tex]g = 9.81\; {\rm N\cdot kg^{-1}}[/tex].)

Explanation:

Assuming that the surface is level. The normal force on Johnny will be equal to the weight of Johnny: [tex]N(\text{John}) = m(\text{John})\, g[/tex]. Similarly, the normal force on Lucy will be equal to weight [tex]N(\text{Lucy}) = m(\text{Lucy})\, g[/tex].

Multiply normal force by the coefficient of kinetic friction to find the friction on each person:

[tex]f(\text{John}) = \mu_{k}\, N(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex].

[tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex].

Again, because the surface is level, the net force on each person after the first [tex]1.0\; {\rm s}[/tex] will be equal to the friction. Divide that the net force on each person by the mass of that person to find acceleration:
[tex]\displaystyle a(\text{John}) = \frac{\mu_{k}\, m(\text{John})\, g}{m(\text{John})} = \mu_{k}\, g[/tex].

[tex]\displaystyle a(\text{Lucy}) = \frac{\mu_{k}\, m(\text{Lucy})\, g}{m(\text{Lucy})} = \mu_{k}\, g[/tex].

(Note that the magnitude of acceleration is independent of mass and is the same for both John and Lucy.)

[tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex].

In other words, after the first [tex]1\; {\rm s}[/tex], both John and Lucy will slow down at a rate of [tex]1.962\; {\rm m\cdot s^{-2}}[/tex].

To find the speed of Lucy immediately after the first [tex]1.0\: {\rm s}[/tex], multiply this acceleration by the time [tex]t = 8.0\; {\rm s}[/tex] it took for Lucy to slow down to [tex]0\; {\rm m\cdot s^{-1}}[/tex]:

[tex]\begin{aligned}& (8.0\; {\rm s})\, (1.962\; {\rm m\cdot s^{-2}}) \\ =\; & (8.0)\, (1.962)\; {\rm m\cdot s^{-1}} \\ =\; & 15.696\; {\rm m\cdot s^{-1}}\end{aligned}[/tex].

Thus, in the first [tex]1.0\; {\rm s}[/tex], Lucy accelerated (from [tex]0\; {\rm m\cdot s^{-1}}[/tex]) to [tex]15.696\; {\rm m\cdot s^{-1}}[/tex].

The average acceleration of Lucy in the first [tex]1.0\; {\rm s}[/tex] would be [tex](15.696) / (1) = 15.696\; {\rm m\cdot s^{-2}}[/tex]. Multiply this average acceleration by the mass of Lucy to find the average net force on Lucy during that [tex]1.0\; {\rm s}[/tex]:

[tex]\begin{aligned}F_{\text{net}}(\text{Lucy}) &= m(\text{Lucy})\, a \\ &= (45)\, (15.696)\; {\rm N} \\ &= 706.320\; {\rm N}\end{aligned}[/tex].

This net force on Lucy during that [tex]1.0\; {\rm s}[/tex] is the combined result of both the push from Johnny and friction:

[tex]F_{\text{net}}(\text{Lucy}) = F(\text{push}) - f(\text{Lucy})[/tex].

Since [tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex]:

[tex]\begin{aligned}F(\text{push}) &= F_{\text{net}}(\text{Lucy}) + f(\text{Lucy}) \\ &= F_{\text{net}}(\text{Lucy}) + \mu_{k}\, m(\text{Lucy})\, g \\ &= (706.320) \; {\rm N}+ (0.2)\, (45)\, (9.81)\; {\rm N} \\ &= 706.320\; {\rm N} + 88.290\; {\rm N} \\ &=794.610\; {\rm N}\end{aligned}[/tex].

In other words, Johnny would have applied a force of [tex]794.610\; {\rm N}[/tex] on Lucy.

By Newton's Laws of Motion, when Johnny exerts this force on Lucy in that [tex]1.0\; {\rm s}[/tex], Lucy would exert a reaction force on Johnny of the same magnitude: [tex]794.610\; {\rm N}[/tex].

Similar to Lucy, the net force on Johnny during that [tex]1.0\; {\rm s}[/tex] will be the combined effect of the push [tex]F(\text{push})[/tex] and friction [tex]f(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex]:

[tex]\begin{aligned}F_{\text{net}}(\text{John}) &= F(\text{push}) - f(\text{John}) \\ &= F(\text{push}) - \mu_{k}\, m(\text{John})\, g\\ &= 794.610\; {\rm N} - (0.2)\, (65)\, (9.81)\; {\rm N} \\ &= 667.080\; {\rm N}\end{aligned}[/tex].

Divide net force by mass to find acceleration:

[tex]\begin{aligned}\frac{667.080\; {\rm N}}{65\; {\rm kg}} \approx 10.2628\; {\rm m\cdot s^{-2}}\end{aligned}[/tex].

In other words, Johnny accelerated at a rate of approximately [tex]10.5406\; {\rm m\cdot s^{-2}}[/tex] during that [tex]1.0\; {\rm s}[/tex]. Assuming that Johnny was initially not moving, the velocity of Johnny right after that [tex]1.0\; {\rm s}\![/tex] would be:

[tex](0\; {\rm m\cdot s^{-1}}) + (10.2628\; {\rm m\cdot s^{-2}})\, (1.0\; {\rm s}) = 10.2628\; {\rm m\cdot s^{-1}}[/tex].

After the first [tex]1.0\; {\rm s}[/tex], the acceleration of both John and Lucy (as a result of friction) would both be equal to [tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex]. Divide initial velocity of Johnny by this acceleration to find the time it took for Johnny to slow down to a stop:

[tex]\displaystyle \frac{10.2628\; {\rm {m\cdot s^{-1}}}}{1.962\; {\rm m\cdot s^{-2}}} \approx 5.23\; {\rm s}[/tex].