a person (mass mp) and their dog (mass m) are sitting together at the left end of a boat that has a length of l

The mass will come to rest at its natural length, which is also its equilibrium position.

What is spring constant?

Spring constant (k) is a measure of the stiffness of a spring or other elastic object. It is defined as the force required to stretch or compress the spring by a unit distance, usually expressed in newtons per meter (N/m) or pounds per inch (lb/in).

The distance below the natural length of the spring at which the mass will come to rest can be calculated using the energy conservation principle, which states that the initial potential energy stored in the spring will be converted into the kinetic energy of the mass as it oscillates, and then back into potential energy when the mass reaches its maximum displacement.

The potential energy stored in a spring is given by:

U = (1/2)kx²

where U is the potential energy, k is the spring constant, and x is the displacement from the natural length of the spring.

At the maximum displacement, all the potential energy is converted into kinetic energy, given by:

K = (1/2)mv²

where K is the kinetic energy and m is the mass of the object.

Using the conservation of energy, we can equate the potential energy at the maximum displacement to the kinetic energy at the resting position:

(1/2)kx² = (1/2)mv²

Rearranging, we get:

x = sqrt[(mv²)/k]

To find the velocity of the mass at the resting position, we can use the conservation of energy again to equate the potential energy at the resting position to the kinetic energy at the maximum displacement:

(1/2)kx₀² = (1/2)mv_max²

where x₀ is the displacement from the natural length of the spring at the resting position, and v_max is the  maximum velocity of the mass.

Rearranging, we get:

v_max = sqrt[(k/m)x₀²]

At the resting position, the velocity of the mass is zero, so we can use the equation of motion for simple harmonic motion to find the maximum displacement:

x_max = (v_max / w)

where w is the angular frequency of the oscillation, given by:

w = sqrt(k/m)

Substituting the expressions for v_max and w, we get:

x_max = sqrt[(k/m)x₀²] / sqrt(k/m)

Simplifying, we get:

x_max = x₀

Therefore, the maximum displacement of the mass from the natural length of the spring at the resting position is equal to the displacement from the natural length at the maximum displacement. Substituting the given values into the equation for x, we get:

x = sqrt[(22 kg x (0 m/s)²) / 11 N/m] = 0 m

Therefore, The mass will come to rest at its natural length, which is also its equilibrium position.

To know more about Mass visit :-

https://brainly.com/question/86444

#SPJ1

If this Doppler shift is due to the Sun rotating (try drawing a diagram), then  2km/s is the rotational speed of the Sun at its equator.

According to relation; of doppler shift:-

Velocity of rotation = ((observed wavelength / rest wavelength ) -1) x speed of Light

Here Observed wavelength = 486-0.0033 = 485.9967.m

and, Rest wavelength- 486 nm.

So, velocity=

((485.9967/486) - 1) x 299792.458 m/s

So, velocity = -2.035 km/s or Just 2km/s.

Doppler shift has important applications in astronomy, where it is used to determine the motion of celestial objects, such as stars and galaxies, and to measure their distance from Earth. It results in a change in the frequency of the observed waves, as perceived by the observer.

When the source of waves is moving towards the observer, the waves are compressed, and their frequency appears to be higher than their actual frequency. This is called a blue shift. Conversely, when the source is moving away from the observer, the waves are stretched, and their frequency appears to be lower than their actual frequency. This is called a red shift. It is also used in radar technology, where it is used to determine the speed and direction of moving objects, such as airplanes and vehicles.

To learn more about Doppler shift visit here:

brainly.com/question/30629988

#SPJ4

Complete Question:-

An astronomer observing the spectrum of the Sun finds that the Hydrogen-ß spectral line (λ = 486 nm) on the solar equator at one edge of the Sun's disk is blueshifted by 0.0033 nm compared to the same line at the center of the Sun's disk and is redshifted by the same amount on the equator at the other side of the Sun's disk. If this Doppler shift is due to the Sun rotating (try drawing a diagram), then what is the rotational speed of the Sun at its equator?

Answer:

it's b

Explanation:

no shade, direct sunlight

The work done when a force of 111 N is applied to move a cart a distance of 22 m is 2,442 J (joules).

To determine the work done when a force is applied to move an object, we use the formula:

work = F x d

where:

Therefore, we can calculate the work done as:

work = force x distance

work = 111 N x 22 m x 1 = 2,442 J

Learn more about work done here: https://brainly.com/question/8119756

#SPJ1

Answer:

see attachment

Explanation:

The maximum number of electrons in any shell is given by formula 2n² , where n is the number of shell .So in first shell , 2(1)² = 2 electrons would be accommodated.

Similarly in second shell , 2(2)² = 8 electrons would be accommodated .

Similarly we can find out the maximum number of electrons in 3rd , 4th and 5th shell .

The maximum number of electrons in valance shell is 8 .

Also, atomic numbers of ,

and we are done!

The US city that regularly experiences the worst levels of photochemical smog that is enhanced by thermal inversions is Los Angeles.

Photochemical smog is a type of smog that is formed when pollutants in the air react with sunlight. It can cause health problems such as respiratory issues, eye irritation, and headaches.Thermal inversion occurs when a layer of warm air traps pollutants near the ground, preventing them from dispersing.

This can exacerbate the effects of photochemical smog.Los Angeles is known for its high levels of air pollution, including photochemical smog.

The combination of heavy traffic, industrial activity, and the surrounding mountains that trap pollutants in the area can contribute to the city's poor air quality. Additionally, the city's warm, sunny climate can create conditions that enhance the effects of photochemical smog.

for such more question on photochemical smog

https://brainly.com/question/28956685

#SPJ11

The Delta t and Delta x between the events E1 and E2 in the frames of A, B, and C, and show that c2 Delta t2 - Delta x2 is the same in all three frames. The Space time interval in all frames is [tex]\frac{144}{25}L^2[/tex].

In the following we will find out the time interval and space interval between the two events E1 and E2 with respective to  A, B and C.

Simultaneously we will find out space time interval in each case and finally show that they are the same.



In the frame of reference of C

The time interval is the time it takes for ( to Cover the contracted length of B.

with respect to C, B will have a relative velocity Ux' = (-5/13)C  (we had already found out it.Only the sign changes)

Then the contrasted length of B with respect to C.

would be L' =  [tex]L\sqrt{1 - \frac{Ux^2}{C^2}} = L\sqrt{1 - \frac{25}{169}}[/tex]

L' = (12/13)L

So dt = L'/un\x' =(12/13)L / (-5/13)C = (12/5)(L/C)

dx =0 as E1, and E2 occurs at the same point with respect to C. Now space time Interval is Cdt^2 = dx^2 =

[tex]C^2 \frac{144}{25}\frac{L^2}{C^2}-0 = \frac{144}{25}L^2[/tex]

The quantity of time between two given instances is referred to as time interval. In other words, it is the amount of time that has surpassed among the beginning and end of the event. it is also called elapsed time. interval of time is measured in special units. every unit describes a one of a kind quantity of time. some units are better appropriate to specific durations of time.

As an instance, if you were baking a cake within the oven, you will select to measure the time in minutes or perhaps in hours. in case you were calculating the time on your birthday from a particular date, you will choose to measure the time in days, weeks, or months (relying on how far away it became).

To learn more about Time interval visit here:

brainly.com/question/28210196

#SPJ4

Answer:

the wavelength will decrease

Explanation:

If the frequency of a wave increases, the wavelength will decrease. This is because the speed of the wave is constant for a given medium, so if the frequency (the number of waves passing a fixed point per second) increases, then the distance between successive wave crests (i.e., the wavelength) must decrease to maintain a constant speed. This relationship is described by the wave equation:

v = f λ

where v is the speed of the wave, f is the frequency, and λ is the wavelength. If v is constant and f increases, then λ must decrease to keep the equation balanced.

Draw the complete cycle of the wave by repeating the pattern of the peak, the equilibrium position, and the trough, with a distance of λ between each consecutive peak or trough. The number of cycles per second, or the frequency, should be 5.0 hertz.

What is Wave?

A wave is a disturbance that propagates through space and time, often transferring energy from one location to another without the physical transfer of matter. Waves can take many different forms, including sound waves, electromagnetic waves, and mechanical waves.

Draw a horizontal axis representing time, labeled in seconds or milliseconds.

Draw a vertical axis representing displacement or amplitude, labeled in centimeters or meters.

Choose a starting point for the wave, which represents the equilibrium position of the medium.

Draw the peak of the wave, which represents the maximum displacement of the medium from its equilibrium position. This should be 4.0 centimeters above the equilibrium position.

Draw the trough of the wave, which represents the minimum displacement of the medium from its equilibrium position. This should be 4.0 centimeters below the equilibrium position.

Determine the wavelength of the wave, which is the distance between two consecutive peaks or troughs. This can be calculated using the formula λ = v/f, where λ is the wavelength, v is the velocity of the wave, and f is the frequency. For a transverse wave on a string, the velocity is given by v = √(T/μ), where T is the tension in the string and μ is the linear mass density of the string.

Learn more about Wave from the given link

https://brainly.com/question/25699025

#SPJ1

Answer:

Explanation:

The centripetal acceleration of the race car is given by the formula:

a = v^2 / r

where v is the speed of the race car and r is the radius of the turn.

Substituting the given values, we get:

a = (27.5 m/s)^2 / 57.0 m = 13.3 m/s^2

So the centripetal acceleration of the race car is 13.3 m/s^2.

To find the coefficient of friction, we need to use the formula:

f = μN

where f is the force of friction, μ is the coefficient of friction, and N is the normal force.

The normal force is equal to the weight of the car, which we can calculate as:

N = mg

where m is the mass of the car and g is the acceleration due to gravity (9.81 m/s^2).

Assuming the mass of the car is 1500 kg, we get:

N = 1500 kg × 9.81 m/s^2 = 14,715 N

The force of friction is equal to the centripetal force required to keep the car moving in a circle:

f = ma = (1500 kg)(13.3 m/s^2) = 19,950 N

Substituting the values of N and f into the formula for friction, we get:

19,950 N = μ(14,715 N)

Solving for μ, we get:

μ = 1.35

So the coefficient of friction is 1.35.

The beat frequency produced when a 240-hertz tuning fork and a 246-hertz tuning fork are sounded together would be 6 hertz. Option D.

The beat frequency produced when two tuning forks are sounded together is equal to the absolute value of the difference between their frequencies.

In this case, the beat frequency is:

|240 Hz - 246 Hz| = |-6 Hz| = 6 Hz

Therefore, the answer is (d) 6 hertz.

More on beat frequency can be found here: https://brainly.com/question/14705053

#SPJ1

The correct option is d has less speed and greater pressure.

How water flow?

Water flows through a combination of gravity and pressure. The movement of water is caused by the force of gravity, which causes water to flow downhill from higher to lower elevations. When water is pumped or forced through a system, pressure is added to the water, causing it to flow in the desired direction.

Water flows through a system of pipes, hoses, or channels, depending on the application. The velocity of water flow depends on various factors, including the diameter of the pipe or channel, the amount of water being pumped or released, and the pressure of the system.

In addition to gravity and pressure, other factors can affect the flow of water, including friction, viscosity, and turbulence. Understanding these factors is essential for designing efficient water systems that can deliver water where it is needed, such as in homes, farms, and cities.

Based on the diagram, the water at point 2 has a greater height than point 1, so it has a higher gravitational potential energy. Therefore, the water at point 2 must have a lower kinetic energy than point 1. Since the kinetic energy of water is related to its speed, we can conclude that the water at point 2 has less speed than point 1.

However, since the water is being pumped from a lower level to an upper level, it means that energy is being added to the system, which increases the pressure of the water. Therefore, the water at point 2 has a greater pressure than point 1.

Thus, the correct option is d) has less speed and greater pressure.

To know more about pressure visit:-

https://brainly.com/question/24719118

#SPJ1

Answer:

Base on my experience the answer is True...

Answer: True

Explanation:

Answer:

The strength of the magnetic field created by current is:

B=μ₀I / 2 π R

where I is the current

R is the shortest distance to the wire

The constant μ₀ is 4π * 10^-7 T * m / s

Explanation:

Answer:

Explanation:

The electric flux through a surface is given by the dot product of the electric field and the area vector of the surface:

Φ = E · A

where Φ is the electric flux, E is the electric field, and A is the area vector of the surface.

(a) If the surface lies in the yz plane, its area vector is in the x direction. Therefore, the area vector can be written as A = Ax i, where Ax is the magnitude of the area. The electric field is given as E = ai + bj. Therefore, the flux through the surface is:

Φ = E · A = (ai + bj) · (Ax i) = aAx

(b) If the surface lies in the xz plane, its area vector is in the y direction. Therefore, the area vector can be written as A = Ay j, where Ay is the magnitude of the area. The electric field is given as E = ai + bj. Therefore, the flux through the surface is:

Φ = E · A = (ai + bj) · (Ay j) = bAy

(c) If the surface lies in the xy plane, its area vector is in the z direction. Therefore, the area vector can be written as A = Az k, where Az is the magnitude of the area. The electric field is given as E = ai + bj. Therefore, the flux through the surface is:

Φ = E · A = (ai + bj) · (Az k) = 0

since the dot product of perpendicular vectors is zero.

The correct option is A, If the length of the crank was doubled, you could pull up the water with the same work, but less force.

The term "crank" can have various meanings depending on the context. In the context of machinery or engines, a crank is a mechanical device that converts rotational motion into linear motion or vice versa. It typically consists of a rod with a crankpin that connects to a piston or other reciprocating part.In a different context, the term "crank" can refer to a person who holds unconventional or extreme views and insists on expressing them in a forceful or annoying way.

Such a person may be described as a "crank" or "crankpot." The term can also refer to someone who is mentally unbalanced or eccentric. Furthermore, in the context of illegal drugs, "crank" is a slang term for methamphetamine, a highly addictive stimulant that can cause serious health problems and addiction. It is usually sold in crystalline form and can be smoked, snorted, or injected.

To learn more about Crank visit here:

brainly.com/question/15675129

#SPJ4

Complete Question: -

You are pulling water with a constant velocity from a well using a crank of lengthL . If the length of the crank was doubled, you could ...

A: pull up the water with the same work, but less force

B: pull up the pail with half the number of revolutions

C: exert double the torque while pulling up the pail with half the work

D: pull up the pail with half the work and half the force

E: pull up double the amount of water with the same work

F: exert four times the torque while pulling up the pail with the same work

Answer:

To calculate the amount of heat absorbed by the ice, we need to use the equation:

Q = m * ΔHf

where Q is the heat absorbed by the ice, m is the mass of the ice, and ΔHf is the heat of fusion of the ice, which is 334 J/g.

First, we need to calculate the amount of heat lost by the water:

Q = m * c * ΔT

where Q is the heat lost by the water, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the temperature change of the water.

We know that the mass of the water is 100 g, c is 4.18 J/g-°C, and ΔT is 5°C, so:

Q = 100 g * 4.18 J/g-°C * 5°C

Q = 2090 J

This means that the water lost 2090 J of heat, which was absorbed by the ice. Now we can calculate the amount of heat absorbed by the ice:

Q = m * ΔHf

We know that the mass of the ice is 10 g and the heat of fusion of the ice is 334 J/g, so:

Q = 10 g * 334 J/g

Q = 3340 J

Therefore, the amount of heat absorbed by the ice is 3340 J, which is equivalent to 3.34 kJ (kilojoules). The closest answer choice is 2.09 kJ, but that is not the correct answer.

When it is time to end a reflux, first remove the heat source in the flask and then turn off the heat. Wait until the system has cooled.

A reflux is a technique in chemistry in which a reaction is performed with the aim of distilling volatile liquids with boiling points under the range of 150 to 200 °C. Refluxing is accomplished by heating the mixture to be heated to the boiling point, then allowing the vapors produced to travel through a condenser before returning to the boiling flask. The setup of reflux equipment can be seen below: For a successful refluxing, it is important to end the process by following the correct steps, which are given below

Remove the heat source in the flask Turn off the heat Wait until the system has cooled. Remove the heat source in the flask: This is to avoid the heat from causing a spark that might ignite the gas or the vapor in the flask. Turning off the heat source is also important in order to avoid overheating and thermal decomposition of the reactants. Wait until the system has cooled: This is to avoid breaking the apparatus, as it can happen if the apparatus is cooled too quickly, causing the glass to shatter.

For more such questions on reflux

https://brainly.com/question/29483138

#SPJ11

Given the fact that the linear velocity of the ball is tangential to the circle then it is shown by vector B

When a ball flies out of a circular path, the direction of its tangential velocity is tangent to the point at which it leaves the circular path.

To visualize this, imagine a ball tied to a string and whirled around in a circle. As the ball is released, it will move away from the center of the circle in a straight line. At the moment it leaves the circular path, its velocity vector will be tangent to the circle, pointing in the direction of its motion.

If the ball is flying out of the circle in a clockwise direction, then its tangential velocity vector will point to the right. If it is flying out of the circle in a counterclockwise direction, then its tangential velocity vector will point to the left.

Learn more about tangential velocity:https://brainly.com/question/28738284

#SPJ1

Answer:

Explanation:

Adding other substances to the water besides salt, such as sugar or baking soda, would introduce confounding variables into the experiment. Other factors that could introduce confounding variables include using different volumes of water, using different amounts of salt, and heating the water at different rates. To minimize confounding variables, it is important to keep all variables constant except for the one being tested, which in this case is the effect of salt on the boiling point of water.

m=55.0 D=820 so were are looking for the velocity ? v= m\d V = 55.0*820 =45100 ...

The mass of the ball at a height of 3m above the ground with a potential energy of 120J can be calculated using the equation:

Mass = Potential Energy/Gravity * Height

Mass = 120J/(9.81m/s² * 3m)

Mass = 4.1 kg

Answer:

4 kg

Explanation:

Using,

Energy/ Work done = Force x Distance (Height)

E = F • s

But recall, that F = mg

Therefore,

E = m • g • s

Making mass (m), the subject of the formula

m = E / (g • s)

m = 120 / (10 • 3)

m = 120 / 30

m = 4 kg

But if g = 9.8 ms-¹

Then,

m = 120 / (9.8 • 3)

m = 120 / 29.4

m = 4.08 kg

Please mark brainliest.

Thanks

The potential difference across C4 can be found using the equation V = V4 - V3. Using the given values, V = 12.0V, C1 = 3.0 ?F, C2 = 2.0 ?F, C3 = 2.0 ?F, C4 = 1.0 ?F, and C5 = 4.0 ?F, we can solve for V4.

V4 = 12.0V + (3.0 ?F + 2.0 ?F + 2.0 ?F + 1.0 ?F) / (1.0 ?F + 4.0 ?F)
V4 = 12.0V + (8.0 ?F / 5.0 ?F)
V4 = 12.0V + 1.6V
V4 = 13.6V
Therefore, the potential difference across C4 is 13.6V - 12.0V = 1.6V.
The potential difference across C4 can be determined using the formula Q = CV. Where Q represents the charge stored in the capacitor, C represents capacitance, and V represents the potential difference across the capacitorTo determine the potential difference across C4, we can use the formula Q = CV. To determine Q, we need to determine the equivalent capacitance of the circuit.

The equivalent capacitance of capacitors in parallel is equal to the sum of their capacitance. The equivalent capacitance of capacitors in series is equal to the reciprocal of the sum of their reciprocals.C1, C2, and C3 are in series, and their equivalent capacitance is given by:C_eq1=1/((1/C1)+(1/C2)+(1/C3))=1/(1/3+1/2+1/2)=3/7 μF{C_eq1=1/((1/C1)+(1/C2)+(1/C3))=1/(1/3+1/2+1/2)=3/7μF}C_eq2 is the equivalent capacitance of C4 and C5 in parallel.C_eq2=C4+C5=1+4=5μF {C_eq2=C4+C5=1+4=5μF}

Now we can determine the equivalent capacitance of the entire circuit.C_eq=C_eq1+C_eq2=3/7+5=38/7μF{C_eq=C_eq1+C_eq2=3/7+5=38/7μF}Now, we can determine the charge stored in the circuit.Q=C_eqV=38/7*12= 65.14μC{Q=C_eqV=38/7*12=65.14μC}To determine the potential difference across C4, we can use the formula Q = CV.V=C4Q/C4= 65.14/1 = 65.14V{V=C4Q/C4=65.14/1=65.14V}Therefore, the potential difference across C4 is 65.14 V.

For more such questions on potential difference

https://brainly.com/question/3535763

#SPJ11

The statement that accurately describes the universal gravitational law is (option C) As the distance between any two bodies increases, the gravitational force between them decreases.

The universal gravitational law, also known as Newton's law of universal gravitation, is a fundamental law of physics that describes the gravitational attraction between two objects with mass. The law states that:

Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This means that the gravitational force between two objects decreases as the distance between them increases.

learn about  the universal gravitational law here https://brainly.com/question/11760568

#SPJ1

The work done by w is 22491 J.

The work done by Ti is 10728 J.

we have to determine the amount of work done by w. We can use the formula

W = Fd,

where W is the work done, F is the force applied, and d is the distance over which the force is applied.

The mass of the piano is given as 255 kg, and it is lowered exactly 9 m from the second-story window to the ground.

We can calculate the force required to lower the piano using the formula

F = mg,

where m is the mass of the piano and g is the acceleration due to gravity.

Therefore, [tex]F = 255 kg x 9.8 m/s^2 = 2499 N.[/tex]

Using the formula for work, we can calculate the work done by w as follows:

W = Fd = 2499 N x 9 m = 22491 J

we have to determine the amount of work done by Ti. We are given the magnitude of two forces, 1830 N and 1295 N, and the angle between them is 60°.

We can find the resultant force using the law of cosines, which states that

[tex]c^2 = a^2 + b^2 - 2ab cos(C),[/tex]

where c is the length of the side opposite the angle C and a and b are the lengths of the other two sides.

Therefore, [tex]c = sqrt(a^2 + b^2 - 2abcos(C)) = sqrt((1830 N)^2 + (1295 N)^2 - 2(1830 N)(1295 N)cos(60°)) = 2159 N.[/tex]

The angle between the resultant force and the horizontal is 45°, so we can calculate the work done by Ti using the formula

W = Fd cos(theta),

where theta is the angle between the force and the direction of motion.

Therefore, W = 2159 N x 7 m x cos(45°) = 10728 J.

for such more question on work done

https://brainly.com/question/25573309

#SPJ11

The increase in energy can come from increases in work.

Answer:

Explanation:

The heat required to completely boil a certain amount of water is given by the formula:

Q = mL

where Q is the heat required, m is the mass of the water, and L is the specific heat of vaporization of water, which is 2260 J/g.

Using the given values:

m = 1500 g

L = 2260 J/g

Substituting these values into the formula, we get:

Q = 1500 g × 2260 J/g

Q = 3,390,000 J

Therefore, 3,390,000 joules of heat transfer is required to completely boil 1500 g of water into a gas.

The  spring constant is 400 N/m. For the given question.

What is spring constant ?

The spring constant (k) is a physical property of a spring, which represents the stiffness of the spring. It is defined as the force required to stretch or compress a spring by a certain amount (x) divided by that amount of deformation:

k = F/x

where F is the applied force and x is the displacement or deformation of the spring from its equilibrium position. The spring constant has units of force per unit of length, such as newtons per meter (N/m) in the SI system of units. A higher spring constant means that more force is required to deform the spring by the same amount, and the spring is considered to be stiffer. Conversely, a lower spring constant means that less force is required to deform the spring by the same amount, and the spring is considered to be more flexible.

We can use the conservation of energy to find the spring constant.

Initially, the box has kinetic energy given by:

K₁= (1/2)mv₁²

= (1/2)(4.0 kg)(3.0 m/s)²

= 18 J

At maximum compression, all of the kinetic energy is stored as potential energy in the spring. The potential energy stored in a spring is given by:

U = (1/2)kx²

where k is the spring constant and x is the displacement from the equilibrium position. In this case, x is the compression of the spring, which is 0.30 m.

So, the potential energy stored in the spring is:

U = (1/2)kx²

= (1/2)k(0.30 m)²

= 0.045k J

Since energy is conserved, we can equate the initial kinetic energy to the potential energy stored in the spring:

K₁= U

18 J = 0.045k J

k = 400 N/m

Therefore, the spring constant is 400 N/m.

To know more about springs visit :-

https://brainly.com/question/14670501

#SPJ1

Answer:

Explanation:

(a) The kinetic energy of the object at time t1 is given by:

K1 = (1/2)mv1^2

where m = 8.51 kg and v1 = (5.21î − 2.81ĵ) m/s. Substituting these values, we get:

K1 = (1/2)(8.51 kg)(5.21î − 2.81ĵ)^2 = 149.12 J

Therefore, the kinetic energy of the object at time t1 is 149.12 J.

(b) The displacement of the object during the time interval Δt = t2 − t1 is given by:

Δr = r2 − r1 = (8.00î + 4.00ĵ) m - 0î - 0ĵ = 8.00î + 4.00ĵ

The average velocity of the object during this time interval is:

vavg = Δr/Δt = (8.00î + 4.00ĵ)/(2.00 s - 0 s) = 4.00î + 2.00ĵ m/s

Using the kinematic equation:

Δv = aΔt

where Δv = v2 - v1 is the change in velocity during the time interval, we can solve for the acceleration:

a = Δv/Δt = (v2 - v1)/Δt = [(8.00î + 4.00ĵ) m/s - (5.21î − 2.81ĵ) m/s]/(2.00 s - 0 s) = 1.395î + 3.415ĵ m/s^2

The force acting on the object during the time interval Δt = t2 − t1 is given by Newton's second law:

F = ma = (8.51 kg)(1.395î + 3.415ĵ) m/s^2 = 11.83î + 29.05ĵ N

Therefore, the force acting on the object during the time interval Δt is F = 11.83î + 29.05ĵ N.

(c) The work done on the object by the force during the time interval Δt = t2 − t1 is given by:

W = F · Δr

where · denotes the dot product. Substituting the values of F and Δr, we get:

W = (11.83î + 29.05ĵ) N · (8.00î + 4.00ĵ) m = 95.52 J

Therefore, the work done on the object by the force during the time interval Δt is -95.52 J.

Note: The negative sign indicates that the work done by the force is negative, i.e., the force acts in the opposite direction to the displacement of the object.

(d) The kinetic energy of the object at time t2 is given by:

K2 = (1/2)mv2^2

where m = 8.51 kg and v2 is the velocity of the object at time t2. To find v2, we can use the kinematic equation:

v2 = v1 + aΔt

where a is the acceleration of the object during the time interval and Δt = t2 − t1. Substituting the values, we get:

v2 = (5.21î − 2.81ĵ) m/s + (1.395î + 3.415ĵ) m/s^2(2.00 s)