An archer shot a 0. 04 kg arrow at a target. The arrow accelerated at 7,000 m/s2 to reach a speed of 60. 0 m/s as it left the bow. How much force did the arrow have? ___N

During a fusion reaction, two statements that describe what happens to the nuclei of atoms are A small amount of mass in the nuclei that combine is converted to energy and Nuclei with small masses combine to form nuclei with larger masses.​ The correct option is B and D.

A small amount of mass in the nuclei that combine is converted to energy. During the fusion reaction, when the smaller nuclei combine, a small amount of mass is converted into a significant amount of energy, as described by Einstein's famous equation E=mc². This energy release is what makes fusion reactions so powerful and a potential source of clean energy.

Nuclei with small masses combine to form nuclei with larger masses. In a fusion reaction, lighter nuclei, typically isotopes of hydrogen like deuterium and tritium, combine under high pressure and temperature to form larger nuclei, such as helium. This process is what powers the Sun and other stars, as they fuse hydrogen into helium, releasing energy in the form of light and heat. The correct option is B and D.

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Complete question:

Which two statements describe what happens to the nuclei of atoms during a fusion reaction?

A. Large nuclei break apart into two or more smaller nuclei.

B. A small amount of mass in the nuclei that combine is converted to energy.

C. Each nucleus formed has fewer protons than each original nucleus had.

D. Nuclei with small masses combine to form nuclei with larger masses.​

The acceleration due to gravity on the surface of the other planet is approximately 77.98 m/s².

To find the acceleration due to gravity on another planet, we can use the formula:

Weight = Mass × Acceleration due to gravity

On Earth, your weight is given as 742.32 N, and the acceleration due to gravity is 9.80 m/s².

We can rearrange the formula to solve for mass:

Mass = Weight / Acceleration due to gravity

So, on Earth, your mass would be:

Mass on Earth = 742.32 N / 9.80 m/s²

Mass on Earth = 75.63 kg

Now, let's consider the surface of another planet where your weight is given as 5900.91 N.

We'll use the same formula and solve for the acceleration due to gravity on that planet:

5900.91 N = Mass × Acceleration due to gravity on the other planet

Substituting the value of mass we calculated earlier:

5900.91 N = 75.63 kg × Acceleration due to gravity on the other planet

Now, we can solve for the acceleration due to gravity on the other planet:

Acceleration due to gravity on the other planet = 5900.91 N / 75.63 kg

Acceleration due to gravity on the other planet ≈ 77.98 m/s²

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The height of the cliff is approximately 490 meters (or about 1,607 feet).

To find the height of the cliff, we can use the kinematic equation:

[tex]h = 1/2 * g * t^2[/tex]

where h is the height of the cliff, g is the acceleration due to gravity (which is approximately 9.8 m/s²), and t is the time it takes for the rock to hit the ground.

In this case, we know that the time it takes for the rock to hit the ground is 10.0 seconds.

So we can plug in the values:

[tex]h = 1/2 * 9.8 m/s^2 * (10.0 s)^2[/tex]

h = 490 m

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We can observe total internal reflection when light travels from air to glass, but not from glass to water or from water to glass. This is because in those cases, the light is traveling from a higher refractive index medium to a lower one, and thus there is no opportunity for internal reflection.

Total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index, and the angle of incidence is greater than the critical angle. In this case, n_glass = 1.50 and n_water = 1.33.
a. From glass to water: Total internal reflection can occur as the light is moving from a higher refractive index (glass) to a lower refractive index (water).
b. From water to glass: Total internal reflection cannot occur as the light is moving from a lower refractive index (water) to a higher refractive index (glass).
c. From air to glass: Total internal reflection cannot occur as the light is moving from a lower refractive index (air) to a higher refractive index (glass).
Therefore, total internal reflection can be observed when light travels from glass to water (option a).

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The angular acceleration required to stop the rotation of the Earth over a period of 30 minutes would be equal to the final angular velocity divided by the time interval.

The Earth's rotation is an example of rotational motion, which is described by angular velocity and angular acceleration. Angular velocity is the rate of change of angular displacement with respect to time, and angular acceleration is the rate of change of angular velocity with respect to time.  

The final angular velocity would be zero, since the Earth would have stopped rotating, and the initial angular velocity can be calculated by dividing the circumference of the Earth (40,075 km) by the time period of 24 hours or 1,440 minutes, which gives a value of approximately 0.28 degrees per minute.

Therefore, the initial angular velocity would be (0.28 degrees/minute)(2pi radians/360 degrees) = 0.00489 radians/minute. Dividing this value by 30 minutes gives an angular acceleration of approximately 0.000163 radians/(minute²).

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The index of refraction of the unknown material is 1.5.

The index of refraction (n) of a medium is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in the medium (v):

n = c / v

In this case, the speed of light in the unknown material is given as 2.00 × [tex]10^8[/tex] m/s. The speed of light in a vacuum is approximately 3.00 × [tex]10^8[/tex] m/s. Substituting these values into the formula:

n = (3.00 × [tex]10^8[/tex] m/s) / (2.00 × [tex]10^8[/tex] m/s)

Simplifying the expression:

n = 1.5

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The total mass of the glass marbles is m = ρV = 2500 kg/m³ × 4.19×[tex]10^{-3}[/tex] m³ = 10.5 g. Susan's box is heavier than Robert's box because it contains more glass mass.

Assuming the density of the glass marbles is constant, the weight of each box will depend on the total mass of glass in the box.

The volume of the single glass marble is (4/3)πr³ = (4/3)π(0.05m)³ = 5.24×[tex]10^{-5}[/tex] m³. The volume of the box is 10 cm × 10 cm × 10 cm = [tex]10^{-3}[/tex] m³.

Therefore, only one glass marble can fit in the box, which has a total mass of m = ρV = 2500 kg/m³ × 5.24×[tex]10^{-5}[/tex] m³ = 0.13 g.

The volume of 1,000 glass marbles is 1000 × (4/3)π(0.01m)³ = 4.19×[tex]10^{-3}[/tex] m³. Therefore, the total mass of the glass marbles is m = ρV = 2500 kg/m³ × 4.19×[tex]10^{-3}[/tex] m³ = 10.5 g.

Thus, Susan's box is heavier than Robert's box because it contains more glass mass.

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The vector (2, y) has an x-component with a length of 2.

A vector with an x-component of length 2 can be represented as:

Vector V = (2, y)

In this representation, the x-component of the vector is 2, and the y-component can have any value since it was not specified.

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The wavelength of the light used in the experiment is approximately 5.69 × [tex]10^{-7}[/tex] meters.

In the two-slit experiment, the distance between the slits is known as the "d" value. The distance from the slits to the screen is known as the "L" value.

The third-order bright fringe is at the center of the third bright band on the screen. Using the formula, d sinθ = mλ, where d is the distance between the slits, θ is the angle between the center of the third-order bright fringe and the horizontal, m is the order of the bright fringe, and λ is the wavelength of light.

We know that d = 0.0236 mm, θ = 2.09°, and m = 3. Rearranging the formula to solve for λ, we get: λ = d sinθ / m

Substituting the values, we get: λ = (0.0236 mm) sin(2.09°) / 3

Converting the distance to meters and the angle to radians, we get: λ = (2.36 × 10^-5 m) sin(0.0364 rad) / 3

Solving this equation gives us: λ = 5.69 × [tex]10^{-7}[/tex] m

Therefore, the wavelength of the light used in the experiment is approximately 5.69 × [tex]10^{-7}[/tex] meters.

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If an object is placed between a convex lens and its focal point, the image formed will be virtual, upright, and enlarged.

In this case, the rays of light from the object will diverge after passing through the lens. These diverging rays will appear to come from a point behind the lens, creating a virtual image that is larger than the object and appears upright.

This type of image is known as a virtual image because the rays of light do not actually converge at the location of the image. Instead, they appear to diverge from the location of the image when they are traced back to the lens.

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In the first scenario, heat transfers from hot water to a cold metal sphere until they reach thermal equilibrium. In the second scenario, heat and radiation occur from a hot rock to a metal container with no air until they reach thermal equilibrium.

For the scenario where a cold metal sphere is placed in hot water:

(a) The type of energy transfer is heat transfer.

(b) The transfer will occur from the hot water to the cold metal sphere, resulting in a decrease in the temperature of the water and an increase in the temperature of the sphere.

(c) The heat energy from the water will flow to the sphere until the two objects reach a state of thermal equilibrium, meaning they are at the same temperature. This occurs because heat naturally flows from hotter objects to cooler ones.

For the scenario where a hot rock is placed in a metal container with all the air removed:

(a) The type of energy transfer is both heat transfer and radiation.

(b) Heat transfer will occur from the hot rock to the metal container, while radiation will occur from the rock to the surrounding environment.

(c) The hot rock will lose heat energy to the metal container until they reach thermal equilibrium. Additionally, as the rock cools, it will emit electromagnetic radiation in the form of infrared waves. Because there is no air in the container, convection, another form of heat transfer, cannot occur.

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Answer: D 400 N/m

Explanation:

His results appear to be accurate and reasonable.

Yes, Chayse's results are reasonable. The frequency and wavelength of ultraviolet (UV) rays fall within the appropriate range of values for this type of electromagnetic radiation.

UV rays have higher frequencies and shorter wavelengths than visible light, and their frequencies typically range from about 7.5 × 10^14 Hz to 3 × 10^16 Hz, while their wavelengths range from about 10 nm to 400 nm. Chayse's measured frequency of 1.53 × 10^16 Hz and wavelength of 1.96 × 10^-8 m are consistent with these values for UV rays.

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As soil particle size decreases from silt to clay, the field capacity typically increases and the available water decreases.

This is because as particle size decreases, the pore spaces between particles also decrease, which in turn decreases the amount of water that can be held in the soil.

However, the smaller pore spaces also increase the surface area available for water to adhere to soil particles, resulting in a higher field capacity.

Field capacity is the amount of water held in the soil after excess water has drained away, and it is affected by factors such as soil texture, structure, and organic matter content.

Available water is the amount of water that plants can extract from the soil, and it is influenced by factors such as the depth of the plant roots and the water-holding capacity of the soil.

Overall, understanding the relationship between soil particle size and water retention is important for effective irrigation and soil management practices.

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You are approximately 1170.96 meters away from the location of the firework.

We know that the time difference between seeing the light and hearing the sound is 3.50 seconds. The speed of sound in air depends on the temperature, so we need to use the temperature information to calculate the speed of sound. The formula for the speed of sound in air at a given temperature is:

v = 331.3 + 0.606T

where v is the speed of sound in meters per second, and T is the temperature in degrees Celsius.

Substituting T = 5.00 degrees Celsius, we get:

v = 331.3 + 0.606 × 5.00

v = 334.56 m/s

Now we can calculate the distance to the firework using the formula:

d = v × t

where d is the distance, v is the speed of sound, and t is the time difference between seeing the light and hearing the sound.

Substituting v = 334.56 m/s and t = 3.50 s, we get:

d = 334.56 × 3.50

d = 1170.96 m

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ultra-violet is NOT considered an example of low Electromagnetic energy. Hence option C is correct.

Electromagnetic waves, which are synchronised oscillations of the electric and magnetic fields, are the traditional form of electromagnetic radiation. The electromagnetic spectrum is created at various wavelengths depending on the oscillation frequency. Electromagnetic waves move at the speed of light, typically abbreviated as c, in a vacuum. The oscillations of the two fields create a transverse wave in homogeneous, isotropic media when they are perpendicular to each other, perpendicular to the direction of energy and wave propagation, and perpendicular to each other. Either an electromagnetic wave's oscillation frequency or its wavelength can be used to describe its location within the electromagnetic spectrum. Because they come from different sources and have different effects on matter, electromagnetic waves of different frequencies are known by various names. These are listed in decreasing wavelength and increasing frequency order: sound waves, lower energy have lower frequency.

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Answer:

Red is your warm front.
Blue is your Cold front
Red and blue is your stationary front

Explanation:

The intensity of sound at 4 m from a 500 W speaker is found using the inverse square law of sound propagation. Therefore, the energy absorbed by the eardrum per minute is approximately 0.107 millijoules.

The intensity of sound is the power per unit area and is given by the formula I = P/A, where I is intensity, P is power and A is the surface area. Given that the speaker has a power of 500 W and the distance is 4 m, we can find the intensity of sound using the inverse square law of sound propagation.

[tex]I = P/(4\pi r^{2} )[/tex]

[tex]I = 500/(4\pi \times 4^{2} )[/tex]

I = 4.93 W/m²

Therefore, the intensity of sound at a distance of 4 m from the speaker is 4.93 W/m².

To calculate the energy absorbed by the eardrum per minute, we need to first convert the intensity to units of energy per time per area, which is given by the formula E = ItA, where E is energy, t is time, and A is the surface area.

The energy absorbed per minute is:

E = ItA

[tex]E = 4.93 W/m^{2} \times 60 s/min \times 600\;mm^{2} \times (1 m / 1000\;mm)^{2}[/tex]

E = 0.107 mJ/min

Therefore, the energy absorbed by the eardrum per minute is approximately 0.107 millijoules.

In summary, the intensity of sound at 4 m from a 500 W speaker is found using the inverse square law of sound propagation. The energy absorbed by the eardrum per minute is calculated by converting the intensity to units of energy per time per area and using the surface area of the ear.

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Brenda needs to walk a distance of 0.3865 meters to reach the next loud spot.

Brenda is walking along a line parallel to the speakers, the sound waves from each speaker will reach her in phase and interfere constructively, producing a loud spot. The distance between consecutive loud spots is equal to half the wavelength, so we can calculate this distance using the wavelength of the sound wave:

Distance between loud spots = 0.5 × wavelength

For an A note with a wavelength of 0.773 m, the distance between consecutive loud spots is:

Distance between loud spots = 0.5 × 0.773 m = 0.3865 m

Since Brenda is 24.0 m away from the speakers and the speakers are 12.0 m apart, she is equidistant from the two speakers and will hear the sound at its maximum intensity.

Therefore, she is currently at a loud spot. To find the next loud spot, she needs to walk a distance equal to the distance between consecutive loud spots:

Distance between loud spots = 0.3865 m

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To determine the minimum horizontal force that must be applied to the top of the container to cause tipping, we need to use the concept of torque. Torque is a force that causes rotation and is defined as the product of the force and the perpendicular distance from the point of rotation. In this case, the point of rotation is the edge of the container in contact with the floor.

Firstly, we need to find the weight of the container which is given by the mass times the acceleration due to gravity (9.8 m/s^2). Thus, the weight of the container is 593.88 N.

Next, we need to find the maximum force of static friction that the floor can exert on the container to prevent it from tipping. This is given by the coefficient of static friction (0.570) times the weight of the container (593.88 N). Thus, the maximum force of static friction is 338.73 N.

To cause tipping, a force must be applied to the container in such a way that it produces torque. This torque must overcome the torque produced by the force of static friction. The torque produced by the force of static friction is equal to the product of the maximum force of static friction and the distance from the point of rotation to the line of action of the force of static friction, which is half the height of the container (0.5 m).

Thus, the minimum horizontal force that must be applied to the top of the container to cause tipping is the force required to produce a torque equal to the torque produced by the force of static friction. This is given by the equation:

force x distance = maximum force of static friction x 0.5
Solving for force, we get:
force = (maximum force of static friction x 0.5) / distance

Substituting the values, we get:
force = (338.73 N x 0.5) / 0.6 m
force = 282.27 N

Therefore, the minimum horizontal force that must be applied to the top of the container to cause tipping is 282.27 N.

In conclusion, the minimum horizontal force required to tip the container depends on the coefficient of static friction and the distance between the point of rotation and the line of action of the force. In this case, the force required is 282.27 N, which must be applied at a distance of 0.6 m from the point of rotation.

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The distance between two consecutive staples in the finished part is approximately 0.28 meters or 28.46 centimeters.

Consider the relative velocity between the staple gun and the parts to be stapled.

The staple gun is rolling to the left at 1.5 m/s, while the parts are rolling to the right at 2.2 m/s. Therefore, the relative velocity between the staple gun and the parts is:

v_rel = v_parts - v_staple_gun = 2.2 m/s - (-1.5 m/s) = 3.7 m/s

The staple gun fires 13 staples per second, so the time between two consecutive staples is:

t = 1/13 s

During this time, the relative velocity between the staple gun and the parts causes the distance between the two consecutive staples in the finished part. Let's call this distance "d".

d = v_rel * t = 3.7 m/s * (1/13 s) = 0.2846 m

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The momentum of an object is defined as the product of its mass and velocity. The momentum of the larger truck is the same as the momentum of the smaller truck, even though the larger truck has more mass and less velocity.

Therefore, the momentum of an object can be calculated using the formula:

momentum = mass x velocity

In this problem, we have two trucks. Let's call the smaller truck A and the larger truck B. We are given that truck A has a velocity of 20 m/s. We are also told that truck B has twice the mass of truck A, but is traveling at half the speed. This means that the velocity of truck B is:

velocity of truck B = 1/2 x 20 m/s = 10 m/s

Using the formula for momentum, we can calculate the momentum of each truck:

momentum of truck A = mass of truck A x velocity of truck A

momentum of truck B = mass of truck B x velocity of truck B

Since truck B has twice the mass of truck A, we can substitute 2m for mB in the second equation:

momentum of truck A = mAx20 m/s = 20mA

momentum of truck B = (2m)x10 m/s = 20m

Comparing the two equations, we see that the momentum of truck B is equal to the momentum of truck A. Therefore, the momentum of the larger truck is the same as the momentum of the smaller truck, even though the larger truck has more mass and less velocity.

In summary, the momentum of an object is the product of its mass and velocity. The momentum of the larger truck is the same as the momentum of the smaller truck, even though the larger truck has more mass and less velocity.

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Answer:

used in the ideal gas law to describe the energy of a gas, and many more situations.

The y component of puck B's momentum after the collision is 0 g cm/s.

Momentum is the quantity of motion of a moving object, measured as a product of its mass and velocity. In physics, it is a conserved quantity, meaning that the total momentum of a closed system remains constant, regardless of the interactions within the system. Momentum can be transferred from one object to another, or between objects and their environment. Momentum is the driving force behind many physical phenomena, including collisions, friction, rocket propulsion, and the orbits of planets and stars.

[tex]p_A[/tex]  (before) = [tex]m_A[/tex] * [tex]v_A[/tex] = 165.0 g * 55.0 cm/s = 9077.5 g cm/s
[tex]v_A[/tex] (x) = [tex]v_A[/tex] * cos(27.0°) = 29.0 cm/s * cos(27.0 °) = 27.61 cm/s
[tex]v_A[/tex] (y) = [tex]v_A[/tex] * sin(27.0 °) = 29.0 cm/s * sin(27.0 °) = 14.26 cm/s
Using these components, we can calculate the momentum of puck A after the collision:
[tex]p_A[/tex]  (after) = [tex]m_A[/tex] * [tex]v_A[/tex] = 165.0 g * 27.61 cm/s = 4562.1 g cm/s
[tex]p_A[/tex] (before) + [tex]p_B[/tex] (before) = [tex]p_A[/tex]  (after) + [tex]p_B[/tex] (after)
9077.5 g cm/s + [tex]p_B[/tex] (before) = 4562.1 g cm/s + [tex]p_B[/tex] (after)
[tex]p_B[/tex] (before) = 4562.1 g cm/s - 4562.1 g cm/s = 0
Since the momentum of puck B before the collision was 0, its momentum after the collision must also be 0. Therefore, the y component of puck B's momentum after the collision is 0 g cm/s.

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The tension on the rope is 886 N. We can use Newton's second law to solve this problem:

ΣF = ma

where

ΣF is the net force acting on the hero,

m is the mass of the hero, and

a is the acceleration of the hero.

In this case, the hero is being pulled upward by a rope, so the net force acting on the hero is the tension in the rope minus the weight of the hero:

ΣF = T - mg

where

T is the tension in the rope and

g is the acceleration due to gravity.

Substituting the given values, we get:

T - mg = ma

T - (75.0 kg)(9.81 m/s²) = (75.0 kg)(2.00 m/s²)

Simplifying, we get:

T = (75.0 kg)(2.00 m/s² + 9.81 m/s²)

T = 75.0 kg × 11.81 m/s²

T = 886 N

Therefore, the tension on the rope is 886 N.

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Birdman needs to fly at a height of 49.05m to hit the bucket placed 118m away from the start of the field, assuming he flies at a speed of 37m/s.

To calculate the height at which Birdman needs to fly to hit the bucket, we need to use the equations of motion and consider the horizontal and vertical components separately.

Let's assume that Birdman is launching himself horizontally from the start of the field and needs to hit the bucket at a distance of 118m. We can use the horizontal distance, speed, and time to calculate the time it takes for Birdman to reach the bucket:

Horizontal distance = 118m

Horizontal speed = 37m/s

Time = Distance / Speed

Time = 118m / 37m/s

Time = 3.189s

Now that we know the time it takes for Birdman to reach the bucket horizontally, we can use the vertical component of motion to calculate the height at which he needs to fly.

We know that the only force acting on Birdman is gravity, and we can use the equation of motion for a vertically launched projectile to calculate the height:

Vertical distance = (Initial vertical velocity x Time) + (0.5 x Acceleration x Time^2)

Assuming that Birdman launches himself vertically with zero initial velocity, the equation simplifies to:

Vertical distance = 0.5 x Acceleration x Time^2

Where Acceleration is the acceleration due to gravity, which is approximately 9.81m/s^2.

Vertical distance = 0.5 x 9.81m/s^2 x (3.189s)^2

Vertical distance = 49.05m

Therefore, Birdman needs to fly at a height of 49.05m to hit the bucket placed 118m away from the start of the field, assuming he flies at a speed of 37m/s.

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Option (D) is correct.

The relation between potential energy(U(x)) and the associated force(F(x)) can be given as,

F(x) = (-)(dU/dx)

Therefore,

[tex]dU = (-) \int\limits^x_0{F(x)} .\, dx[/tex]

On putting, F(x) = (-)kx^3, and integrating, we have

[tex]U = \frac{1}{4}.k.x^{4}[/tex]

So, a possible energy function U for this force is, U = ((k.x^4)/4).

Thus, option (D) is correct.

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Because the planet is so far away from Earth, we will assume that it has no effect on Corus. The satellite radius will be 121943.5927 km.

The mass of the Corus is precisely equivalent to the mass of the Mars, we take it M. We see that the rocket makes a total rotation about the planet in only 15 days, so we expect that the rocket was spinning all over the world about a radius r. In this way, the satellite will move with at his range in the wake of detaching from the rocket.

We know T = 2πr/v

mv²/ r = GMm/r ²

where m = mass of satellite

r = GMm/ mv² = GM /v²

r = GMT²/ 4 π²r²   , putting the value of v

                               r³ = (GM / 4 π²r²) T²

                               r³ = ( GM / 4π² ) ¹/³ T²/³

G = 6.67 × 10 ⁻¹¹

M = 6.39 × 10 ²³ kg

T = 1296000

                                       r = 10258.621 × 11886.94

                                            r = 121943.5927 km

gravitational acceleration towards the core of corner = GM/ r²

                     a = 6.67 × 10 ⁻¹¹ ×6.39 ×10 ²³/ (121943592.7) ²

                      a = 2.89 × 10 ⁻³ m/s²

force between satellite and the Corus =mass of the satellite × acceleration of the satellite

iv) minimum speed = [GM/r(1+e)]¹/²  e is the eccentricity of the satellite

Gravitational speed increase is portrayed as the article getting a speed increase because of the power of gravity following up on it. It is measured in m/s2, and its symbol is g. Gravitational acceleration is a vector quantity with a magnitude and a direction.

The universe is governed by a force known as gravity, also referred to as gravitation. For any two items or particles having nonzero mass, the power of gravity will in general draw in them toward one another. Everything from subatomic particles to galaxies in a cluster is affected by gravity.

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A woman lifts up a laundry basket 1.5m and carries it 20m across the room in 15s.

Work is done on the laundry basket both in lifting the basket and in walking across the room during the entire 15s.

Lifting the basket:

The work done in lifting the basket is equal to the force applied multiplied by the distance lifted. The force applied is the weight of the basket, which is equal to the mass of the basket multiplied by the acceleration due to gravity (9.8 m/s^2).

Let's assume the mass of the basket is 'm'. The work done in lifting the basket vertically is given by:

Work_lift = force_lift × distance_lift = (m × 9.8) × 1.5

Carrying the basket horizontally:

When the woman carries the basket across the room, work is done against the force of friction between the basket and the floor. The work done is equal to the force of friction multiplied by the distance traveled horizontally.

The force of friction can be calculated using the coefficient of friction (μ) and the normal force (N). The normal force is equal to the weight of the basket since it is on a horizontal surface.

Let's assume the coefficient of friction between the basket and the floor is 'μ'. The work done in moving the basket horizontally is given by:

Work_horizontal = force_friction × distance_horizontal = (μ × m × 9.8) × 20

The total work done on the basket during the entire 15s is the sum of the work done in lifting and the work done horizontally:

Total work = Work_lift + Work_horizontal

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The height of the hill will be h' = (1/2)(v_i^2)/g

The spread halfway down is equal to half of the initial potential energy of the roller coaster at the top of the hill.

The spread halfway down is equal to half of the initial potential energy of the roller coaster at the top of the hill.

To solve this energy problem, we can utilize the principles of conservation of energy. The total mechanical energy of the roller coaster, consisting of its potential energy (PE) and kinetic energy (KE), remains constant throughout the ride.

A) To determine the height of the hill, we can equate the initial and final mechanical energies of the roller coaster at the top and bottom of the hill, respectively.

At the top of the hill:

Initial mechanical energy (E_i) = PE_i + KE_i = mgh + (1/2)mv_i^2

At the bottom of the hill:

Final mechanical energy (E_f) = PE_f + KE_f = mgh' + (1/2)mv_f^2

Since the roller coaster is at the top of the hill, its final kinetic energy (KE_f) is zero because it has come to a stop momentarily. Therefore, we have:

E_i = PE_i + KE_i = PE_f + KE_f = mgh + (1/2)mv_i^2 = mgh'

We are given that the roller coaster's initial velocity at the top of the hill (v_i) is 2.7 m/s, and its final velocity at the bottom (v_f) is 14 m/s.

Substituting these values into the equation, we get:

(1/2)mv_i^2 = mgh'

Simplifying and solving for h', the height of the hill, we have:

h' = (1/2)(v_i^2)/g

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

B) To find the spread halfway down, we need to calculate the difference in potential energy between the top and halfway down the hill.

The potential energy at the top of the hill (PE_i) is given by mgh, and the potential energy halfway down (PE_half) is given by (1/2)mgh.

The spread halfway down is the difference between these two potential energies:

Spread halfway down = PE_i - PE_half = mgh - (1/2)mgh = (1/2)mgh

Therefore, the spread halfway down is equal to half of the initial potential energy of the roller coaster at the top of the hill.

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