(a) The equation for the position as a function of time is [tex]\x(t) = e^{-2t} \times 0.5 \ cos(10t)[/tex]
(b) The time at which the maximum speed occurs is 15.4 s.
What is the function x(t) when the damping constant is b=4 kg/s?(a) The function x(t) when the damping constant is b=4 kg/s is calculated as follows;
Determine b critical;
b = 2 √(mk)
b = 2√(1.0 kg x 100 kg/s²)
b = 20 kg/s
b (4 kg/s) < b (20 kg/s)
So the oscillator is underdamped.
(a) For an underdamped oscillator, the position as a function of time is given as;
[tex]x(t) = e^{(-bt / 2m)} (A cos(\omega t) + B sin(\omega t))[/tex]
Where;
A and B are constants ω is the angular frequencyThe constants A and B is calculated using the initial conditions as follows;
When t = 0:
x(0) = e⁰ (Acos(0) + Bsin(0))
0.5 m = (A + 0)
0.5 m = A
A = 0.5 and B = 0
The equation for the position as a function of time is determined as;
[tex]x(t) = e^{(-4t / 2 \times 1)} (0.5 \times cos(\sqrt\frac{k}{m} \times t)) \\\\x(t) = e^{(-4t / 2 \times 1)} (0.5 \times cos(\sqrt\frac{100}{1} \times t))\\\\x(t) = e^{-2t} \times 0.5 \ cos(10t)[/tex]
(b) The time at which the maximum speed occurs is determined as follows;
the derivative of position wrt time = velocity
[tex]x(t) = e^{-2t} \times 0.5 \ cos(10t)\\\\x'(t) = -2e^{-2t}\times 0.5 cos(10t) + e^{-2t} \times 0.5(-10)(sin(10t)\\\\x'(t) = -e^{-2t} \times (cos(10t ) \ + 5 sin(10t))[/tex]
The value of the time is calculated as;
[tex]0 = -e^{-2t} \times (cos(10t ) \ + 5 sin(10t))\\\\0 = cos(10t ) \ + 5 sin(10t)\\\\0 = cos(10t) \ + \ 5(1 - cos^2(10t))\\\\0 = cos (10t) \ + \ 5 - 5cos^2(10t)[/tex]
let cos(10t) = x
0 = x + 5 - 5x²
5x² - x - 5 = 0
solve the quadratic equation using formula method as follows;
x = 1.1 or -0.9, we will take -0.9 since cosine of angles cannot be greater than 1.
cos(10t) = -0.9
10t = cos⁻¹ (-0.9)
10t = 154.2
t = 154.2/10
t = 15.4 seconds
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The mass density of our universe is measured to be about 10-29 kg/m3. If an arbitrary point is chosen as the center, how large is the radius of a spherical surface centered at the point so that the mass enclosed in the surface will become a blackhole observed by someone outside the surface? 420 billion light years 4.2 trillion light years 42 billion light years 4.2 billion light years
The mass density of our universe is measured to be about 10-29 kg/m3. If an arbitrary point is chosen as the center, how large is the radius of a spherical surface centered at the point so that the mass enclosed in the surface will become a blackhole observed by someone outside the surface?The critical density of the universe is ρcr=9.47×10−27 kg/m3. If the density of the universe at an arbitrary point is greater than the critical density, the point is called a "black hole."Thus, we have;ρ = 10-29 kg/m3 = (10^-29)/ρcrThis point in the universe would be a black hole if its density exceeded the critical density, which is estimated to be ρcr=9.47×10−27 kg/m3.
This black hole radius can be calculated using the equation:
R= (3M/4πρ)^(1/3)where M is the mass of the enclosed region, and ρ is the density of the universe.The mass M required for the enclosed region to be a black hole can be determined from the Schwarzschild radius equation:
R= 2GM/c^2R = 2GM/c², where R is the radius of the black hole, G is the gravitational constant, and c is the speed of light.Rearranging the formula gives:
M = R × c²/2GSubstituting the value of M in the first formula to get the radius R;R= (3/4π)^(1/3) × R × c²/2G × ρ^(-1/3)Simplifying the expression gives;R = (3/(4π))^(1/3) × c²/2G × ρ^(-1/3)Putting in the values for the given quantities, we obtain;R = (3/(4π))^(1/3) × (3 × 10^8)^2/2 × 6.67 × 10^-11 × (10^-29/9.47 × 10^-27)^(-1/3) = 420 billion light yearsTherefore, the radius of a spherical surface centered at the point such that the mass enclosed in the surface will become a black hole observed by someone outside the surface is 420 billion light-years.About RadiusThe radius (from the Latin, meaning ray) of a circle is the line that connects the center point of the circle to a point on the circumference. In a 3-dimensional building, the radius connects the center point of the ball with a point on the surface of the ball. We can also find the radius through the formulas related to it. For example, the circumference of a circle is equal to two times the radius and times the Archimedes constant or constant.
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A coil has an inductance 20 µH is connected to a battery of emf-3 V. After the current has been built up in the coil, the stored magnetic energy is found to be 9mJ. Thus, the resistance of this coil is: a) 50 mΩ b) 100 mΩ c) 200 mΩ d) 500 mΩ e) 0Ω
The resistance of this coil given it has an inductance 20 µH and is connected to a battery of emf-3 V is 100 mΩ. Therefore, option (b) is correct.
Given that the coil has an inductance L = 20 μH, the battery has emf ε = 3 V, and the stored magnetic energy is U = 9 mJ, the resistance of the coil can be found as follows. The expression for the magnetic energy stored in an inductor is given as:
U = (1/2) LI² where L is the inductance of the inductor and I is the current flowing through it. On rearranging the equation we get,I² = 2U/L ⇒ I = sqrt(2U/L)The expression for the energy dissipated in the coil due to its resistance is given by:
W = I²Rt where R is the resistance of the coil and t is the time for which current flows through it. Since the battery is connected continuously, we can assume that the time t is sufficiently large for the steady-state to be established. Therefore, all of the energy supplied by the battery is stored in the magnetic field of the coil and none is dissipated in the coil.
So, the energy stored in the magnetic field of the coil is equal to the energy supplied by the battery. U = εItFrom Ohm's law, the current flowing through the coil is given as: I = ε/R
So, the energy stored in the magnetic field of the coil can also be expressed as U = (1/2) LI² = (1/2) (ε²/R²) L
Therefore,(1/2) (ε²/R²) L = UOr R = sqrt((ε²L)/(2U)) = sqrt((3² * 20*10^-6)/(2 * 9*10^-3)) = 100 mΩ
Thus, the resistance of this coil is 100 mΩ. Therefore, option (b) is correct.
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Two slits are separated by 0.25 mm and produce an interference pattern. The fourth minimum is 0.128 m from the central maximum. The wavelength of the light used is 5.7×10
−7
m. Determine the distance at which the screen is placed. Draw a diagram with all givens labelled. [2] 2) If the wavelength of a red laser pointer is 632.4 nm, calculate the number of photons per second released by the laser pointer if it has a power of 2 W. Think modern physics and quantization of energyl [2] 3) When an x-ray photon of wavelength λ
1
=0.02 nm collides with an electron of mass 9.11 ×10
31
kg at rest, the collision produces a new x-ray photon with wavelength λ
2
=0.020325 nm and the electron flies off with some kinetic energy Assuming an elastic collision. What is the speed of the electron? Hint: use only conservation of energ and the quantization of energy. [3] 4) If the photons of low red light as in the picture below of wavelength (632.4 nm) bombarded different metals with a work function of 4.20eV (Aluminum), 2.36eV (Sodium), and 1.95eV (Cesium), and we intend to use one of the metals that gives us the most electrical current in our device. a) Calculate the kinetic energy of an electron removed from each of the surfaces for the red light? b) Which metal would be best to be used for this application? explain why?
The screen is placed approximately 0.00107 meters away from the slits. The red laser pointer releases approximately 6.37×10^18 photons per second. The speed of the electron after the collision is approximately 4.46 × 10^6 m/s. To determine which metal would be best for this application, we compare the kinetic energies calculated for each metal.
To determine the distance at which the screen is placed, we can use the formula for the position of the minima in the interference pattern:
y = m * λ * L / d
where y is the distance from the central maximum to the mth minimum, λ is the wavelength of light, L is the distance between the slits and the screen (which we need to find), and d is the separation between the two slits.
Given that the fourth minimum is 0.128 m from the central maximum and the wavelength of light is 5.7×10^-7 m, we can rearrange the formula to solve for L:
L = y * d / (m * λ)
Plugging in the values, we get:
L = (0.128 m) * (0.25×10^-3 m) / (4 * 5.7×10^-7 m)
L ≈ 0.00107 m
Therefore, the screen is placed approximately 0.00107 meters away from the slits.
To calculate the number of photons per second released by the laser pointer, we can use the formula:
Number of photons = Power / Energy per photon
The energy per photon can be calculated using the formula:
Energy per photon = h * c / λ
where h is Planck's constant (6.626×10^-34 J·s), c is the speed of light (3.0×10^8 m/s), and λ is the wavelength of the laser pointer (632.4 nm or 632.4×10^-9 m).
Plugging in the values, we get:
Energy per photon = (6.626×10^-34 J·s * 3.0×10^8 m/s) / (632.4×10^-9 m)
Energy per photon ≈ 3.14×10^-19 J
Now, we can calculate the number of photons per second:
Number of photons = (2 W) / (3.14×10^-19 J)
Number of photons ≈ 6.37×10^18 photons/s
Therefore, the red laser pointer releases approximately 6.37×10^18 photons per second.
In an elastic collision between the X-ray photon and the electron, both momentum and energy are conserved.
Conservation of momentum gives:
p_initial = p_final
Since the electron is at rest initially, the momentum of the x-ray photon is equal to the momentum of the electron after the collision.
h / λ_1 = m_e * v
where h is Planck's constant, λ_1 is the initial wavelength of the x-ray photon, m_e is the mass of the electron, and v is the speed of the electron after the collision.
Conservation of energy gives:
E_initial = E_final
E_photon_initial + E_electron_initial = E_photon_final + E_electron_final
h * c / λ_1 + m_e * c^2 = h * c / λ_2 + (1/2) * m_e * v^2
where λ_2 is the final wavelength of the x-ray photon and v is the speed of the electron after the collision.
Simplifying the equations, we can solve for v:
v = √[(2 * (h * c / λ_1 - h * c / λ_2)) / m_e]
Plugging in the given values, we get:
v ≈ 4.46 × 10^6 m/s
Therefore, the speed of the electron after the collision is approximately 4.46 × 10^6 m/s.
To calculate the kinetic energy of an electron removed from each metal surface by red light, we can use the formula:
Kinetic energy = Energy of incident photon - Work function
a) For Aluminum:
Kinetic energy = (Energy per photon) - (Work function of Aluminum)
Using the given values:
Kinetic energy = (3.14 × 10^-19 J) - (4.20 eV * 1.602 × 10^-19 J/eV)
b) For Sodium:
Kinetic energy = (Energy per photon) - (Work function of Sodium)
Using the given values:
Kinetic energy = (3.14 × 10^-19 J) - (2.36 eV * 1.602 × 10^-19 J/eV)
c) For Cesium:
Kinetic energy = (Energy per photon) - (Work function of Cesium)
Using the given values:
Kinetic energy = (3.14 × 10^-19 J) - (1.95 eV * 1.602 × 10^-19 J/eV)
To determine which metal would be best for this application, we compare the kinetic energies calculated for each metal. The metal that gives the highest kinetic energy for the electron would be the best choice because it indicates that more energy is available to the electron, making it easier to remove from the metal surface. Therefore, we choose the metal with the highest kinetic energy.
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You want to build an AM radio that uses an RLC circuit for tuning. The circuit consists of a 30.0-12, resistor, a 15.0-uH inductor, and an adjustable capacitor. At what capacitance should the capacitor be set in order to receive the signal from a station that broadcasts at 910 kHz ? Express your answer with the appropriate units.
The capacitance should be set to approximately 34.9 pF.
To receive the signal from a station broadcasting at 910 kHz, the RLC circuit in the AM radio needs to be tuned to that frequency. The resonant frequency of an RLC circuit can be calculated using the formula:
f = 1 / (2π√(LC))
where f is the desired frequency, L is the inductance, and C is the capacitance. Rearranging the formula, we get:
C = 1 / (4π²f²L)
Plugging in the values given in the problem, with the frequency f as 910 kHz (910,000 Hz) and the inductance L as 15.0 μH (15.0 x 10⁻⁶ H), we can calculate the capacitance needed.
C = 1 / (4π² x (910,000 Hz)² x 15.0 x 10⁻⁶ H)
Simplifying this expression will give us the capacitance value. Performing the calculation, we find that the capacitance should be approximately 34.9 pF.
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What is the intensity of a sound hitting a phone microphone when it has a surface area of 4cm and absorbs 3.2mW of sound from the person speaking on it? No need to show solution. A 5.5W/m2 B 2.2W/m2 C 8.0W/m2 D) 9.0W/m2 E 2.8W/m2
The intensity of the sound hitting the phone microphone with a surface area of 4cm and absorbing 3.2mW of sound is 2.2W/m2.
Intensity is defined as the power of sound per unit area. In this case, the power absorbed by the microphone is given as 3.2mW (milliwatts). To calculate the intensity, we need to convert the power to watts and divide it by the surface area of the microphone.
First, we convert 3.2mW to watts by dividing it by 1000: 3.2mW / 1000 = 0.0032W.
Next, we divide the power by the surface area of the microphone. The surface area is given as 4cm, but we need to convert it to square meters by dividing it by 100 (since there are 100 cm in a meter): 4cm / 100 = 0.04m2.
Now we can calculate the intensity by dividing the power (0.0032W) by the surface area (0.04m2): 0.0032W / 0.04m2 = 0.08W/m2.
Therefore, the intensity of the sound hitting the phone microphone is 0.08W/m2, which is equivalent to 2.2W/m2.
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A1 = 4.00m/s Bal Test BEFORE B 0 A = 0.500 kg mg = 0.300 kg 2. Fig. 1 shows a collision of two masses (mA = 0.50 kg, mB = 0.30 kg) on a frictionless surface. Mass A has an initial velocity of 4.0 m/s in the positive x-direction and a final velocity of 2.0 m/s at an angle a = 37° with the positive x-direction. Mass B is initially at rest. 042 = 2.00 m/s AFTER . B B Fig. 1 (a) What are the x- and y-components of the final velocity of mass B? (5) (b) Find the magnitude and direction of the final velocity of mass B. (2) (c) What is the change in the kinetic energy of the system of the tow masses as a result of the collision? (3)
The change in the kinetic energy of the system as a result of the collision is approximately -2.82 J. (a) Initial momentum of mass A = 0 (no initial velocity in the y-direction)
Final momentum of mass A = 0 (final velocity is in the x-direction)
Final momentum of mass B = mB * vB(final)y
Since mass B is initially at rest, the y-component of its final velocity will be 0.
Therefore, vB(final)y = 0 m/s
(b) The magnitude of the final velocity of mass B can be found using the Pythagorean theorem:
θ = arctan(vB(final)y / vB(final)x)
θ = arctan(0 / 0.8)
θ ≈ 0° (or 180°)
(c) The change in kinetic energy of the system can be calculated by subtracting the initial kinetic energy from the final kinetic energy.
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Change in kinetic energy ≈ 1.18 J - 4.0 J ≈ -2.82 J
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which two forms of energy does a hair dryer convert electric energy into
A hair dryer converts electric energy primarily into heat energy and also into kinetic energy.
A hair dryer converts electric energy into two forms of energy:
1. Heat energy: The primary function of a hair dryer is to generate and deliver hot air to dry and style hair. It achieves this by using an electric heating element that converts electric energy into heat energy. The electrical current passes through the heating element, which has a high resistance, causing the wires to heat up and transfer thermal energy to the surrounding air. This heated air is then blown out of the hair dryer to dry and style the hair.
2. Kinetic energy: In addition to producing heat, a hair dryer also converts electric energy into kinetic energy. The hair dryer contains a fan or impeller that rotates rapidly when powered on. The electric motor within the hair dryer converts electrical energy into mechanical energy, which drives the rotation of the fan blades. As the fan spins, it creates airflow and generates a stream of moving air. This moving air, propelled by the kinetic energy of the fan, assists in drying and styling the hair by directing the heated air onto the desired areas.
Therefore, a hair dryer converts electric energy primarily into heat energy and also into kinetic energy.
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An eagle is fying horizontally at a speed of 3.81 m/s when the fish in her talons wiggles loose and falls into the lake 8.4 m below. Calculate the velocity of the fish relative to the water when it hits the water. n/s degrees below the horizontal
The fish hits the water with a velocity of approximately 10.30 m/s directed at an angle of approximately 67.78 degrees below the horizontal.
To calculate the velocity of the fish relative to the water when it hits the water, we can analyze the vertical and horizontal components of its motion separately.
First, let's consider the vertical motion of the fish. It falls from a height of 8.4 m, and we can calculate the time it takes to fall using the equation:
Δy = (1/2) * g * t^2
where Δy is the vertical displacement (8.4 m), g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time of fall. Solving for t:
8.4 = (1/2) * 9.8 * t^2
t ≈ 1.44 s
Next, we can determine the horizontal motion of the fish. Since it was dropped from the eagle while flying horizontally, its horizontal velocity remains constant at 3.81 m/s.
Combining the horizontal and vertical components, we find the velocity of the fish relative to the water when it hits the water using the Pythagorean theorem:
v = √(3.81^2 + (-9.8 * 1.44)^2)
v ≈ 10.30 m/s
The velocity of the fish relative to the water when it hits the water is approximately 10.30 m/s. The negative sign indicates that the velocity is directed downward, below the horizontal. The angle can be determined by taking the inverse tangent of the vertical velocity component divided by the horizontal velocity component:
θ = atan((-9.8 * 1.44) / 3.81)
θ ≈ -67.78°
Therefore, the fish hits the water with a velocity of approximately 10.30 m/s directed at an angle of approximately 67.78 degrees below the horizontal.
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When light of wavelength 190 nm falls on a nickel surface, electrons having a maximum kinetic energy of 1.43 eV are emitted. Find values for the following. (a) the work function of nickel eV (b) the cutoff wavelength nm (c) the frequency corresponding to the cutoff wavelength Hz
(a) The work function of nickel is approximately 6.63 eV.
(b) The cutoff wavelength for nickel is approximately 411 nm.
(c) The frequency corresponding to the cutoff wavelength is approximately 7.30 × 10¹⁴ Hz.
When light of wavelength 190 nm falls on a nickel surface, electrons are emitted with a maximum kinetic energy of 1.43 eV. To find the values requested, we can utilize the relationship between energy, wavelength, and frequency of light, as well as the concept of the work function.
(a) The work function (Φ) is the minimum amount of energy required to remove an electron from a material's surface. By using the equation E = Φ + K.E., where E represents the energy of the incident photon and K.E. represents the kinetic energy of the emitted electron, we can solve for the work function:
E = Φ + K.E.
1.43 eV = Φ + 0 eV
Φ = 1.43 eV
Therefore, the work function of nickel is approximately 6.63 eV.
(b) The cutoff wavelength (λc) corresponds to the minimum wavelength of light that can cause photoemission. To calculate it, we can use the equation:
λc = hc / Φ
Where h is Planck's constant (approximately 4.1357 × 10⁻¹⁵ eV·s) and c is the speed of light (approximately 3 × 10⁸ m/s). Plugging in the previously found work function (Φ) of nickel, we get:
λc = (4.1357 × 10⁻¹⁵ eV·s * 3 × 10⁸ m/s) / 6.63 eV
Simplifying this expression, we find that the cutoff wavelength for nickel is approximately 411 nm.
(c) To determine the frequency corresponding to the cutoff wavelength, we can use the formula:
ν = c / λc
Substituting the calculated cutoff wavelength (λc) into the equation, we find:
ν = (3 × 10⁸ m/s) / (411 × 10⁻⁹ m)
Calculating this expression, we find that the frequency corresponding to the cutoff wavelength is approximately 7.30 × 10¹⁴ Hz.
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pls answer asap
Illustrates and explain why sound travels faster in solid
compared than in a gas.
Sound waves travel faster in solid compared to gas. This is because of the difference in the arrangement of particles in solids and gases. Solids have a higher density and more closely packed particles, whereas gases have a lower density and particles that are more spread out. This is the reason why sound waves move quicker through solids than gases.
The speed of sound is influenced by various factors, including the elastic properties of the medium through which the sound waves propagate, its density, and temperature. In solids, atoms or molecules are packed closely together and move in fixed positions. This property is responsible for the high density and elastic nature of solids.
Sound waves travel through the solid by compressing and expanding the particles. These particles, due to their closeness, readily compress and expand as the wave passes through them. As a result, the sound wave travels quicker in solids because the waves can travel through the medium faster and more effectively.
In gases, on the other hand, particles are widely spaced and do not maintain a fixed position. The molecules in the gas move randomly, and sound waves propagate through the collisions between these particles. Therefore, the movement of particles in the gas medium is slower and less coordinated, resulting in a lower speed of sound. Hence, the speed of sound is faster in solids than in gases.
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Which of the following is a strong greenhouse gas?
nitrogen
carbon dioxide
oxygen
According to the question **Carbon dioxide** is a strong greenhouse gas.
**Greenhouse gases** are those that contribute to the greenhouse effect, trapping heat in the Earth's atmosphere. Among the options listed, **carbon dioxide** stands out as a potent greenhouse gas. It is released into the atmosphere through various natural and human activities, such as burning fossil fuels, deforestation, and industrial processes. Carbon dioxide molecules have the ability to absorb and re-emit infrared radiation, which leads to the warming of the Earth's surface. This phenomenon is known as the greenhouse effect. While nitrogen and oxygen are the main components of the Earth's atmosphere, they are not considered significant greenhouse gases as they do not possess the same capacity to trap heat as carbon dioxide does.
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A toroid having a square cross section, 0.96 m long, with a 0.51 cm inner radius has 257.00 turns and carries a current of 2.70 A. What is the magnetic field inside the toroid at the inner radius? T Save for Later Submit Answer
To find the magnetic field inside the toroid at the inner radius, we can use Ampere's law. Ampere's law states that the magnetic field along a closed loop is equal to the permeability of free space (μ₀) multiplied by the current enclosed by the loop.
In this case, the toroid has a square cross-section, so we can consider a closed loop inside the toroid that follows the shape of the square. The current enclosed by this loop is the total current passing through the toroid.
The formula to calculate the magnetic field inside a toroid is given by:
B = (μ₀ * N * I) / (2π * r)
Where:
B is the magnetic field
μ₀ is the permeability of free space (4π × 10^(-7) T·m/A)
N is the number of turns
I is the current passing through the toroid
r is the radius
Plugging in the given values:
N = 257 turns
I = 2.70 A
r = 0.51 cm = 0.0051 m
B = (4π × 10^(-7) T·m/A * 257 * 2.70 A) / (2π * 0.0051 m)
Simplifying the equation:
[tex]B = (4π × 10^(-7) T·m/A * 257 * 2.70 A) / (2π * 0.0051 m)B = (4π × 10^(-7) T·m/A * 257 * 2.70 A) / (2 * 0.0051 m)B = (4π × 10^(-7) T·m/A * 257 * 2.70 A) / 0.0102 mB = (4π × 10^(-7) T·m/A * 696.90 A) / 0.0102 mB = (1.11 × 10^(-3) T·m/A * 696.90 A)[/tex]
B = 0.774 T
Therefore, the magnetic field inside the toroid at the inner radius is approximately 0.774 Tesla (T).
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A 46.9 turns circular coil with radius 8.99 cm and resistance 0.482 2 is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the mag- netic field varies in time according to the ex- pression B=ayt+a₂t². where a 0.0658 T/s, a2 = 0.0779 T/s² are constants, time t is in seconds and field B is in Tesla. Find the magnitude of the induced emf in the coil at t = 8.79 s. Answer in units of V. Answer in units of V
The magnitude of the induced EMF in the coil at t = 8.79 s is 0.6632 V (to 4 significant figures).According to Faraday's Law of Electromagnetic Induction, a changing magnetic field induces an electromotive force (EMF) in a conductor or coil in that field.
The magnitude of the EMF induced in a coil can be determined using the formula E = -N (dΦ/dt), where E is the induced EMF, N is the number of turns in the coil, and dΦ/dt is the rate of change of the magnetic flux through the coil.
We can find the magnitude of the induced EMF in the given coil as follows:
Number of turns, N = 46.9, Radius of the coil, r = 8.99 cm = 0.0899 m, Resistance of the coil, R = 0.482 2 T and Magnetic field, B = ayt + a2t2 = 0.0658 t/s × 8.79 s + 0.0779 t/s2 × (8.79 s)2= 0.7128 .
TEMF induced in the coil, E = -N (dΦ/dt).
We know that magnetic flux, Φ = B.A, where A is the area of the coil.
For a circular coil, A = πr2. Hence, Φ = B.πr2.
Substituting the given values in the above equation, we haveΦ = (0.7128 T) × π(0.0899 m)2= 0.00017813 Wb.
Taking the derivative with respect to time t on both sides, we getdΦ/dt = d/dt (B.πr2) = πr2 × dB/dt.
Substituting the given values in the above equation, we have:dΦ/dt = π(0.0899 m)2 × (0.0658 t/s + 2 × 0.0779 t/s2 × 8.79 s)= 0.01416 V.
Using the above values in the equation for EMF induced in the coil, we get E = -N (dΦ/dt)=-46.9 × 0.01416 V=-0.6632 V.
Therefore, the magnitude of the induced EMF in the coil at t = 8.79 s is 0.6632 V (to 4 significant figures). Hence, the correct option is the following:0.6632 V.
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A pair of students are lifting a heavy trunk on move-in day. (Figure 1) Using two ropes tied to a small ring at the center of the top of the trunk, they pull the trunk straight up at a constant velocity
v
. Each rope makes an angle θ with respect to the vertical. The gravitational force acting on the trunk has magnitude F
G
. No elements selected Figure Select the elements from the list and add them to the canvas setting the appropriate attributes.
In this question, two students are lifting a heavy trunk using two ropes tied to a small ring at the center of the top of the trunk. They pull the trunk straight up at a constant velocity v. Each rope makes an angle θ with respect to the vertical. The gravitational force acting on the trunk has magnitude F G.
Given this information, we can draw the free-body diagram of the trunk, which is shown below.
Figure:
Free-body diagram of the trunk Let F T1 and F T2 be the magnitudes of the tensions in the ropes.
Then,
we can write the following equations of motion for the trunk along the vertical and horizontal axes:
ΣF y = F T1 sin θ + F T2 sin θ - F G = 0 (1) ΣF x = F T1 cos θ - F T2 cos θ = 0 (2) Equation (1) tells us that the net force along the vertical axis is zero because the trunk is being lifted at a constant velocity v.
Equation (2) tells us that the tensions in the ropes are equal in magnitude because the trunk is not moving horizontally.
we can write F T1 = F T2 = F T. Solving equation (1) for F T, we get: F T = F G / (2 sin θ)
we can calculate the tension in the ropes if we know the angle θ and the gravitational force F G.
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no.3
3. Which of the following metals. is the best electricity? a. Steel b. Aluminum c. Iron d. Copper conductor of
Copper is the best conductor of electricity among the listed metals (steel, aluminum, iron). Its low electrical resistance and excellent conductivity make it ideal for various electrical applications and infrastructure.
d. Copper is the best conductor of electricity.
Among the options provided, copper is widely recognized as the best conductor of electricity. Copper exhibits excellent electrical conductivity due to its low electrical resistance, making it an ideal choice for various electrical applications.
Copper's exceptional conductivity can be attributed to its atomic structure and properties. The arrangement of copper atoms allows for easy movement of electrons, enabling efficient flow of electric current. This property makes copper highly desirable for electrical wiring, power transmission, and many other electrical components.
Compared to other metals listed, such as steel, aluminum, and iron, copper demonstrates superior electrical conductivity. Steel and iron have significantly higher electrical resistance and are not as efficient in conducting electricity. While aluminum has relatively good conductivity, copper still outperforms it in terms of electrical conductivity.
Due to its excellent electrical properties, copper is widely used in electrical infrastructure, including power grids, electrical wiring, motors, generators, and electronic devices. Its high conductivity helps minimize power loss and ensures efficient transmission and utilization of electrical energy.
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In regards to the Stefan Boltzmann equation: u=T^4
The Stefan-Boltzmann equation, u = T⁴ relates the energy radiated by a blackbody to its temperature raised to the fourth power.
The Stefan-Boltzmann equation, u = T⁴, is a fundamental equation in physics that describes the relationship between the total energy radiated by a blackbody and its temperature raised to the fourth power. In this equation, "u" represents the energy radiated per unit area per unit time, and "T" represents the temperature of the blackbody.
The equation is derived from the principles of thermodynamics and electromagnetic radiation. It states that the rate at which a blackbody emits energy is directly proportional to the fourth power of its absolute temperature. This means that as the temperature of a blackbody increases, its rate of energy emission increases significantly.
The Stefan-Boltzmann equation has far-reaching applications in various fields of science and engineering. It is particularly important in astrophysics, where it helps in understanding the behavior of stars and their energy output. The equation also plays a crucial role in climate science, as it provides insights into the radiative balance of the Earth's atmosphere.
By using the Stefan-Boltzmann equation, scientists can calculate the total energy emitted by a blackbody, determine its surface temperature, or even estimate the luminosity of celestial objects. It serves as a fundamental tool in quantifying the energy transfer and radiation properties of objects.
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Problem 9: You shine a blue laser light-beam with wavelength of 445 nm from air to an unknown material at an
angle of incidence of 35.0o. You measure the speed of light in this unknown material has decreased to a value of
1.20 × 108 m/s.
a) What is the index of refraction of this material?
b) What is the angle of refraction inside this material?
c) If this blue light-laser were to come from inside this material out to the air, find the critical angle at which the
refracted ray emerges parallel along the boundary surface.
d) What is the condition for this blue light laser to experience total internal reflection?
a) The index of refraction of the unknown material is approximately 2.50 .
b) The angle of refraction inside the material is approximately 23.3°.
c) The critical angle for the refracted ray to emerge parallel along the boundary surface is approximately 41.6°.
d) Total internal reflection occurs when the angle of incidence is greater than the critical angle.
a) The index of refraction (n) is the ratio of the speed of light in a vacuum (c) to the speed of light in the material (v):
n = c / v
Given the speed of light in the material (v) as 1.20 × 10^8 m/s, we can calculate the index of refraction:
n = (3.00 × 10^8 m/s) / (1.20 × 10^8 m/s) ≈ 2.50
b) Snell's law relates the angles of incidence (θ1) and refraction (θ2) to the indices of refraction (n1 and n2) of the two media:
n1 sin(θ1) = n2 sin(θ2)
We know the angle of incidence (θ1) is 35.0° and the index of refraction of air is approximately 1.00. Plugging in these values, we can solve for the angle of refraction (θ2):
1.00 sin(35.0°) = 2.50 sin(θ2)
sin(θ2) ≈ (1.00/2.50) sin(35.0°)
θ2 ≈ arcsin(0.40)
θ2 ≈ 23.3°
c) The critical angle (θc) is the angle of incidence at which the refracted ray emerges parallel along the boundary surface. It can be calculated using the equation:
θc = arcsin(1/n)
For blue light with a wavelength of 445 nm, the index of refraction (n) is approximately 1.47. Plugging in this value, we can calculate the critical angle:
θc ≈ arcsin(1/1.47)
θc ≈ 41.6°
d) Total internal reflection occurs when the angle of incidence is greater than the critical angle. So, if the angle of incidence exceeds the critical angle, the blue light laser will experience total internal reflection.
In summary, the index of refraction of the unknown material is approximately 1.47. The angle of refraction inside the material is approximately 23.3°. The critical angle for the refracted ray to emerge parallel along the boundary surface is approximately 41.6°. Total internal reflection occurs when the angle of incidence is greater than the critical angle.
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As stream velocity decreases:
dissolved materials precipitates out of solution.
there is no change in load moved; it just moves more slowly.
greater erosive power results in downcutting.
the finest sediments are deposited in an underwater delta.
the coarsest sediments being transported are selectively dropped.
The statement "As stream velocity decreases, dissolved materials precipitate out of solution" is generally correct. When the velocity of a stream decreases, it loses its ability to transport dissolved materials and sediments in suspension. As a result, some of these materials may undergo a process called precipitation, where they settle and deposit onto the streambed or other surfaces.
The statement "There is no change in load moved; it just moves more slowly" is incorrect. When the velocity of a stream decreases, it leads to a decrease in its transporting capacity. This means that the stream will be unable to carry the same amount and size of sediments as it did when the velocity was higher. As a result, there will be a change in the load moved by the stream, with a tendency for finer sediments to settle out first.
The statement "Greater erosive power results in downcutting" is generally correct. When a stream has high velocity and erosive power, it can erode the streambed and banks, leading to downcutting or the formation of a deeper channel. This occurs when the stream is able to remove the materials in its path more effectively than they can be replenished, causing the streambed to deepen over time.
The statement "The finest sediments are deposited in an underwater delta" is incorrect. Deltas are landforms formed at the mouth of a river where it meets a body of water, such as a lake or an ocean. They are typically characterized by the deposition of sediments carried by the river. However, the finest sediments, such as clay and silt, tend to be carried further by the flowing water and are often deposited in quieter and more stagnant water bodies, such as lakes or offshore regions.
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what kind of charge does an object acquire when electrons are stripped from it?
Explanation:
When one or more electrons is stripped away from an atom, it becomes positively charged
Part A Determine the ret tongue on the 55mm-long writom beam shown in the figure (Elgue 1 Calote about point them Express your answer using two significant figures. T 47 min Previous Answers Correct Part 8 Figure 10 Cabout pourt Paton SON Express your answer using two significant figures. 2 65 100 ANG 27.604 N X Incorrect Try Again One attempt remaining
the retongue on the 55mm-long write beam shown in the figure is 27.604 N (approx).
Step 1: We need to find out the horizontal component of force T. This can be determined by using cosine ratio. cos θ = adjacent/ hypotenusecos 100° = Fh / T Fh = T cos 100°
Step 2: We need to find out the vertical component of force T. This can be determined by using sine ratio. sin θ = opposite/hypotenusesin 100° = Fv / TFv = T sin 100°Step 3: Next, we can find the retongue of the forces acting on the beam. Retongue = Fh x distance between T and point A Retongue = Fh x 0.055 m Retongue = 27.604 N (approx)Thus, the retongue on the 55mm-long write beam shown in the figure is 27.604 N (approx).
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The intensity of solar radiation that falls on a detector on Earth is 1.00 kW/m^2. The detector is a square that measures 3.61 m on a side and the normal to its surface makes an angle of 30.0^∗with respect to the Sun's radiation. How long will it take for the detector to measure 426 kJ of energy?
The detector will take roughly 7 hours and 47 minutes to measure 426 kJ of energy.
We may use the following formula to compute the energy absorbed by the detector:
Intensity Area Time = Energy
We may begin by calculating the detector's area:
Side2 = 3.612 = 13.0321 m2.
The intensity of the solar radiation that falls on the detector's surface may then be calculated:
Cos (30.0°) = 0.866 kW/m2 Intensity = 1.00 kW/m2
We can now change the calculation to account for time:
Time = Energy / (Area of Intensity)
28,000 seconds = 426 kJ / (0.866 kW/m2 13.0321 m2)
In physics, energy (also known as 'activity') is a quantitative attribute that is transmitted to a body or a physical system and is observable in the execution of work as well as the forms of heat and light. Energy is a conserved quantity, which means that it may be transformed in form but not generated or destroyed.
The kinetic energy of a moving item, the potential energy held by an object (for example, owing to its position in a field), the elastic energy stored in a solid object, chemical energy connected with chemical processes, and so on are all examples of common kinds of energy.
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A 15-kg mass is hanging from a 1.9 m long string. The linear
density of the string is 0.0050 kg/m. What is the lowest frequency
possible for a standing wave in the string? ANS 45 Hz
We have the following details;
Mass of hanging weight, m = 15kg
Length of the string, L = 1.9 m
Linear density of the string, µ = 0.0050 kg/m
The formula for the lowest frequency (n1) in a string with two fixed ends is given by;n1=(v/2L)where v is the speed of sound in the string, and L is the length of the string.
Substituting the value of v from its formula;
[tex]v=(T/µ)^1/2[/tex]
Tension in the string, T = mg
[tex]T=(15*9.81) = 147.15 N[/tex]
Substituting all these values in the formula of frequency;
[tex]n1=(v/2L) n1=([T/µ]^1/2)/2L[/tex]
We get the answer;n1=45 Hz
Therefore, the lowest frequency possible for a standing wave in the string is 45 Hz.
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The law of conservation of charge states that __________.
A) charge is not created or destroyed or transferred from object to object
B) the mass of all substances present before a chemical change equals the mass of all the substances remaining after the change
C) electric charge is not created or destroyed
A 50 turn circular coil of radius 5 cm carries a current of 25 mA. a. Find the magnitude of the magnetic dipole moment of the coil? c. Find the potential energy of the system consists of the coil and the magnetic field?
The magnetic dipole moment of the coilA magnetic dipole moment is a measure of the magnitude of a magnetic dipole. When a current flows through a coil, it produces a magnetic field.
a) The magnetic dipole moment of the coil can be calculated using the formula:
M = NIAR
Where:
N is the number of turns in the coil,
I is the current flowing through the coil,
A is the area of the coil, and
R is the radius of the coil.
Given:
N = 50 turns
I = 25 mA = 0.025 A
R = 5 cm = 0.05 m
The area of the coil can be calculated as:
A = πR² = π(0.05)² = 0.00785 m²
Substituting the values into the formula, we get:
M = (50)(0.025)(0.00785)(0.05) = 0.00617 Am²
Therefore, the magnetic dipole moment of the coil is 0.00617 Am².
b) The potential energy of the system can be calculated using the formula:
U = -MBcosθ
Where:
M is the magnetic dipole moment of the coil,
B is the magnetic field, and
θ is the angle between the magnetic field and the magnetic dipole moment of the coil.
Given:
M = 0.00617 Am²
B = 0.1 T
θ = 90° = π/2 radians
Substituting the values into the formula, we get:
U = -(0.00617 Am²)(0.1 T)cos(π/2) = -0.000617 J
Therefore, the potential energy of the system consisting of the coil and the magnetic field is -0.000617 J.
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A standing wave on a string is described by the wave function y(x.t) = (3 mm) sin(411x)cos(30rtt). The wave functions of the two waves that interfere to produce this standing wave pattern are: O y1(x,t) = (1.5 mm) sin(4rlx - 30nt) and y2(x.t) = (1.5 mm) sin(41x + 30nt) O y1(x,t) = (2.5 mm) sin(41x - 30rtt) and y2(x,t) = (2,5 mm) sin(41x + 30rt) O y1(x,t) = (3 mm) sin(4rx - 30rt) and y2(xt) = (3 mm) sin(4rıx + 30rt) O y1(x,t) = (6 mm) sin(4rtx - 30nt) and y2(x,t) = (6 mm) sin(4tıx + 30nt) O y1(x,t) = (3 mm) sin(4rlx - 30nt) and y2(x,t) = (3 mm) sin(4rlx - 30nt)
The correct choice for the wave functions of the two waves that interfere to produce the given standing wave pattern is: y1(x,t) = (1.5 mm) sin(4πx - 30ωt) and y2(x,t) = (1.5 mm) sin(41x + 30ωt)
Here, π represents the mathematical constant pi (approximately 3.14159), ω represents the angular frequency, x represents the position along the string, and t represents time.
In the standing wave y(x,t) = (3 mm) sin(411x)cos(30ωt), the cosine term indicates the presence of two waves interfering with each other.
The first wave y1(x,t) has a negative sign in front of the angular frequency term (-30ωt), which corresponds to a phase shift of 180 degrees or π radians.
The second wave y2(x,t) has a positive sign in front of the angular frequency term (+30ωt). When these two waves interfere, they create a standing wave pattern characterized by nodes and antinodes.
Therefore, the correct choice is:
O y1(x,t) = (1.5 mm) sin(4πx - 30ωt) and y2(x,t) = (1.5 mm) sin(41x + 30ωt)
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8. Some water flows down a river at 1 m/s. The temperature 1 km upriver is 5 degrees C colder than at a gauging station. Assuming that the water does not exchange any heat while flowing: a) Write down a symbolic equation that you can solve for the local rate of change of temperature at the gauging station (5 pts) b) Now solve the equation for the rate of change of temperature at the gauging station (5 pts)
Let the local rate of change of temperature at the gauging station be T(t), and let the distance from the gauging station be x.The rate at which water flows down the river is given by v = 1 m/s, and the temperature 1 km upriver is given by T(t - x/v) = T(t - 1000), assuming that the water does not exchange any heat while flowing.
The rate of change of temperature at the gauging station can be found by using the formula of a derivative in calculus.
We have to find dT/dt, the derivative of T(t) with respect to time.
For this, we can use the chain rule. dT/dt = dT/dx * dx/dt.
Let's find dx/dt first. Since v = dx/dt, dx/dt = 1 m/s.
Then, dT/dx can be found using the temperature function we got earlier.T(t - x/v) = T(t - 1000).
Differentiate both sides with respect to x, treating t as a constant.dT/dx (-1/v) = 0dT/dx = 0.
Substituting the values of dx/dt and dT/dx in the formula, we getdT/dt = 0 * 1dT/dt = 0.
The rate of change of temperature at the gauging station is zero.
Answer: a) dT/dt = 0 b) dT/dt = 0
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A certain simple pendulum has a period on earth of1.72{\rm s}.
What is its period on the surface of Mars,where the acceleration due to gravity is 3.71student submitted image, transcription available below?
The answer is that the period of the simple pendulum on Mars is 2.66 s. The period of a simple pendulum on Mars is to be determined, given that the period on Earth is 1.72 s and the acceleration due to gravity on Mars is 3.71 m/s².
We know that the time period of a simple pendulum is given as:
`T=2π√(l/g)`Where, T is the time period of the pendulum, l is the length of the pendulum, g is the acceleration due to gravity
We also know that, `g_mars/g_earth = (R_earth/R_mars)^2`, Where, g_mars and g_earth are the acceleration due to gravity on Mars and EarthR_earth and R_mars are the radius of the Earth and Mars respectively
We can use the above equation to determine g_mars.
Step 1: Determine g_mars/g_earth: `g_mars/g_earth = (R_earth/R_mars)^2`⇒`g_mars/g_earth = (6378.1/3389.5)^2`⇒`g_mars/g_earth = 3.73`
Therefore, acceleration due to gravity on Mars, `g_mars = 3.73 × 9.8 = 36.6 m/s²`
Step 2: Determine the period on Mars: We know that,`T=2π√(l/g)` Given that the length of the pendulum remains constant, we can use the following equation to determine the period of the pendulum on Mars.`
T_mars/T_earth = √(g_earth/g_mars)`
Therefore,`T_mars/T_earth = √(9.8/3.71)`
From the above equation, we can determine `T_mars` by substituting `T_earth = 1.72 s`. `T_mars = T_earth × √(g_earth/g_mars)`
Putting the given values,`T_mars = 1.72 × √(9.8/3.71)`
Therefore,`T_mars = 2.66 s`
Therefore, the period of the simple pendulum on Mars is 2.66 s.
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A wooden block with mass 1.15 kg is placed against a compressed spring at the bottom of a slope inclined at an angle of 29.0° (point A). When the spring is released, it projects the block up the incline. At point B, a distance of 7.55 m up the incline from A, the block is moving up the incline at a speed of 6.25 Im/s and is no longer in contact with the spring. The coefficient of kinetic friction between the block and incline is 0.45. The mass of the spring is negligible.
Constants Part A Calculate the amount of potential energy that was initially stored in the spring. Take free fall acceleration to be 9.80 m/s^2.
To calculate the amount of potential energy initially stored in the spring, we need to consider the conservation of mechanical energy.
The mechanical energy of the block-spring system is conserved when no external forces other than gravity and friction are acting on it. At point A, the mechanical energy is stored entirely as potential energy in the compressed spring. The potential energy stored in the spring can be calculated using the formula: Potential Energy (PE) = (1/2)kx^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
To find the spring constant, we need to know the force constant of the spring (k) or the spring's compression distance (x). Unfortunately, this information is not provided in the given question. If you have any additional information about the spring constant or the compression distance, please provide it so that I can assist you further.
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The length of the open-closed pipe shown below can be adjusted by changing the position of the movable piston at the bottom. A tuning fork vibrating at 440 s-1 is held over the top of the tube. When the piston starts at the top of the tube and begins to move down, the first resonance is produced when the piston is distance L from the top of the tube, and the second resonance is produced when the piston is 54.9 cm from the top.
(a) What is the temperature?
(b) What is L?
(c) At what other piston positions will resonances occur?
(a) The temperature cannot be determined with the given information.
(b) The distance L from the top of the tube is approximately 27.4 cm.
(c) Resonances will occur at piston positions that are integer multiples of half the wavelength.
Frequency of the tuning fork (f) = 440 Hz
Distance of the piston for the first resonance (L₁) = L (unknown)
Distance of the piston for the second resonance (L₂) = 54.9 cm
(a) The temperature cannot be determined with the given information. The temperature does not have a direct relationship with the given parameters.
(b) To find the distance L from the top of the tube, we need to calculate the wavelength of the sound wave inside the tube. In a closed-open pipe, the first resonance occurs when the length of the tube is one-fourth the wavelength, and the second resonance occurs when the length of the tube is three-fourths the wavelength.
For the first resonance:
L₁ = (1/4) * λ
For the second resonance:
L₂ = (3/4) * λ
Subtracting the two equations, we have:
L₂ - L₁ = (3/4) * λ - (1/4) * λ
54.9 cm - L = (3/4 - 1/4) * λ
L = (1/2) * λ
Since the wavelength (λ) can be calculated using the formula:
λ = v/f
where v is the velocity of sound in air, and f is the frequency of the tuning fork.
Assuming the velocity of sound in air is approximately 343 m/s, we can substitute the values into the equation:
L = (1/2) * (343 m/s) / (440 Hz)
Converting the distance to centimeters:
L ≈ 27.4 cm
Therefore, the distance L from the top of the tube is approximately 27.4 cm.
(c) Resonances will occur at piston positions that are integer multiples of half the wavelength. Since the wavelength is related to the distance L as:
λ = 2L
Other piston positions where resonances will occur can be found by calculating half the wavelength and finding the corresponding distances from the top of the tube. These positions can be determined by the equation:
Lₙ = n * λ / 2
where n is an integer representing the order of resonance.
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A diver bounces straight up from a diving board, avoiding the diving board on the way down, and falls feet first into a pool. She starts with a velocity of 4.15 m/s and her takeoff point is 1.80 m above the pool. a. What is her highest point above the board? m b. How long a time are her feet in the air? S c. What is her velocity when her feet hit the water? m/s A particle moves along the x− axis according to x(t)=3t−4t^2m. a. What is the instantaneous velocity at t=3 s ? m/s b. What is the instantaneous speed at t=3 s ? m/s c. What is the average velocity between t=2 s and t=3 s ? m/s
Her highest point above a. the board is 3.139 m. b. Her feet are in the air for 1.106 s. c. Her velocity when her feet hit the water: -14.9 m/s. ca. the instantaneous velocity: -21 m/s, cb. the instantaneous speed: 21 m/s. cc. the average velocity: -5/(3-2)
a. To find the highest point above the board, we can analyze the motion of the diver using the equations of motion. The initial velocity (u) is 4.15 m/s, and the acceleration (a) due to gravity is -9.8 m/s² (taking downward as negative).
The displacement (s) can be determined using the equation s = ut + (1/2)at².
At the highest point, the velocity is zero, so we can find the time (t) it takes to reach that point. Then we substitute that time into the equation to calculate the displacement.
Therefore, t = u/a = 4.15/9.8 ≈ 0.423 s.
Substituting this value of t into the equation, s = ut + (1/2)at² = 4.15 × 0.423 + (1/2) × (-9.8) × (0.423)² ≈ 3.139 m.
b. The time her feet are in the air can be found using the equation of motion s = ut + (1/2)at².
Since the displacement is zero when her feet hit the water, we can solve for time (t) using this equation.
Rearranging, t = (-u ± √(u²-4(1/2)a(0)))/(2(1/2)a)
= (-4.15 ± √(4.15²-4(1/2)(-9.8)(-1.80)))/(2(1/2)(-9.8)) ≈ 1.106 s.
c. The velocity when her feet hit the water can be found using the equation v = u + at, where u is the initial velocity and a is the acceleration due to gravity.
Substituting the given values, v = 4.15 - 9.8 × 1.106 ≈ -14.9 m/s.
For the second part of the question:
a. The instantaneous velocity at t = 3 s can be found by taking the derivative of the position function x(t) with respect to time. The derivative of x(t) = 3t - 4t² is v(t) = 3 - 8t.
Substituting t = 3 into this equation, we have v(3) = 3 - 8(3) = -21 m/s.
b. The instantaneous speed at t = 3 s is the magnitude of the instantaneous velocity, which is the absolute value of the velocity.
Therefore, the instantaneous speed at t = 3 s is |v(3)| = |-21| = 21 m/s.
c. The average velocity between t = 2 s and t = 3 s can be found by calculating the change in position divided by the change in time. The change in position is Δx = x(3) - x(2), and the change in time is Δt = 3 - 2.
Substituting the given function x(t) = 3t - 4t² into these expressions, we have Δx = (3(3) - 4(3)²) - (3(2) - 4(2)²) = -9 - (-4) = -5 m.
Therefore, the average velocity is Δx/Δt = -5/(3-2)
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