an astronaut who is repairing the outside of her spaceship accidentally pushes away a 96.5 cm long steel rod, which flies off at 11.7 m/s, never to be seen again. as it happens, the rod is oriented perpendicularly to the magnetic field in that region of space. the rod is moving perpendicularly to its length as well as to the direction of the magnetic field. the magnetic field strength there is 8.03 mt. what is the magnitude of the emf, in millivolts, induced between the ends of the rod?

Answer:

4.18 joules of heat energy to raise a gallon of water by 1 degree celcius

Explanation:

If a person could not see violet but could see infrared light, the middle of their hypothetical visible spectrum would be green.

The middle of the hypothetical visible spectrum in this scenario would be red, as it is the middle of the traditional visible spectrum (ROYGBIV). Infrared light is not part of the visible spectrum, so it would not factor into the middle of the hypothetical visible spectrum.

The traditional visible spectrum is made up of red, orange, yellow, green, blue, indigo, and violet (ROYGBIV).
The visible spectrum of light is the part of the electromagnetic spectrum that is visible to the human eye. The spectrum ranges from red to violet, with red light having the longest wavelength and violet light having the shortest wavelength. If a person could not see violet, but they could see infrared light, the middle of their hypothetical visible spectrum would be green.Infrared light has a longer wavelength than visible light, which means it has a lower frequency. It is not visible to the human eye. The color green, on the other hand, has a wavelength of about 520-570 nm, which is in the middle of the visible spectrum.

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The magnitude of the resultant displacement is 334.4 m and the direction of the resultant displacement is 53.1° North of East.

To draw the displacement vector diagram, we start at point A and draw a vector from A to B, representing the athlete's displacement of 120 m South.

We then draw a vector from the end of the first vector (B) to the end of the second vector (C), representing the athlete's displacement of 200 m East. Finally, we draw a vector from the end of the second vector (C) to point D, representing the athlete's displacement of 270 m North. The diagram should form a closed triangle.

To find the resultant displacement of the athlete, we use the Pythagorean theorem and trigonometry. Let's call the displacement from A to B "vector AB," the displacement from B to C "vector BC," and the displacement from C to D "vector CD." The magnitude of the resultant displacement (R) is given by:

R = √(AB² + BC² + CD²)

R = √(120² + 200² + 270²) = 334.4 m (rounded to one decimal place)

To find the direction of the resultant displacement, we use trigonometry. We can find the angle between the resultant displacement and the North direction using the following formula:

θ = tan⁻¹(CD/BC)

Where CD is the Northward component of the displacement vectors and BC is the Eastward component of the displacement vectors.

θ = tan⁻¹(270/200) = 53.1° (rounded to one decimal place)

Therefore, the magnitude of the resultant displacement is 334.4 m and the direction of the resultant displacement is 53.1° North of East.

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The similarities  and the differences between the velocity and acceleration have been shown below.

Similarities:

Both are vectors, which means they have both magnitude and direction.

Both are measures of motion and are expressed in units of distance and time.

Both have the same units of distance per time, such as meters per second (m/s).

Differences:

Velocity measures the rate at which an object changes position over time, while acceleration measures the rate at which an object changes its velocity over time.

Velocity has direction and magnitude, while acceleration only has magnitude.

Velocity can be positive, negative or zero, depending on the direction of the object's motion, while acceleration can be positive or negative, depending on whether the object is speeding up or slowing down.

The SI unit of velocity is meters per second (m/s), while the SI unit of acceleration is meters per second squared (m/s^2).

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The scenario that correctly illustrates heat is: "a bathtub full of hot water gets even warmer in a cold room."

Heat is a form of energy that is transferred between two or more objects or systems due to a difference in temperature. Heat energy flows from hotter objects or systems to colder ones until thermal equilibrium is reached, meaning that the temperatures of the objects or systems become equal.

The  scenario of a bathtub full of hot water gets even warmer in a cold room  illustrates heat because heat always flows from hotter objects to colder objects until they reach thermal equilibrium. In this scenario, the hot water in the bathtub is at a higher temperature than the cold room, so heat flows from the water to the surrounding air, causing the water to become even warmer.

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Answer:

These are called Sankey Diagrams.

These summarise all the energy transfers taking place in a process. Sankey diagrams are drawn to scale - the thicker the line or arrow, the greater the amount of energy involved. Usually, the top line/arrow shows the amount of useful energy transferred from the total input and the one which curves shows the wasted/dissipated energy of the total input energy. It helps to show efficiency in this manner.

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The marble keeps going when the wagon stops because of the Inertia.

Inertia is defined as the property of an object to remain in a state of rest or uniform motion in a straight line unless acted upon by an external force. Newton's first law of motion, often known as the law of inertia, explains this.

Inertia is a fundamental concept in physics, and it applies to everything, from electrons to massive objects like planets and galaxies.

When a small toy wagon with a rolling marble is abruptly stopped by a rock under one of its wheels, the marble moves forward since it has a forward velocity due to its motion when the wagon was rolling, and because of inertia.

As a result, when the wagon comes to a halt, the marble is still moving forward, allowing it to keep going until it eventually reaches the front of the wagon.

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If the box slides forward another meter, the amount of work the person does is 1600 joules.

The work done by the person can be calculated as the product of the applied force and the distance moved by the box in the direction of the force:

W = Fd

where W is the work done, F is the applied force, and d is the distance moved in the direction of the force.

In this case, the person applies a force of 400 newtons and pushes the box for a distance of 4 meters. After he stops pushing, the box slides forward another meter, but since the person is not applying any force at that point, no work is done by him.

Therefore, the work done by the person is:

W = Fd = (400 N)(4 m) = 1600 J

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The wavelength of the wave will increase to 10 meters. Option B.

The speed of a wave (v) is equal to the product of its wavelength (λ) and frequency (f):

v = λ × f

Rearranging this equation to solve for frequency:

f = v/λ

In the first scenario, where the wave has a speed of 20 m/s and a wavelength of 5 meters, we can calculate the frequency as:

f1 = v/λ1 = 20/5 = 4 Hz

In the second scenario, the frequency is reduced by half, so the new frequency is:

f2 = f1/2 = 4/2 = 2 Hz

To find the new wavelength (λ2), we can rearrange the original equation to solve for wavelength:

λ = v/f

Substituting the values for the second scenario, we get:

λ2 = v/f2 = 20/2 = 10 meters

Therefore, if the same wave was created in the same medium with half of the original frequency, the wavelength would double and become 10 meters.

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Answer:

To solve this problem, we need to use the chain rule to find the derivative of the distance between the rocket and the observer with respect to time.

Let's call the distance between the rocket and the observer "d" and the height of the rocket "h". We know that the rocket is rising at a constant velocity of 30 miles per second, so its height can be expressed as:

h = 30t

where t is the time since the rocket started rising.

Using the Pythagorean theorem, we can express the distance between the rocket and the observer as:

d^2 = h^2 + 112^2

Differentiating both sides with respect to time, we get:

2d * (dd/dt) = 2h * (dh/dt)

where (dd/dt) is the rate of change of the distance between the rocket and the observer and (dh/dt) is the rate of change of the height of the rocket.

Solving for (dd/dt), we get:

(dd/dt) = (h/d) * (dh/dt)

Substituting the expressions for h and d, we get:

(dd/dt) = (30t) / sqrt((30t)^2 + 112^2)

When the rocket is at a height of 15 miles, we can find the corresponding value of t as follows:

15 miles = 15 * 5280 feet = 79200 feet

79200 feet / 5280 feet per second = 15 seconds

Substituting t = 15 seconds into the expression for (dd/dt), we get:

(dd/dt) = (30 * 15) / sqrt((30 * 15)^2 + 112^2)

= 450 / sqrt(22500 + 12544)

= 450 / sqrt(35044)

= 2.409

Therefore, the rate of change of the distance between the rocket and the observer when the rocket is at a height of 15 miles is approximately 2.409 miles per second.

5.45 cm to the right of the second lens is where the final image is created.

The image will get smaller and smaller as we move the object further and further away. The focal point will draw the image's location ever-closer. The light would be concentrated at the focal point if the object were extremely far away, such as the sun.

Using the thin lens equation, we have: 1/f = 1/di + 1/do

For the first lens, we have:

f1 = -6.00 cm (negative because the lens is diverging)

do1 = -10.0 cm (negative because the object is to the left of the lens)

Solving for di1, we get: 1/di1 = 1/f1 - 1/do1

di1 = -15.0 cm (negative because the image is to the left of the lens)

The first lens creates a virtual, upright image whose magnification is determined by: m1 = -di1/do1 = 1.50

As there are 4.50 cm between the first and second lenses, the location of the thing that the second lens sees is:

do2 = di1 - 4.50 cm = -19.5 cm

For the second lens, we have:

f2 = 6.00 cm (positive because the lens is converging)

do2 = -19.5 cm (negative because the object is to the left of the lens)

Solving for di2, we get:

1/di2 = 1/f2 - 1/do2

di2 = 5.45 cm

The final image is real and inverted, and its magnification is given by:

m = -di2/do2 = 0.279

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Answer:

The statement is True, waves propagate faster in a less dense medium if the stiffness is the same.

Waves can be defined as a disturbance that travels through space or matter, transferring energy from one point to another without transporting matter.

There are various kinds of waves, and they all exhibit similar properties.

Waves propagate faster in a less dense medium if the stiffness is the same .

Waves move at various speeds in different media. The speed of a wave is determined by the nature of the medium and the frequency and wavelength of the wave.

In general, waves propagate faster in less dense media than in denser ones.

This holds true if the stiffness is constant. For example, sound waves travel quicker in air than in water because air is less dense than water. In brief, waves propagate faster in a less dense medium if the stiffness is constant.

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The magnetic force acting on a current-carrying wire is determined by the direction of the external magnetic field and the direction of the current flowing through the wire.

If the length of the wire is oriented either parallel or antiparallel to the external magnetic field lines at the location of the wire, then the magnetic force acting on the wire will be zero.

This is because the magnetic field lines would be either in the same direction as the current or in the opposite direction, but the two forces would cancel each other out.

Additionally, if the current is carried by equal numbers of positive and negative charges that flow in opposite directions along the wire, then the magnetic force acting on the wire will also be zero. This is because the two opposite magnetic forces created by the two opposite charges will cancel each other out.

Lastly, if the magnetic field is generated by a second wire carrying a current in the opposite direction, then the two magnetic fields will cancel each other out and no force will be acting on the wire.

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The time taken by the rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of 3.45 rad/s² is 5.30 s. Hence, option (b) is the correct answer.

We are given the initial angular velocity, ω1 = 15.0 rad/s, final angular velocity, ω2 = 33.3 rad/s, and angular acceleration, α = 3.45 rad/s². We are supposed to find the time, t, taken by the rotating object to speed up from ω1 to ω2.

We can use the following kinematic equation to solve this problem:

ω2 = ω1 + αt

Rearranging this equation, we get:

t = (ω2 - ω1) / α

Substituting the given values in the above equation, we get:

t = (33.3 - 15.0) / 3.45

t = 5.30 s

So, the correct option is B.

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(a) The period for this motion is  1.48 × 10^-7 s

(b) The speed of the proton is   2.79 × 10^6 m/s

(c) The kinetic energy of the proton is  6.86 × 10^-14 J.

a) To determine the period of motion, we can use the formula for the magnetic force on a charged particle in a circular motion F = qvB, where F is the magnetic field, q is the charge of the particle which is the proton in this case, v is the speed of the particle and B is the magnitude of the magnetic field.

The magnetic force is always perpendicular to both the direction of motion of the particle and the direction of the magnetic field, since the proton is moving in a circle, there must be a net force acting toward the center of the circle to keep it in its orbit.The force is the centripetal force, which is provided by the magnetic force in this case:

F = ma = mv^2/r, where is and m is the radius of the circle and mass of the proton.Comparing those equations we get qvB = mv^2/r.Now solving for the period T = 2πr/v, we get: T = 2πm/(qB), substituting the values we get:T = 2π(1.67 × 10^-27 kg)/(1.6 × 10^-19 C)(0.700 T) = 1.48 × 10^-7 s

b) we use  the formula for the circumference of  a circle :C = 2πr, as the proton completed one full revolution in the period T, its v is equal to the circumference divided by period v = C/T = 2πr/T, after substituting the values we get : v = (2π)(0.650 m)/(1.48 × 10^-7 s) = 2.79 × 10^6 m/s

(c)  for calculating the kinetic energy, we use the formula:KE = 1/2 mv^2, plugging the know values we get:

KE = (1/2)(1.67 × 10^-27 kg)(2.79 × 10^6 m/s)^2 = 6.86 × 10^-14 J

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Tthe work done by the person on the block is 677.4 J.

The work done on an object by a force is defined as the product of the force and the displacement of the object in the direction of the force.

In this case, the force is constant, so we can use the formula:

work = force × displacement × cos θ

Where θ is the angle between the force and the displacement vectors. Since the force is applied horizontally and the displacement is also horizontal, θ = 0 and cos θ = 1. Therefore:

work = force × displacement

We need to determine the force that is parallel to the displacement, which is the force of friction opposing the motion. The force of friction is given by:

friction = μN

Where

Since the object is on a flat surface and is not accelerating vertically, the normal force is equal in magnitude to the weight of the object:

N = mg

Where

Therefore:

friction = μmg

Substituting the given values:

friction = (0.25)(5 kg)(9.81 m/s²) = 12.26 N

Since the force of friction opposes the motion, its direction is opposite to the force applied by the person. Therefore, the net force on the object is:

F_net = F - friction = 80 N - 12.26 N = 67.74 N

The displacement of the object is D = 10 m. Therefore, the work done on the object is:

work = F_net × D = (67.74 N)(10 m) = 677.4 J

Therefore, the work done by the person on the block is 677.4 J.

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An electric motor is an electrical machine that converts electrical energy into mechanical energy.

It works on the principle of electromagnetic induction, where a current-carrying conductor placed in a magnetic field produces a mechanical force.

Electric motors are commonly used in applications such as household appliances, power tools, and vehicle propulsion systems. They come in a variety of sizes and power ratings, and are used in industrial, automotive, and consumer applications. The most common type of electric motor is the induction motor, which utilizes a rotating magnetic field to produce torque.

Other types of electric motors include brushless DC motors, direct current motors, and synchronous motors. Electric motors are able to generate high torque even at low speeds, making them efficient for a variety of applications. Their versatility and easy maintenance make them popular for both consumer and industrial applications.

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It took 1.82 seconds for the rock to come to a stop when the force of -1.1 N was applied.

What is force?

We can use the formula:

force = mass x acceleration

To find the acceleration of the rock when the force is applied. Since we know the mass of the rock is 1kg, we can rearrange the formula to solve for acceleration:

acceleration = force / mass

We know that the force applied to the rock is -1.1 Newtons (N), since it is bringing the rock to a stop. Thus:

acceleration = -1.1 N / 1 kg = -1.1 m/s²

The negative sign indicates that the acceleration is in the opposite direction of the motion of the rock.

Next, we can use the formula:

velocity = initial velocity + acceleration x time

To find the time it takes for the rock to come to a stop. We know the initial velocity of the rock is 2 m/s and the final velocity is 0 m/s (since it comes to a stop). Thus:

0 m/s = 2 m/s + (-1.1 m/s²) x time

Solving for time:

time = (0 m/s - 2 m/s) / (-1.1 m/s²) = 1.82 seconds

Therefore, it took 1.82 seconds for the rock to come to a stop when the force of -1.1 N was applied.

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According to the given statement The acceleration of the car is 32 m/s²

Every procedure where velocity varies is referred to as acceleration. There are only two ways to accelerate: changing your speed , changing your direction, or changing both. This is due to the fact that velocity is both a speed or a direction.

The distance in feet must first be converted to metres:

212.0 ft = 64.6216 m

The kinematic equation can then be applied:

v² = u² + 2as

where,

The final velocity is v. (60 m/s),

u is the starting speed  (0 m/s),

a denotes the acceleration,

s denotes the distance. (64.6216 m).

Solving for a, we get:

a = (v² - u²) / (2s)

a = (60² - 0²) / (2 * 64.6216)

a ≈ 32 m/s²

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Answer:

50 minutes / 3000 seconds

Explanation:

Energy = power x time

time = energy / power

time = 900,000 / 300

time = 3000s

This can also be written as 50 minutes

The solution is D) 3.68 N (rounded to two significant figures).

Let's find the system's acceleration first. The difference between the force due to the 710 g mass and the force due to the 385 g mass is the net force acting on the system. Considering upward as favourable,

net force = (710 g)(9.8 m/s^2) - (385 g)(9.8 m/s^2) = 3.213 N

The 385 g mass produces an anticlockwise torque, but the 710 g mass produces a clockwise (negative) torque. (positive). When we apply the torque formula, = I, where, the pulley's moment of inertia and is its angular acceleration, we get

τ = (0.0125 kg ? m^2)(α)

The torque due to the 710 g mass is given by

τ = (0.710 kg)(9.8 m/s^2)(0.15 m) = 1.038 Nm

The torque due to the 385 g mass is given by

τ = (0.385 kg)(9.8 m/s^2)(0.15 m) = 0.567 N

2 / 2

First, let's find the acceleration of the system. The net force on the system is the difference between the force due to the 710 g mass and the force due to the 385 g mass. Taking upward as positive,

net force = (710 g)(9.8 m/s^2) - (385 g)(9.8 m/s^2) = 3.213 N

The torque due to the 710 g mass is clockwise (negative), while the torque due to the 385 g mass is counterclockwise (positive). Using the formula for torque, τ = Iα, where I is the moment of inertia of the pulley and α is its angular acceleration, we have

τ = (0.0125 kg ? m^2)(α)

The torque due to the 710 g mass is given by

τ = (0.710 kg)(9.8 m/s^2)(0.15 m) = 1.038 Nm

The torque due to the 385 g mass is given by

τ = (0.385 kg)(9.8 m/s^2)(0.15 m) = 0.567 Nm

Thus, the net torque is

net torque = (1.038 N ? m) - (0.567 N ? m) = 0.471 Nm

Using the formula for torque, τ = Fr, where F is the tension in the string and r is the radius of the pulley, we have

τ = (F)(0.15 m)

F = net torque / r = (0.471 Nm) / (0.15 m) = 3.14 N

T = (710 g)(9.8 m/s^2) - F = 6.96 N - 3.14 N = 3.82 N

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The measure of exterior angle is 130 degrees. Option D is correct.

Since triangle jkl is equilateral, all its interior angles have a measure of 60 degrees. The sum of an exterior angle and an interior angle of a triangle is always 180 degrees. Therefore, the measure of exterior angle 1 is the sum of angle j and angle k. Since angle j is 60 degrees, angle k must be 180 - 60 = 120 degrees, because the sum of angles in a triangle is always 180 degrees.

Therefore, the measure of exterior angle 1 is 60 + 120 = 180 degrees. However, an exterior angle is defined as the angle formed by a side of a triangle and the extension of an adjacent side. Therefore, the actual exterior angle 1 is the supplement of 180 degrees, which is 180 - 1 = 179 degrees. So the answer is (D) 130 degrees.

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The resulting vector has a magnitude of roughly  5.9 m.

When two or more vectors are added while adhering to the vector addition rules, the resultant vector is the resultant vector. When two vectors are supplied as R1 and R2, the resulting vector is given as R=R1+R2. This holds true not just for forces but also for every vector.

Let's first break down vector A into its constituent parts. It intersects the positive x-axis at an angle of 90° - 37° = 53°. The x-component of A is thus:

Ax = A cos(53°) = 2.5 cos(53°) ≈ 1.62 m

And the y-component of A is:

Ay = A sin(53°) = 2.5 sin(53°) ≈ 1.95 m

Let's now break down vector B into its component parts. It intersects the positive x-axis at a 20° angle. As a result, B's x-component is:

Bx = B cos(20°) = 3.5 cos(20°) ≈ 3.31 m

And the y-component of B is:

By = B sin(20°) = 3.5 sin(20°) ≈ 1.20 m

The vector sum of A and B is the resulting vector R. Using the Pythagorean theorem, we can determine the size of R:

|R| = sqrt(Rx² + Ry²)

where Rx and Ry are, respectively, R's x- and y-components. We combine the corresponding x- and y-components of A and B to yield Rx and Ry:

Rx = Ax + Bx ≈ 4.93 m

Ry = Ay + By ≈ 3.15 m

Now, we can find the magnitude of R:

|R| = sqrt(Rx² + Ry²) ≈ sqrt((4.93 m)² + (3.15 m)²) ≈ 5.85 m

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Answer:

To determine how much the height of the Washington Monument changes due to the increase in temperature, we can use the formula:

ΔL = αLΔT

where ΔL is the change in length, α is the coefficient of linear thermal expansion, L is the original length, and ΔT is the change in temperature.

In this case, we want to find the change in height, which is the same as the change in length along the vertical direction. Therefore, we can use the same formula to find the change in height:

Δh = αhΔT

where Δh is the change in height, α is the coefficient of linear thermal expansion, h is the original height, and ΔT is the change in temperature.

Substituting the given values, we get:

Δh = (2.5 × 10^-6/°C) × (170 m) × (35.0°C)

Δh ≈ 0.15 m

Therefore, the height of the Washington Monument increases by approximately 0.15 m when the temperature increases from 0°C to 35.0°C.

Tuning fork is 0.223 m below the point of release when waves with frequency of 480 Hz reaches the release point.

Number of waves that pass a fixed point in the unit time is known as frequency.

y = 1/2 * a * t²

y is distance traveled, a is acceleration, and t is time.

t = √(2y/a)

v = f *  λ

v is speed of sound, f is the frequency, and  λ is wavelength.

λ = v/f = 340 m/s / 480 Hz = 0.708 m

y = n *  λ/2

where n is the number of half-wavelengths traveled.

y = λ/2 = 0.354 m

t = √(2y/a) --> y = 1/2 * a * t² = 1/2 * 9.80 m/s² * (2y/9.80 m/s²) = y

t = √(2y/a) --> t = √(2y/a) = √(2 * 0.354 m / 9.80 m/s²) = 0.212 s

Therefore, tuning fork falls for 0.212 seconds before the sound wave reaches the release point. During that time, the tuning fork travels a distance of:

y = 1/2 * a * t² = 1/2 * 9.80 m/s² * (0.212 s)² = 0.223 m

Therefore, tuning fork is 0.223 m below the point of release when waves with a frequency of 480 Hz reach the release point.

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The minimum coefficient of friction needed for the button to stay on the turntable is 0.22, assuming the turntable is a flat disk and neglecting air resistance.

When an object is placed on a rotating turntable, there must be a sufficient coefficient of friction between the object and the turntable to prevent it from sliding off. In this scenario, the button is placed at the rim of a turntable with a radius of 15.0 cm and is rotating at 45.0 rpm (revolutions per minute).

To determine the minimum coefficient of friction required for the button to stay on the turntable, we can use the centripetal force equation. The centripetal force required to keep the button in circular motion is provided by the force of friction acting on the button. If the frictional force is less than the centripetal force required, the button will slip off the turntable.

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The nucleotides included in DNA allow for the joining of the two strands to form a double helix shape. The nucleotides are cytosine, adenine, guanine, and thiamine. The Strand of DNA unzips during transcription to enable the production of mRNA.

Why is knowledge of DNA's architecture as well as replication process crucial?

The plan for existence on this planet is included in it. enables humans and organisms to defend themselves against sickness and helps us grasp faults.

Biological physics makes use of physics' methods and instruments to comprehend the internal world of the mechanism present in living things on length different scales first from molecule to the macroscopic.

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Varbara have to react , before the volleyball hits the ground in 0.71 s.

Solve for the time that Barbara has to react before the volleyball hits the ground, we need to use the formula below:

t = (v_f - v_i)/g

Where:v_f = final velocity (when the ball hits the ground, its velocity is 0) v_i = initial velocity (given as 23 ft/s, upwards) g = acceleration due to gravity (constant at 32.2 ft/s²)

So we have:v_i = 23 ft/st = ?g = 32.2 ft/s²

Using these values:

0 = 23 - 32.2t

t = 23/32.2

t ≈ 0.71 seconds.

Rounding to two decimal places, Barbara has approximately 0.71 seconds to react before the volleyball hits the ground.

Therefore, the correct option is 0.71 seconds.

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Equations/Concepts Used:

Kinetic Energy => [tex]K=\frac{1}{2}mv^2[/tex]

Gravitational Potential Energy => [tex]U_{g}=mgy[/tex]

Mechanical Energy => [tex]E_{Mech.}=U+K[/tex]

Conservation of Energy => [tex]E_{0}=E_{f}[/tex]

At point 1

[tex]m=60 \ kg[/tex]

[tex]v=8 \ m/s[/tex]

PE ==> [tex]U_{g}=mgy \Rightarrow =(60)(9.8)(0) \Rightarrow U_{g}= 0 \ J[/tex]

KE ==> [tex]K=\frac{1}{2}mv^2 \Rightarrow =\frac{1}{2}(60)(8)^2 \Rightarrow K=1920 \ J[/tex]

ME==> [tex]E_{Mech.}=U+K \Rightarrow = 0+1920 \Rightarrow E_{Mech.}=1920 \ J[/tex]

At point 2

[tex]y=1 \ m[/tex]

Find the velocity of the skater at point 2 using conservation of energy.

We already found the total energy at point 1, which was 1920 Joules.

==> [tex]E_{1}=E_{2} \Rightarrow 1920=U_{g_{2}}+K_2 \Rightarrow 1920=(60)(9.8)(1)+\frac{1}{2}(60)v^2[/tex]

[tex]\Rightarrow 1920=588+30v^2 \Rightarrow 1332=30v^2 \Rightarrow v^2=44.4 \Rightarrow v=6.66 \ m/s[/tex]

From the equation above we answered the following,

[tex]v=6.66 \ m/s[/tex]

[tex]U_g=588 \ J[/tex]

We know the velocity at point 2, find KE then ME.

[tex]K=\frac{1}{2}(60)(6.66)^2 \Rightarrow K=1331 \ J[/tex]

[tex]E_{Mech.}=588+1331 \Rightarrow E_{Mech.}= 1919 \ J[/tex]

Notice how mechanical energy remains constant, this is because energy is a conserved quantity.

At point 3

Use conservation of energy again, using points 1 and 3.

==> [tex]E_1=E_3 \Rightarrow 1920=U_{g_3}+K_3 \Rightarrow 1920=(60)(9.8)h+0[/tex]

At point 3 the skaters velocity will go to 0 and all energy will be potential.

So, [tex]v=0 \ m/s[/tex]

[tex]\Rightarrow 1920=588h \Rightarrow h=3.27 \ m[/tex]

==> [tex]U_g=(60)(9.8)(3.27) \Rightarrow U_g=1923 \ J[/tex]

Answers:

Point 1, PE=0 J, KE=1920 J, ME=1920J

Point 2, PE=588 J, KE= 1331 J, ME= 1919 J, v=6.66 m/s

Point 3, PE=1923 J, KE=0 J ,ME= 1923 J, v=0 m/s, h=3.27 m