which of the following is largest? which of the following is largest? 1 light-year distance to the nearest star (other than our sun) size of pluto's orbit size of a typical galaxy

It is demonstrated that for each k, there exists a set of k positive integers whose difference has an even number of prime components, counted with multiplicity.

Let's construct the set of positive integers using the following algorithm:

Start with the first k prime numbers: 2, 3, 5, 7, 11, ..., p_k.

Define the i-th integer in the set as the product of the first i prime numbers: a_i = 235*...*p_i.

Claim: For any two integers in the set, their difference has an even number of prime factors, counted with multiplicity.

Proof:

Let a_i and a_j be two integers in the set such that i < j. We want to show that the difference a_j - a_i has an even number of prime factors, counted with multiplicity.

Notice that a_j - a_i can be written as:

a_j - a_i = 235*...p_i(235*...*p_j/2 - 1)

The first term on the right-hand side has i prime factors, and the second term has an odd number of prime factors. To see why the second term has an odd number of prime factors, notice that all the prime factors of 235*...p_j are also prime factors of 235...*p_i, so they cancel out in the difference.

a_j - a_i has i prime factors plus an odd number of prime factors, which is an even number of prime factors. This completes the proof.

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When a weak stimulus is applied in rapid succession, it will often reach threshold due to Temporal summation.

Stimulus is the effect that cause a change in the external or internal environment which brings physiological responses.

Here,

The weak stimulus applied is converted into a large signal so that it could reach the threshold.

Temporal summation is the phenomenon that could convert a series of weak stimuli at a specific frequency into a large signal and to achieve action potential.

Hence,

When a weak stimulus is applied in rapid succession, it will often reach threshold due to Temporal summation.

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b in terms of a that satisfies the given condition is, [tex]b = \ln(3e^a - 2)[/tex].

The equation of the curve is y = e^x.

The shaded region A is the area under the curve between x = 0 and x = a, so its area is given by,

[tex]A = \int_0^a e^x dx = e^a - 1[/tex]

The area labeled B is four times the area labeled A, so its area is given by,

B = 4A = 4(e^a - 1)

To express b in terms of a, find the value of b that satisfies,

[tex]\int_0^a e^x dx = 3(e^a - 1)[/tex]

Using the formula for the integral of e^x, we get:

[tex]e^b - e^a = 3(e^a - 1)[/tex]

Solving for b, we get:

[tex]b = \ln(3e^a - 2)[/tex]

So the area labeled B is [tex]4(e^a - 1)[/tex], and the value of b that satisfies the given condition is [tex]b = \ln(3e^a - 2)[/tex].

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The structure of the given paragraph about Ernest Cline is sequence writing.

The correct option is B.

Writing that organizes information or events in a certain order is known as sequence writing. In order for the reader to understand and follow the flow of events, sequence writing aims to convey information or events in a straightforward and logical manner.

For instance, the phases in a procedure are described sequentially in instructional writing to aid the reader in comprehending how to carry out a task. Events in narratives are presented chronologically to make it easier for the reader to follow the plot.

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Electric field change if a third object with a charge +q is placed at point P2 is the magnitude of the electric field will decrease by 25%.

Electric field is a vector quantity. It has a direction towards the negative charge and away from a positive charge.

Due to the 2 negatice charges, electric field E's verticle component will add up and horizontal component will cancel each other.

Electric field due to a charge particle = K q / r^2

K = constant

q = charge

r = distance of the point from the charge

Angle made by electric field = 45 degrees

Verticle component of the electric field due to two -q = E =2 *  K(-q) cos 45 / (V2 R)^2 \

= -V2 Kq/2R^2 = -0.71 Kq/R^2

New electric field after a charge +q is introduced = initial E + E due to +q

E due to +q = Kq / (2R)^2 = 0.25 Kq/R^2

Percentage Change in electric field = (Change in electric field / Initial electric field) * 100 = 0.25 *100 / -0.71 = -25%.

Therefore, the magnitude of the electric field will decrease by 25%.

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Answer:1.13m

Explanation:

Potential Energy= mgh

m= mass(kg)

g= gravitational field strength(9.81m/s^2)

h= height

PE=mgh

PE= 2J

therefore: 2J= 0.180kg x 9.81 m/s^2 x h

       h= 2/(1.764)

       h= 1.13m

The top of the 12 ft ladder that leans against the house, and the bottom of the ladder is 9 ft from the side of the house = 7.9 ft

The triangle formed by the ladder must be a right angle with a hypotenuse (Side opposite to right angle) of 12 feet since the side of the house must be vertical to the ground to form a right angle.

Let x represent how high the ladder's top is above the ground.

We have the following using the Pythagoras theorem of right triangles:

c² = a² + b²

12² = 9² + x²

x² = 144 - 81

= 63

Hence,

x = √63

= 7.9 ft

The question is incomplete, it should be:

A 12 - ft ladder leans against the side of a house. the bottom of the ladder is 9 ft from the side of the house. how high is the top of the ladder from the ground? if necessary, round your answer to the nearest tenth.

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Answer:

A

Explanation:

The track in relatively brief speed skating has straight stretches and semicircles with a 16-minute diameter. This 68 kg skate makes the maneuver at a steady 12 m/s while exerting 1224 N.

The pace at which a distance changes over time is referred to as speed. It has a dimension of time-distance. As a consequence, the fundamental unit of time as well as the basic measure of distances are combined to form the Special name of speed. Thus, the meter per second (m/s) is the Unit of measure of speed.

r = 8 m.

tangential velocity = 12 m/s.

angular speed is,

ω = v/r

= (12 m/s)/(8 m)

= 1.5 rad/s

centripetal acceleration,

a = r*ω²

= (8 m)*(1.5 rad/s)²

= 18 m/s².

force,

F = m*a

= (68 kg)*(18 m/s²)

= 1224 N

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Jake's velocity at the lowest point of the swing is 7 m/s.

Potential energy is the energy stored in any object or system due to the position or arrangement of its parts. It is, however, unaffected by factors outside of the object or system, such as air or height. Kinetic energy, on the other hand, is the energy of moving particles in an object or system.

To solve this problem, we need to use the conservation of energy principle, which states that the total energy in a system remains constant.

At the highest point, Jake has potential energy (due to his position above the ground), but no kinetic energy (since he is not moving).

As he swings down, his potential energy is converted to kinetic energy, and at the lowest point of the swing, he has the maximum kinetic energy and minimum potential energy. Then, as he swings back up, the process is reversed.

To calculate Jake's potential energy at the top of the swing, we use the formula:

PE = mgh

where m is the mass of Jake, g is the acceleration due to gravity (9.8 m/s²), and h is the height above the ground. Substituting in the values, we get:

PE = (20 kg)(9.8 m/s²)(2 m) = 392 J

This is Jake's potential energy at the top of the swing. At this point, he has no kinetic energy.

At the lowest point of the swing, Jake has converted all of his potential energy into kinetic energy. We can use the conservation of energy principle to find his kinetic energy at this point:

KE = PE

where KE is kinetic energy and PE is potential energy. Substituting in the values, we get:

KE = 392 J

This is Jake's kinetic energy at the lowest point of the swing.

To find Jake's velocity at the lowest point, we can use the formula for kinetic energy:

KE = (1/2)mv²

where v is velocity. Rearranging the formula to solve for v, we get:

v = √((2KE)/m)

Substituting in the values, we get:

v = √((2(392 J))/(20 kg)) = 7.0 m/s

This is Jake's velocity at the lowest point of the swing.

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The bolt's initial velocity, V, is zero when it is released from the container at a height of 94 meters.

A height of 0.26*94 = 24.4 m marks the final 26% of its descent.

By then, the bolt had dropped 94 - 24.4 = 69.56 m.

The last 26% of the bolt's fall takes 0.6122 seconds to complete.

(b) The bolt travels at a speed of 36.92 m/s for the final 26% of its descent (nearest hundredth).

(c) The bolt's speed is 42.94 m/s just before it touches the earth (nearest hundredth).

The change in displacement is measured in terms of velocity. The rate of velocity that changes is known termed accelerated. Due to the magnitude it contains, velocity is a vector quantity.

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(a) The final speed of the cooler is 5.77 m/s.

(b) In the presence of friction, the final speed of the cooler is 3.43 m/s.

The final speed of the cooler is calculated by applying work energy theorem as shown below;

Change in kinetic energy of the cooler = work done on the cooler

¹/₂mv² = fd

v² = (2fd) / m

v = √(2fd) / m

where;

The final speed of the cooler is calculated as;

v = √(2 x 30 x 2) / 3.6

v = 5.77 m/s

At the given coefficient of friction, the final speed is calculated as;

v = √(2Ffd) / m

where;

Ff = μW

where;

v = √(2μWd) / m

v = √(2 x 0.3 x 3.6 x 9.8 x 2) / 3.6

v = 3.43 m/s

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A student wants to launch a toy dart As time t = 0, release a ball that has been resting near the Earth's surface. Allow a cart to leave rest and move down a 40°40° inclination with regard to the ground.

The graphs which could represent the vertical component of the velocity as a function of time for the dart and the target immediately after the dart is launched and the target begins to fall is Dart: above x-axis, decreasing; Target: below x-axis, decreasing.

Velocity is the directional speed of a moving object as an indicator of the rate at which its position changes as perceived from a given frame of reference and measured by a particular standard of time. The idea of ​​speed is important in kinematics, the part of classical mechanics that explains the motion of things.

Velocity is a physical vector quantity that requires both magnitude and direction to be determined. Velocity is a scalar absolute value of speed, a consistently derived unit whose quantity is measured in SI (metric system) in meters per second.

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The above question is incomplete, the complete question is-

A student wants to launch a toy dart toward a target that hangs from a light string. At time t=0, the dart is launched with an initial speed v0 at an angle θ0 above the horizontal ground. At the instant the dart is launched, the string is cut such that the target begins to fall straight down. The positive horizontal direction is considered to be to the right, and the positive vertical direction is considered to be up.

Which of the following graphs could represent the vertical component of the velocity as a function of time for the dart and the target immediately after the dart is launched and the target begins to fall?

The variables in the kinematic equations for motion in one dimension with constant acceleration are time, initial position, initial velocity, initial acceleration, velocity, and position.

The following are accurate definitions for the variables in the kinematic equations for motion in one dimension with constant acceleration:

A. x(t) represents the particle's position at time t as determined by its initial position x. (0).

B. v(t) represents the particle's speed at time t as determined by its initial speed v. (0).

C. The particle's starting position, or the position at time t=0, is represented by the number x(0).

D. The initial velocity of the particle, or its speed at time t=0, is known as v(0).

E. an is the particle's constant acceleration.

It's vital to remember that the x and v subscripts denote the time at which the location or velocity is being measured. As an illustration, x(0) is the particle's position at time.

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All of the electrical energy held in the capacitor is released as heat when it is fully discharged through the resistor.

Two conductors that are separated from one another and positioned close to one another make up a capacitor, an electrical energy storage device.A straightforward illustration of one such storage device is a parallel-plate capacitor.

The following formula can be used to determine how much energy was lost inside the resistor:

= ½ × ×

A parallel-plate capacitor's capacitance is determined by:

= × 0 × /,

Since the capacitor is charged until the stored electric energy is , we can write:

½ × × =

Solving for gives:

= 2 × /

Substituting this expression for into the formula for, we get:

½ × × ² × ( − 1)

(B) The potential difference across the capacitor will rise as long as the charging battery is still attached and the dielectric is inserted because of the higher capacitance. The following formula can be used to compute the new potential difference:

The capacitor's newly accumulated electrical energy is:

½ × × × ²

= ½ × × ² × ( − 1)

= ½ × × ² × ( − 1)

= ½ × × (² − ²)

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Question:

A parallel-plate vacuum capacitor is connected to a battery and charged until the stored electric energy is . The battery is removed, and then a dielectric material with dielectric constant is inserted into the capacitor, filling the space between the plates. Finally, the capacitor is fully discharged through a resistor (a resistor is connected across the capacitor terminals).

A) Find , the energy dissipated in the resistor.

Express your answer in terms of and other given quantities.

B)Consider the same situation as in the previous part, except that the charging battery remains connected while the dielectric is inserted. (Part B figure) The battery is then disconnected and the capacitor discharged. For this situation, what is , the energy dissipated in the resistor?

Express your answer in terms of and other given quantities.

Average acceleration is a) 2.22 [tex]m/s^2[/tex] b) hot rod travels 78.4 m c) time is 20 seconds

(a) To find the average acceleration, we need to use the formula: average acceleration = change in velocity / time.

First, we convert 60 km/h to m/s:

60 km/h x 1000 m/km x 1 h/3600 s = 16.67 m/s

So the change in velocity is:

16.67 m/s - 0 m/s = 16.67 m/s

And the time is:

7.5 s

Therefore, the average acceleration is:

average acceleration = (16.67 m/s) / (7.5 s) = 2.22 m/s^2

(b) To find the distance traveled, we can use the formula: distance = initial velocity x time + 0.5 x acceleration x time^2.

The initial velocity is 0 m/s, so the formula becomes:

distance = 0.5 x acceleration x time^2

Plugging in the values, we get:

distance = 0.5 x 2.22 m/s^2 x (7.5 s)^2 = 78.4 m

Therefore, the hot rod will travel 78.4 meters during the 7.5 seconds.

(c) We can use the formula: distance = 0.5 x acceleration x time^2, and solve for time.

Plugging in the values, we get:

0.40 km x 1000 m/km = 400 m

acceleration = 2.22 m/s^2

So the formula becomes:

400 m = 0.5 x 2.22 m/s^2 x time^2

Solving for time, we get:

time = sqrt(400 m / (0.5 x 2.22 m/s^2)) = 20 s

Therefore, it would take 20 seconds for the hot rod to travel 0.40 km with constant acceleration of 2.22 m/s^2.

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The energy of each photon in electron volts is, 2.07.

Stopping potential is the minimum negative voltage applied to the anode to stop the photocurrent. The maximum kinetic energy of the electrons equal the stopping voltage, when measured in electron volt.

Stopping potential = Energy of light wave falling - energy of each photon

Energy of light wave falling = [tex]\dfrac{hc}{\lambda}[/tex]

Energy of light wave falling = [tex]\dfrac{hc}{\lambda}[/tex]

Energy in electron volts for a wave of wavelength 520 nm is, 2.37 eV.

Now,

Stopping potential = Energy of light wave falling - energy of each photon

0.3V = 2.37 - energy of each photon

Energy of each photon = 2.07

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a) Travelled 20.57 meters before coming to a stop.

b) When travelled half the distance, the speed is 2.6 m/s, which is less than 6 m/s.

Speed is a measure of how fast an object is moving, without regard to the direction of its motion. It is defined as the distance travelled by an object per unit of time. Speed is a scalar quantity, which means it only has magnitude (i.e., numerical value) and not direction. The standard unit of speed is meters per second (m/s) in the International System of Units (SI), although other units such as kilometres per hour (km/h) and miles per hour (mph) are also commonly used. It can be calculated as the ratio of distance traveled by an object to the time taken to cover that distance.

For example, if a car travels 100 meters in 10 seconds, its speed can be calculated as 100 meters divided by 10 seconds, which equals 10 meters per second. Speed can also be constant, if the object is moving at a constant rate without changing its speed, or variable, if the object's speed changes over time.

(a) To determine how far travelled before stopping, use the kinematic equation:

[tex]v^2 = u^2 + 2as[/tex]

where,

v = final velocity (0 m/s, since you come to a stop)

u = initial velocity (12 m/s)

a = acceleration (-3.5 m/s^2, since decelerating)

s = distance traveled (unknown)

Rearranging the equation to solve for s, we get:

[tex]s = (v^2 - u^2) / (2a)[/tex]

Substituting in the known values, we get:

[tex]s = (0 - (12 m/s)^2) / (2*(-3.5 m/s^2)) = 20.57 m[/tex]

Therefore, travelled 20.57 meters before coming to a stop.

(b) To determine speed, travelled half the distance, use the kinematic equation:

[tex]s = ut + 0.5at^2[/tex]

where,

t = time elapsed (unknown, but  use the fact that have travelled half the distance)

s = distance traveled (half of 20.57 m = 10.285 m)

u = initial velocity (12 m/s)

a = acceleration (-3.5 m/s^2)

Substituting the known values,

[tex]10.285 m = (12 m/s)t + 0.5(-3.5 m/s^2)t^2[/tex]

Simplifying and rearranging, a quadratic equation is obtained:

[tex]1.75t^2 - 12t + 10.285 = 0[/tex]

Solving  t using the quadratic formula,

t = 2.52 s (rounded to two decimal places)

Now, to find your speed at this point, we can use the kinematic equation:

[tex]v = u + at[/tex]

Substituting the known values,

[tex]v = 12 m/s + (-3.5 m/s^2)(2.52 s) = 2.6 m/s[/tex]

Therefore, when travelled half the distance, the speed is 2.6 m/s, which is less than 6 m/s.

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Answer:

ETo calculate the duration of the hammer's collision with the nail, we need to find the time it takes for the hammer to come to a stop and then reverse direction.

The hammer's initial momentum before the collision is given by its mass and velocity:

p_i = m * v_i = 1.2 kg * 15 m/s = 18 kg m/s

The hammer's final velocity after the collision can be found using the law of conservation of momentum, which states that the total momentum of an isolated system remains constant. We can write this as:

p_f = m * v_f = m * (-v_i * 0.6) = m * (-15 m/s * 0.6) = 9 kg m/s

The force experienced by the nail during the collision can be calculated as:

F = ma = (m * (v_f - v_i)) / t = (m * (9 kg m/s - 18 kg m/s)) / t

Since we know the force experienced by the nail is 9 000 N, we can use this to solve for t:

9 000 N = (1.2 kg * (9 kg m/s - 18 kg m/s)) / t

t = (1.2 kg * (9 kg m/s - 18 kg m/s)) / 9000 N

t = 0.00133 seconds

So the hammer is in contact with the nail for approximately 0.00133 seconds.

variable:

m: mass of the hammer, 1.2 kg

v_i: initial velocity of the hammer before the collision, 15 m/s

v_f: final velocity of the hammer after the collision, -9 m/s (negative sign indicates the direction of motion is opposite to the initial velocity)

t: duration of the hammer's collision with the nail, approximately 0

Adam lifts a book from the floor, carries it across the room, and places it on a high shelf, he does work only while he carries the book across the room, which is in option B, as work = force x distance.

According to the equation for work (W = FD), work is only done when a force is applied to an object and it moves some distance as a result of that force, when Adam lifts the book from the floor, he applies force to it, but the book does not move any distance, so no work is done when the object doesn't move, but when he moved the book, work was done.

Hence, he does work only while he carries the book across the room, which is in option B, as he lifts a book from the floor and carries it across the room.

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The frequency and wave speed after the water was high in Toledo and low in Buffalo is 2.3 * 10^-5 Hz and 18 m/s

Period = 12 hours

= 43,200 seconds.

The number of cycles or vibrations that a body in periodic motion experiences during one unit of time, as well as the number of waves that pass a fixed point in a unit of time. After moving through a series of situations or positions and then returning to its initial position, a body in periodic motion is said to have experienced one cycle or one vibration.

Frequency= 1/T

= 1/43200

= 2.3 * 10^-5 Hz

The amount of space a wave covers in a certain amount of time, such as the number of metres it covers in a second, is known as its wave speed. Speed = Wavelength x Frequency is an equation that describes how wave speed relates to wave wavelength and wave frequency. When the wavelength and frequency are known, this equation can be used to determine the wave speed.

Wave speed

= wavelength*(1/period)

= wavelength*frequency

= 800,000 m * 2.3 * 10^-5 Hz

= 18 m/s

Therefore, wave speed of the water be 18 m/s.

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Answer:

C. The production of an electric current by a moving magnet

B. The production of a magnetic field by an electric current

What is electromagnetism?

The " real definition " of electromagnetism is the following:

Electromagnetism is the physical interaction among electric charges, magnetic moments, and the electromagnetic field.

There are different " types " of electromagnetic fields

The last thing that YOU should know is that electromagnetic waves are generally known as light and obey the laws of optics. Which means in similar terms they just obey the laws of optics.

Thus, the answers to your question is, C & B

The speed of the GPS satellite can be calculated using the following formula:

v = √(GM/r)

r = RE + h = 6.37 x 10^6 m + 20,200,000 m = 2.06 x 10^7 m

v = √(GM/r) = √((6.67 x 10^-11 Nm^2/kg^2)(5.97 x 10^24 kg)/(2.06 x 10^7 m)) ≈ 3,870 m/s

Therefore, the speed of the GPS satellite is approximately 3,870 m/s.

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Step-by-step Explanation

Given: Diameter (D) of the capillary tube =  m

Surface Tension (T) =

The angle of contact () =

Acceleration due to gravity (g) =

The temperature of water =

To Find: The height (h) of rising of water in the capillary tube

Solution:

Formula to find the height of the rise

The following expression is used to find the height (h) of rising of water in the capillary tube;

Where,  is the density of water, and  is the radius of the capillary tube.

Calculating the height of the rise in the capillary tube

Since the diameter of the capillary tube is , the radius of the tube will be;

And, at , the density of water is

Substituting all the required values in the above formula, we get;

Hence, the water will rise in a capillary tube to a height of

If the mass of an object is tripled while the force acting upon it is kept constant, the acceleration of the object will be reduced to one-third of its original value.

This can be explained using Newton's second law of motion, which states that the acceleration of an object is directly proportional to the force applied to it and inversely proportional to its mass.

In this scenario, the force is held constant, but the mass is increased by a factor of three. As a result, the acceleration will decrease by a factor of three, since the same force is now acting on a much heavier object. Therefore, the object will experience a lower acceleration than before, making it harder to move or change its speed or direction.

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The initial upward acceleration of the rocket is 15.75 m/s. The result is obtained using Newton's second law.

A rocket with a mass of 1.80 × 10⁴ kg is moving upward. The initial upward force produced by its engine (the thrust) is 4.60 × 10 N.

Find the initial upward acceleration without neglecting the gravitational force on the rocket!

We have

When the rocket moves upward, the acceleration can be found by Newton's second law equation.

F = ma

Without neglecting the gravitational force in the opposite direction, it will be

F - W = ma

F - mg = ma

With g = 9.8 m/s², the initial upward acceleration is

ma = F - mg

1.80 × 10⁴a = 4.60 × 10⁵ - 1.80 × 10⁴ × 9.8

1.80 × 10⁴a = 4.60 × 10⁵ - 1.764 × 10⁵

1.80 × 10⁴a = 2.836 × 10⁵

a =  1.575 × 10¹

a = 15.75 m/s

Hence, the rocket's initial upward acceleration is 15.75 m/s.

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The muzzle velocity (VM) of a ball can be found by using a ballistic pendulum.

The basic principle of a ballistic pendulum is that the momentum of the ball before it hits the pendulum equals the momentum of the pendulum and the ball immediately after the collision.

The equation for this is

[tex]m_1v_1 = (m_1 + m_2) v_2[/tex]

, where m₁ is the mass of the ball, v₁ is the initial velocity of the ball, m₂ is the mass of the pendulum, and v₂ is the velocity of the pendulum and the ball immediately after the collision.

Solving for v₁, we get

[tex]VM = v_1 = (m_1 + m_2) \dfrac{v_2}{m_1}[/tex]

By measuring the height to which the pendulum rises and the mass of the pendulum, we can determine the velocity of the ball.

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When the angle is 90 degrees up, the only force acting in a downward direction is gravitational force, with magnitude mg, where m is mass and g is gravitational acceleration.

m = 2.25 kg, g = 9.8 /s²

F = mg = 2.25 × 9.8

F = 22.05 N.

An item falling freely in a vacuum is said to accelerate gravitationally in physics. This is the constant acceleration brought on just by the gravitational pull.

By putting an object in a vacuum chamber and monitoring the object's speed as a function of time as it accelerates, one can calculate the gravitational acceleration. This is the approach that Galileo popularized.

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