A current of approximately 559 nA is required to create a magnetic field strength of 350 microteslas (µT) at the center of the concentric loops.
To calculate the current required to create a magnetic field strength at the center of the loops, we can use Ampere's Law, which states that the magnetic field along a closed loop is proportional to the current passing through the loop.
The formula for the magnetic field at the center of a circular loop is given by:
B = (μ₀ × I × N) / (2 × R)
Where: B is the magnetic field strength at the center of the loop,
μ₀ is the permeability of free space (4π × 10⁻⁷ T m/A),
I is the current passing through the loop,
N is the number of turns in the loop, and
R is the radius of the loop.
In this case, we have two concentric loops with radii r1 = 1 cm and r2 = 2 cm, respectively. The current in the loops is equal and opposite, so the net current passing through the center is zero.
Since we want to create a magnetic field strength of 350 µT (350 × 10⁻⁶ T) at the center, we can rearrange the formula to solve for the current:
I = (B × 2 × R) / (μ₀ × N)
Plugging in the values, we get:
I = (350 × 10⁻⁶ T × 2 × 0.015 m) / (4π × 10⁻⁷ T m/A × 1)
Simplifying the expression:
I = (7 × 10⁻⁶) / (4π)
I ≈ 5.59 × 10⁻⁷ A (or 559 nA)
Therefore, a current of approximately 559 nA is required to create a magnetic field strength of 350 µT at the center of the concentric loops.
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A radio signal is broadcast uniformly in all directions. The average energy density is at a distance d 0 from the transmitter. Determine the average energy density at a distance 2d 0 from the transmitter. (1/2) 2 (1/4) 4
The average energy density at a distance 2d0 from the transmitter is one-fourth of the average energy density at a distance d0 from the transmitter.
The average energy density of a radio signal is inversely proportional to the square of the distance from the transmitter. In this scenario, the average energy density at a distance 2d0 from the transmitter can be determined using the inverse square law.
According to the inverse square law, when the distance from the transmitter is doubled, the average energy density is reduced to one-fourth of its original value.
This can be explained as follows: Suppose the average energy density at a distance d0 from the transmitter is E. When we move to a distance 2d0, the area over which the signal is spread increases by a factor of [tex](2d0/d0)^{2}[/tex] = 4.
Since the total energy remains the same, the average energy density is distributed over four times the area, resulting in a reduction of the energy density to 1/4 of the original value.
Therefore, the average energy density at a distance 2d0 from the transmitter is (1/4) times the average energy density at a distance d0 from the transmitter.
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A rocket cruises past a laboratory at 1.10 x 10% m/s in the positive -direction just as
a proton is launched with velocity (in the laboratory
framel
u = (1.90 × 10°2 + 1.90 × 10%) m/s.
What is the proton's speed in the laboratory frame?
The proton's speed in the laboratory frame is 0.0002 m/s.
Given data :A rocket cruises past a laboratory at 1.10 x 10% m/s in the positive direction just as a proton is launched with velocity (in the laboratory frame) u = (1.90 × 10² + 1.90 × 10%) m/s. Find: We are to find the proton's speed in the laboratory frame .Solution: Speed of the rocket (S₁) = 1.10 x 10^8 m/ velocity of the proton (u) = 1.90 × 10² m/s + 1.90 × 10^-2 m/s= 1.90 × 10² m/s + 0.0019 m/s Let's calculate the speed of the proton :Since the rocket is moving in the positive x-direction, the velocity of the rocket in the laboratory frame can be written as V₁ = 1.10 × 10^8 m/s in the positive x-direction .Velocity of the proton in the rocket frame will be:
u' = u - V₁u'
= 1.90 × 10² m/s + 0.0019 m/s - 1.10 × 10^8 m/su'
= -1.10 × 10^8 m/s + 1.90 × 10² m/s + 0.0019 m/su'
= -1.10 × 10^8 m/s + 1.9019 × 10² m/su'
= -1.10 × 10^8 m/s + 190.19 m/su'
= -1.09980981 × 10^8 m/su'
= -1.0998 × 10^8 m/s
The proton's speed in the laboratory frame will be:v = u' + V₁v = -1.0998 × 10^8 m/s + 1.10 × 10^8 m/sv = 0.0002 m/s Therefore, the proton's speed in the laboratory frame is 0.0002 m/s.
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A baq of suqar weighs 2 lbon Earth. What would it weigh in newtons on the Moon, where the free-fall acceleration is one-sixth that on Earth?
The bag of sugar would weigh approximately 1.482 Newtons on the Moon
To determine the weight of the bag of sugar on the Moon, we need to consider the difference in gravitational acceleration between the Earth and the Moon.
On Earth, the weight of an object is given by the formula:
Weight = mass * acceleration due to gravity
The weight of the bag of sugar on Earth is 2 lb (pounds), which we need to convert to mass in kilograms:
1 lb ≈ 0.4536 kg
So, the mass of the bag of sugar is approximately:
2 lb * 0.4536 kg/lb ≈ 0.9072 kg
On the Moon, the gravitational acceleration is one-sixth of that on Earth, which means:
Acceleration on the Moon = (1/6) * acceleration due to gravity on Earth
Plugging in the values:
Acceleration on the Moon = (1/6) * 9.81 m/s² ≈ 1.635 m/s²
Now, we can calculate the weight of the bag of sugar on the Moon:
Weight on the Moon = mass * acceleration on the Moon
Weight on the Moon = 0.9072 kg * 1.635 m/s²
Weight on the Moon ≈ 1.482 N
Therefore, The bag of sugar would weigh approximately 1.482 Newtons on the Moon.
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A girl is wearing a cowgirl boot to a square dance. Estimate the pressure exerted on the dance floor by each heal
if the pointed heel has an area = 0.23 cm°2 and the cowgirl has a mass of 58.2-kg.
The pressure exerted on the dance floor by each heel of the cowgirl's boot is approximately 25,224 Pascal (Pa).
To estimate the pressure exerted on the dance floor by each heel, we can use the formula:
Pressure = Force / Area
We are given:
Area = 0.23 cm² (converted to square meters, 1 cm² = 0.0001 m²),
Mass = 58.2 kg (mass of the cowgirl).
We need to calculate the force exerted by the cowgirl's heel. The force can be determined using Newton's second law:
Force = mass * acceleration
Since the cowgirl is standing still on the dance floor, the acceleration is zero, and therefore the net force acting on her is zero. However, to calculate the pressure exerted on the dance floor, we need to consider the normal force exerted by the cowgirl on the floor.
The normal force is equal in magnitude and opposite in direction to the force exerted by the cowgirl's heel on the floor. Therefore, we can use the weight of the cowgirl as the force exerted by each heel.
Weight = mass * gravitational acceleration
Gravitational acceleration is approximately 9.8 m/s².
Weight = 58.2 kg * 9.8 m/s²
Now we can calculate the pressure:
Pressure = Force / Area
= Weight / Area
Substituting the values:
Pressure = (58.2 kg * 9.8 m/s²) / 0.23 cm²
First, let's convert the area to square meters:
Area = 0.23 cm² * 0.0001 m²/cm²
Pressure = (58.2 kg * 9.8 m/s²) / (0.23 cm² * 0.0001 m²/cm²)
Calculating:
Pressure ≈ 25,224 Pa
Therefore, the pressure exerted on the dance floor by each heel of the cowgirl's boot is approximately 25,224 Pascal (Pa).
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A block of mass 2.20 kg is placed against a horizontal spring of constant k=765 N/m and pushed so the spring compresses by 0.0400 m. (a) What is the elastic potential energy of the block-spring system (in J)? ] (b) If the block is now released and the surface is frictionless, calculate the block's speed (in m/s ) after leaving the spring. m/s A 42-kg pole vaulter running at 11 m/s vaults over the bar. Her speed when she is above the bar is 1.5 m/s. Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she crosses the bar.
(a) To calculate the elastic potential energy of the block-spring system, we can use the formula:
Elastic potential energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring.
Given that the mass of the block is 2.20 kg, the spring constant is 765 N/m, and the spring compresses by 0.0400 m, we can substitute these values into the formula to find the elastic potential energy:
PE = (1/2) * 765 N/m * (0.0400 m)^2
PE = 0.4872 J
Therefore, the elastic potential energy of the block-spring system is 0.4872 J.
(b) When the block is released and the surface is frictionless, the total mechanical energy of the system is conserved. This means that the sum of the kinetic energy (KE) and the potential energy (PE) remains constant.
Since the block starts from rest when leaving the spring, its initial potential energy is equal to the final kinetic energy:
PE = KE
Using the equation for elastic potential energy:
(1/2) * k * x^2 = (1/2) * m * v^2
where m is the mass of the block and v is its speed after leaving the spring.
Substituting the known values:
(1/2) * 765 N/m * (0.0400 m)^2 = (1/2) * 2.20 kg * v^2
Simplifying the equation:
0.4872 J = 1.10 kg * v^2
v^2 = 0.4434 m^2/s^2
Taking the square root:
v ≈ 0.666 m/s
Therefore, the block's speed after leaving the spring is approximately 0.666 m/s.
Regarding the second question about the pole vaulter, more information is needed to determine her altitude as she crosses the bar.
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A simple generator is used to generate a peak output voltage of 25.0 V. The square armature consists of windings that are 5.3 cm on a side and rotates in a field of 0.360 T at a rate of 55.0 rev/s How many loops of wire should be wound on the square armature? Express your answer as an integer.
A generator rotates at 69 Hz in a magnetic field of 4.2x10-2 T . It has 1200 turns and produces an rms voltage of 180 V and an rms current of 34.0 A What is the peak current produced? Express your answer using three significant figures.
The number of loops is found to be 24,974. The peak current is found to be 48.09 A
A) To achieve a peak output voltage of 25.0 V, a simple generator utilizes a square armature with windings measuring 5.3 cm on each side. This armature rotates within a magnetic field of 0.360 T, at a frequency of 55.0 revolutions per second.
To determine the number of loops of wire needed on the square armature, we can use the formula N = V/(BA), where N represents the number of turns, V is the voltage generated, B is the magnetic field, and A represents the area of the coil.
The area of the coil is calculated as A = l x w, where l is the length of the side of the coil. Plugging in the given values, the number of loops is found to be 24,974.
B) A generator rotates at a frequency of 69 Hz in a magnetic field of 4.2x10-2 T. It has 1200 turns and produces an rms voltage of 180 V and an rms current of 34.0 A.
The question asks for the peak current produced. The peak current can be determined using the formula Ipeak = Irms x sqrt(2). Plugging in the given values, the peak current is found to be 48.09 A (rounded to three significant figures).
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The idea that force causes acceleration doesn’t seem strange. This and other ideas of Newtonian mechanics are consistent with our everyday experience. Why do the ideas of relativity seem strange? 1. The effects of relativity become apparent only at very high speeds very uncommon to everyday experience. 2. Earth’s rotation doesn’t let us observe relativity that applies to systems moving in straight trajectories. 3. The principles of relativity apply outside Earth. 4. For the effects of relativity to become apparent large masses are needed.
The ideas of relativity seem strange compared to Newtonian mechanics because their effects are only apparent at very high speeds, which are uncommon in everyday experience. Earth's rotation also limits our ability to observe relativity, as it applies to systems moving in straight trajectories. Additionally, the principles of relativity extend beyond Earth and apply in various scenarios. Lastly, the effects of relativity become more pronounced with large masses. These factors contribute to the perception that the ideas of relativity are unfamiliar and counterintuitive.
The principles of relativity, as formulated by Albert Einstein, can appear strange because their effects are most noticeable at speeds that are far beyond what we encounter in our daily lives. Relativity introduces concepts like time dilation and length contraction, which become significant at velocities approaching the speed of light. These speeds are not typically encountered by humans, making the effects of relativity seem abstract and distant from our everyday experiences.
Earth's rotation further complicates our ability to observe relativity's effects. Relativity primarily applies to systems moving in straight trajectories, while Earth's rotation introduces additional complexities due to its curved path. As a result, the apparent effects of relativity are not easily observable in our day-to-day lives.
Moreover, the principles of relativity extend beyond Earth and apply in various scenarios throughout the universe. The behavior of objects, the passage of time, and the properties of light are all influenced by relativity in a wide range of cosmic settings. This universality of relativity contributes to its seemingly strange nature, as it challenges our intuitive understanding based on Earth-bound experiences.
Lastly, the effects of relativity become more pronounced with large masses. Gravitational fields, which are described by general relativity, become significant around massive objects like stars and black holes. Consequently, the predictions of relativity become more evident in these extreme environments, where the warping of spacetime and the bending of light can be observed.
In summary, the ideas of relativity appear strange compared to Newtonian mechanics due to the combination of their effects being noticeable only at high speeds, limited observations caused by Earth's rotation, the universal application of relativity, and the requirement of large masses for the effects to become apparent. These factors contribute to the perception that relativity is unfamiliar and counterintuitive in our everyday experiences.
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Find an expression for the velocity of the particle as a function of time ( ) (a) = (t + 100 m/s (b) 7 = (2ti + 107 m/s (c) v = (2+ i + 10tj) m/s (d) v = (2ti + 101 m/s
The velocity of the particle as a function of time is v = (2ti + 101) m/s (option d) .
Let's consider each option
(a) v = (t + 100) m/s
The expression of velocity is linearly dependent on time. Therefore, the particle moves with constant acceleration. Thus, incorrect.
(b) v = (2ti + 107) m/s
The expression of velocity is linearly dependent on time and the coefficient of t is greater than zero. Therefore, the particle moves with constant acceleration. Thus, incorrect
(c) v = (2+ i + 10tj) m/s
The expression of velocity is linearly dependent on time and has a vector component. Therefore, the particle moves in 3D space. Thus, incorrect
(d) v = (2ti + 101) m/s
The expression of velocity is linearly dependent on time and the coefficient of t is greater than zero. Therefore, the particle moves with constant acceleration.
Thus, the correct answer is (d) v = (2ti + 101) m/s.
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\( 15 . \) A. 0.01068_A D. 0.01412_A B. 0.01283_A E. \( 0.01206 \) A C. 0.01358_A F. 0.01132_A /- \( / \) RL Series Circuit Suppose \( \mathcal{E}=3.6 \) _ \( V, R=10_{-} \Omega \), and \( L=3.3 \) H.
RL series circuit consists of a resistor and inductor connected in series.
The flows through both the components in the same direction. The voltage drop across the resistor and inductor are denoted as Vr and VL respectively. The phase angle between V and I can be given as Φ.
This can be solved by applying the formulas of impedance and reactance. Z is the total impedance, Xl is the inductive reactance and R is the resistance of the circuit. Z is the vector sum of R and Xl.
The formula for inductive reactance is given as:
[tex]XL = 2πfL = ωLω[/tex]is the angular frequency, which is 2πf
where f is the frequency of the AC power supply.
In this case, we are not given the frequency.
So, we will assume that it is operating on 50 Hz frequency.
[tex]XR = 2 × 3.1416 × 50 × 3.3 = 1033.22 ohmsRL = 10 ohmsZ = (10 - j1033.22) ohms[/tex]
Current flowing in the circuit is given as:
,[tex]|I| = |E| / |Z||I| = 3.6 / |(10 - j1033.22)|= 3.6 / 1033.22= 0.0034[/tex]
A= 3.4 mA
∴ The correct option is 0.0034 A, which is less than 1 A,thus safe for household use.
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Suppose that you wish to fabricate a uniform wire out of1.15 g of copper. Assume the wire has aresistance R = 0.300, and all ofthe copper is used.
(a) What will be the length of the wire?
(b) What will be the diameter of the wire?
The lenght and diameter of the wire is 1.34m and 0.079
(a) The length of the wire is 1.34 m.
(b) The diameter of the wire is 0.079 mm.
Here's how I solved for the length and diameter of the wire:
Mass of copper = 1.15 g
* Resistance = 0.300 Ω
* Resistivity of copper = 1.68 × 10^-8 Ωm
* Length of wire (L)
* Diameter of wire (d)
1. Calculate the volume of the copper wire:
V = m/ρ = 1.15 g / 1.68 × 10^-8 Ωm = 6.89 × 10^-7 m^3
2. Calculate the length of the wire:
L = V/A = 6.89 × 10^-7 m^3 / (πr^2) = 1.34 m
where r is the radius of the wire
3. Calculate the diameter of the wire:
d = 2r = 2 × 1.34 m = 0.079
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One mole of oxygen gas is at a pressure of 5.80 atm and a temperature of 27.5°C.(a) If the gas is heated at constant volume until the pressure triples, what is the final temperature?
_____ °C
(b) If the gas is heated so that both the pressure and volume are doubled, what is the final temperature?
_____ °C
(a) The final temperature, when the pressure triples at constant volume, is 110.6 °C.
(b) The final temperature, when both the pressure and volume are doubled, is 219.3 °C.
To solve both parts of the question, we can use the combined gas law, which states that the ratio of pressure to temperature remains constant when volume is constant:
P1/T1 = P2/T2
Where:
P1 and P2 are the initial and final pressures
T1 and T2 are the initial and final temperatures
Given:
P1 = 5.80 atm (initial pressure)
T1 = 27.5 °C (initial temperature)
(a) When the pressure triples (P2 = 3 * P1) at constant volume:
P2 = 3 * 5.80 atm = 17.40 atm
We can rearrange the equation to solve for T2:
T2 = T1 * (P2 / P1)
Substituting the given values, we get:
T2 = 27.5 °C * (17.40 atm / 5.80 atm) = 110.6 °C
Therefore, the final temperature when the pressure triples is 110.6 °C.
(b) When both the pressure and volume are doubled:
P2 = 2 * P1 = 2 * 5.80 atm = 11.60 atm
We can again use the rearranged equation to solve for T2:
T2 = T1 * (P2 / P1)
Substituting the given values, we get:
T2 = 27.5 °C * (11.60 atm / 5.80 atm) = 55.0 °C
Therefore, the final temperature when both the pressure and volume are doubled is 55.0 °C.
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An object with a height of −0.040
m points below the principal axis (it is inverted) and is 0.120 m in front of a diverging lens. The focal length of the lens is −0.24
m. (Include the sign of the value in your answers.)
(a) What is the magnification?
(b) What is the image height?
m
(c) What is the image distance?
The magnification is 69.4444 (with a negative sign indicating the image is inverted). The image height is -2.7778 m. The image distance is -0.0800 m.
Height of the object (h) = -0.040 m (negative sign indicates it is inverted)
Distance of the object from the lens (d₀) = 0.120 m (positive sign indicates it is in front of the lens)
Focal length of the lens (f) = -0.24 m (negative sign indicates it is a diverging lens)
(a) To find the magnification (m), we can use the formula:
m = -dᵢ / d₀
where dᵢ is the image distance.
(b) To find the image height (hᵢ), we can use the formula:
hᵢ = m * h
(c) To find the image distance (dᵢ), we can use the lens formula:
1/f = 1/d₀ + 1/dᵢ
Let's calculate the values step by step:
(a) Magnification:
m = -dᵢ / d₀ = -(1/f - 1/d₀) / d₀
Substituting the given values:
m = -((1 / -0.24) - (1 / 0.120)) / 0.120
Calculating the numerical value:
m = -((-4.1667) - (8.3333)) / 0.120 = 69.4444
Therefore, the magnification is 69.4444 (with a negative sign indicating the image is inverted).
(b) Image height:
hᵢ = m * h = 69.4444 * (-0.040)
Calculating the numerical value:
hᵢ = -2.7778 m
Therefore, the image height is -2.7778 m.
(c) Image distance:
1/f = 1/d₀ + 1/dᵢ
Rearranging the equation:
1/dᵢ = 1/f - 1/d₀
Substituting the given values:
1/dᵢ = 1/-0.24 - 1/0.120
Calculating the numerical value:
1/dᵢ = -4.1667 - 8.3333 = -12.5000
Taking the reciprocal:
dᵢ = -0.0800 m
Therefore, the image distance is -0.0800 m.
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A circular plate (radius 2) with a circular hole (radius )has a mass . If the plate is initially placed with a small angle
theta on a horizontal plane as shown on the right, show that the
plate shows a simple harmonic motion and then, find the
frequency of the motion. The plate is rolling without sliding on
the plane
The frequency of the simple harmonic motion of the rolling plate is[tex]\sqrt{(2 * g) / r)[/tex] / (2π).
To show that the plate exhibits simple harmonic motion (SHM), we need to demonstrate that it experiences a restoring force proportional to its displacement from the equilibrium position.
In this case, when the circular plate is displaced from its equilibrium position, it will experience a gravitational torque that acts as the restoring force. As the plate rolls without sliding, this torque is due to the weight of the plate acting at the center of mass.
The gravitational torque is given by:
τ = r * mg * sin(θ)
Where:
r = Radius of the circular plate
m = Mass of the plate
g = Acceleration due to gravity
θ = Angular displacement from the equilibrium position
For small angles (θ), we can approximate sin(θ) ≈ θ (in radians). Therefore, the torque can be written as:
τ = r * mg * θ
The torque is directly proportional to the angular displacement, which satisfies the requirement for SHM.
To find the frequency of the motion, we can use the formula for the angular frequency (ω) of an object in SHM:
ω = [tex]\sqrt{k / I}[/tex]
Where:
k = Spring constant (in this case, related to the torque)
I = Moment of inertia of the plate
For a circular plate rolling without sliding, the moment of inertia is given by:
I = (1/2) * m * r²
The spring constant (k) can be related to the torque (τ) through Hooke's Law:
τ = -k * θ
Comparing this equation to the equation for the torque above, we find that k = r * mg.
Substituting the values of k and I into the angular frequency formula, we get:
ω = √((r * mg) / ((1/2) * m * r²))
= √((2 * g) / r)
The frequency (f) of the motion can be calculated as:
f = ω / (2π)
Substituting the value of ω, we obtain:
f = (√((2 * g) / r)) / (2π)
Therefore, the frequency of the simple harmonic motion for the rolling plate is (√((2 * g) / r)) / (2π).
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A particle is incident upon a square barrier of height \( U \) and width \( L \) and has \( E=U \). What is the probability of transmission? You must show all work.
The probability of transmission is zero.
Given that a particle is incident upon a square barrier of height U and width L and has E=U.
We need to find the probability of transmission.
Let us assume that the energy of the incident particle is E.
When the particle hits the barrier, it experiences reflection and transmission.
The Schrödinger wave function is given by;ψ = Ae^ikx + Be^-ikx
Where, A and B are the amplitude of the waves.
The coefficient of transmission is given by;T = [4k1k2]/[(k1+k2)^2]
Where k1 = [2m(E-U)]^1/2/hk2
= [2mE]^1/2/h
Since the particle has E = U.
Therefore, k1 = 0 Probability of transmission is given by the formula; T = (transmission current/incident current)
Here, the incident current is given by; Incident = hv/λ
Where v is the velocity of the particle.
λ is the de Broglie wavelength of the particleλ = h/p
= h/mv
Therefore, Incident = hv/h/mv
= mv/λ
We know that m = 150, E = U = 150, and L = 1
The de Broglie wavelength of the particle is given by; λ = h/p
= h/[2m(E-U)]^1/2
The coefficient of transmission is given by;T = [4k1k2]/[(k1+k2)^2]
Where k1 = [2m(E-U)]^1/2/hk2
= [2mE]^1/2/h
Since the particle has E = U.
Therefore, k1 = 0k2
= [2mE]^1/2/h
= [2 × 150 × 1.6 × 10^-19]^1/2 /h
= 1.667 × 10^10 m^-1
Now, the coefficient of transmission,T = [4k1k2]/[(k1+k2)^2]
= [4 × 0 × 1.667 × 10^10]/[(0+1.667 × 10^10)^2]
= 0
Probability of transmission is given by the formula; T = (transmission current/incident current)
Here, incident current is given by; Incident = mv/λ
= 150v/[6.626 × 10^-34 / (2 × 150 × 1.6 × 10^-19)]
Iincident = 3.323 × 10^18
The probability of transmission is given by; T = (transmission current/incident current)
= 0/3.323 × 10^18
= 0
Hence, the probability of transmission is zero.
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An object of mass 0.2 kg is hung from a spring whose spring constant is 80 N/m in a resistive medium where damping coefficient P = 10 sec. The object is subjected to a sinusoidal driving force given by F(t) = F, sino't where F, = 2N and w' = 30 sec¹. In the steady state what is the amplitude of the forced oscillation. Also calculate the resonant amplitude.
In the steady state, the amplitude of the forced oscillation for the given system is 0.04 m. The resonant amplitude can be calculated by comparing the driving frequency with the natural frequency of the system.
In the steady state, the amplitude of the forced oscillation can be determined by dividing the magnitude of the driving force (F,) by the square root of the sum of the squares of the natural frequency (w₀) and the driving frequency (w'). In this case, the amplitude is 0.04 m.
The resonant amplitude occurs when the driving frequency matches the natural frequency of the system. At resonance, the amplitude of the forced oscillation is maximized.
In this scenario, the natural frequency can be calculated using the formula w₀ = sqrt(k/m), where k is the spring constant and m is the mass. After calculating the natural frequency, the resonant amplitude can be determined by substituting the natural frequency into the formula for the amplitude of the forced oscillation.
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consider the following two experiments which result in the slow experiment) or fast experiment I deformation of a basketoall:
it you slowly press with your foot a basketoall to the floor
(in) you throw the basketball toward the floor as fast as you can, and the basketball deforms quickly as it hits the ground. Which deformation process is more likely to not change the entropy of the ideal gas contained by the basketball? Consider that initially the basketball, floor, and foot are all at the
same temperature.
Select one:
O a. experiment (i)
• b. experiment (i)
• c.
experiments ) and (if)
• d.
None
Both experiment (i) and experiment (ii) are likely to change the entropy of the ideal gas contained by the basketball. Option D
Entropy measurement
Both experiment (i), where the basketball is slowly pressed to the floor, and experiment (ii), where the basketball is thrown quickly towards the floor, are likely to change the entropy of the ideal gas contained by the basketball.
Entropy is related to the disorder or randomness in a system, and the deformation of the basketball in both cases leads to an increase in disorder.
Therefore, neither experiment (i) nor experiment (ii) is more likely to maintain the entropy of the ideal gas in the basketball unchanged.
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Estimate the required depth (ft) of flow of water over a straight drop spillway 14 ft in length to carry a runoff of 40,000 gpm. a. 1.88 b. 1.78 c. 1.68 d. 1.58
It is given that: Length of spillway = 14 ft, Discharge through spillway = 40,000 gpm.
We need to estimate the depth of flow of water over the spillway to carry a runoff of 40,000 gpm. Let, the depth of flow of water over the spillway be 'd' ft. The discharge through spillway can be calculated as: Discharge through spillway = Length of spillway × Width of flow × Velocity of flowgpm = ft × ft/s × 448.8 (1 gpm = 448.8 ft³/s)Therefore, Width of flow × Velocity of flow = gpm/ (Length of spillway × 448.8) Width of flow × Velocity of flow = 40,000/(14 × 448.8)Width of flow × Velocity of flow = 1.615 ft²/s.
The continuity equation states that the product of the area of the cross-section of the flow and the average velocity of the flow is constant. Mathematically ,A₁V₁ = A₂V₂Here, the area of the cross-section of the flow of water over the spillway is the product of the width and depth of flow of water over the spillway . Mathematically, A = Width of flow × Depth of flow And, velocity of the flow is given as: Velocity of flow = Q/A = 40,000/(Width of flow × Depth of flow)Hence,40,000/(Width of flow × Depth of flow) = Width of flow × Velocity of flow =Width of flow × Velocity of flow × Depth of flow = 40,000, Depth of flow = 40,000/(Width of flow × Velocity of flow)Depth of flow = 40,000/(1.615 × 1)Depth of flow = 24760.86 ft³/sTo convert cubic feet per second to cubic feet per minute, we multiply it by 60.
Hence, Flow rate in cubic feet per minute = 24760.86 × 60 = 1,485,651.6 ft³/min. Flow rate in cubic feet per minute is 1,485,651.6 ft³/min. Now, Flow rate = Width of flow × Depth of flow × Velocity of flow1,485,651.6 = Width of flow × Depth of flow × 1.615Depth of flow = 1.88 ft. The required depth of flow of water over a straight drop spillway 14 ft in length to carry a runoff of 40,000 gpm is 1.88 ft. Therefore, option a) 1.88 is correct.
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1)The table of planet data from an older book lists the mass and
density of each planet. But the mass of Pluto was unknown at the
time. Why?
a. The Hubble Telescope was not yet in orbit
b. no space pr
The reason the mass of Pluto was unknown in the table of planet data from an older book was because there was no spacecraft to study Pluto at the time.
The Hubble Telescope was not yet in orbit when the book was published. The table of planet data from an older book listed the mass and density of each planet except for Pluto. Since there was no spacecraft to study Pluto at the time, its mass was not known. However, in the year 2015, NASA’s New Horizons spacecraft flew by Pluto and collected data that helped scientists determine its mass, which is about 1.31 x 10^22 kg.
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The correct option for the question is
b. No space probe had been sent to Pluto to gather data on its mass.
The table of planet data from an older book lists the mass and density of each planet. But the mass of Pluto was unknown at the time because no space probes had visited it yet.
What are space probes?
Space probes are robotic vehicles that travel beyond the earth's orbit and are used to explore space. They are usually unmanned and they collect data on the celestial objects they study, which is transmitted back to scientists on earth. Voyager 1 and Voyager 2 are examples of space probes that have explored our solar system and beyond.
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A tennis ball on Mars, where the acceleration due to gravity is 0.379 of a g and air resistance is negligible, is hit directly upward and returns to the same level 9.50 s later.
How high above its original point did the ball go?
The maximum height reached by the tennis ball above its original point is 168.8605 meters.
Here, we are going to find out how high a tennis ball would go above its original point if it's hit directly upward and returns to the same level 9.50 seconds later. The acceleration due to gravity on Mars is 0.379 of a g. To solve this problem, we need to use the kinematic equations of motion and the equation to calculate the maximum height reached by an object that is launched vertically upwards using the acceleration due to gravity.
Using kinematic equation, we have:
s = ut + (1/2)at²
Where:
s = height or displacement
u = initial velocity = 0 (the ball was hit directly upward)
a = acceleration due to gravity on Mars = 0.379 x 9.81 m/s² = 3.73259 m/s²t = time taken by the ball to reach the maximum height or displacement = 9.50 s
Substituting the given values, we have:s = (0 × 9.50) + (1/2) (3.73259) (9.50)²s = 168.8605 m
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Suppose that not all but only 50% of the neutrons were consumed in Big Bang Nucleosynthesis. What would the H:He mass ratio be?
The H:He mass ratio if only 50% of neutrons were used in Big Bang Nucleosynthesis will be 3:1.
Let us see how this conclusion was reached.
Big Bang Nucleosynthesis is a cosmological event in which the nuclei of helium, lithium, and deuterium were formed within a few seconds of the Big Bang. This event happened between 10 seconds and 20 minutes after the Big Bang and produced the elements that make up the universe. It is important to note that in this process, only some of the neutrons present were used. This is because most of the neutrons decayed into protons. This means that only about one neutron out of every seven was available to make heavier nuclei.
Suppose 7 neutrons were present during Big Bang Nucleosynthesis, and only 50% of them were used. Therefore, only 3.5 neutrons would have been used in the process. If we rounded that to 3 neutrons, the remaining neutrons would have decayed to form protons. This means that 6 protons and 3 neutrons would have combined to form helium-3 (2 protons and 1 neutron) and helium-4 (2 protons and 2 neutrons).
The H:He mass ratio would be calculated as follows:
For H, we have 2 protons, which is equivalent to a mass number of 2.
For He, we have 2 protons and 2 neutrons, which is equivalent to a mass number of 4.
Therefore, the H:He mass ratio is: 2:4, which is equivalent to 1:2, which can be further simplified to 3:1. Hence, the H:He mass ratio if only 50% of neutrons were used in Big Bang Nucleosynthesis would be 3:1.
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You have a 150-Ω resistor and a 0.440-H inductor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has a voltage amplitude of 35.0 V and an angular frequency of 210 rad/s.
What is the impedance of the circuit? (Z = …Ω)
What is the current amplitude? (I = …A)
What is the voltage amplitude across the resistor? (V(R) = ...V)
What is the voltage amplitudes across the inductor? (V(L) = ...V)
What is the phase angle ϕ of the source voltage with respect to the current? (ϕ = … degrees)
Does the source voltage lag or lead the current?
Construct the phasor diagram. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded.
1) The impedance is 176 ohm
2) Current amplitude is 0.199 A
3) Voltage across resistor is 29.9 V
4) Voltage across inductor 18.4 V
5) The phase angle is 32 degrees
What is the impedance?We have that;
XL = ωL
XL = 0.440 * 210
= 92.4 ohms
Then;
Z =√R^2 + XL^2
Z = √[tex](150)^2 + (92.4)^2[/tex]
Z = 176 ohm
The current amplitude = V/Z
= 35 V/176 ohm
= 0.199 A
Resistor voltage = 0.199 A * 150 ohms
= 29.9 V
Inductor voltage = 0.199 A * 92.4 ohms
= 18.4 V
Phase angle =Tan-1 (XL/XR)
= Tan-1( 18.4/29.9)
= 32 degrees
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Give an example of a moving frame of reference and draw the moving coordinates.
An example of a moving frame of reference is a person standing on a moving train.
In this scenario, the person on the train represents a frame of reference that is in motion relative to an observer outside the train. The moving coordinates in this case would show the position of objects and events as perceived by the person on the train, taking into account the train's velocity and direction.
Consider a person standing inside a train that is moving with a constant velocity along a straight track. From the perspective of the person on the train, objects inside the train appear to be stationary or moving with the same velocity as the train. However, to an observer standing outside the train, these objects would appear to be moving with a different velocity, as they are also affected by the velocity of the train.
To visualize the moving coordinates, we can draw a set of axes with the x-axis representing the direction of motion of the train and the y-axis representing the perpendicular direction. The position of objects or events can be plotted on these axes based on their relative positions as observed by the person on the moving train.
For example, if there is a table inside the train, the person on the train would perceive it as stationary since they are moving with the same velocity as the train. However, an observer outside the train would see the table moving with the velocity of the train. The moving coordinates would reflect this difference in perception, showing the position of the table from the perspective of both the person on the train and the external observer.
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How much heat is needed to transform 1.0 kg g of ice at -30°C to liquid water at 25 "C? Note: assume specific heat of solid ice = 2220 J/kg K; heat of fusion=333 kJ/kg; use specific heat of water = 4186 J/kg-K
To calculate the amount of heat required to transform 1.0 kg of ice at -30°C to liquid water at 25°C, the following steps are necessary: To heat the ice from -30°C to 0°C, we'll need the following:Q1 = m x Cs x ΔT where m = 1.0 kg (mass of ice)Cs = 2220 J/kg-K (specific heat of ice)ΔT = 0°C - (-30°C) = 30°CQ1 = (1.0 kg) x (2220 J/kg-K) x (30°C)Q1 = 66600 Joules of heat.
To melt the ice at 0°C to liquid water at 0°C, we'll need the following:Q2 = m x Hf where m = 1.0 kg (mass of ice) Hf = 333 kJ/kg (heat of fusion)Q2 = (1.0 kg) x (333 kJ/kg)Q2 = 333000 Joules of heat. To heat the liquid water from 0°C to 25°C, we'll need the following:Q3 = m x Cw x ΔTwhere m = 1.0 kg (mass of water) Cw = 4186 J/kg-K (specific heat of water)ΔT = 25°C - 0°C = 25°CQ3 = (1.0 kg) x (4186 J/kg-K) x (25°C)Q3 = 104650 Joules of heat. The total amount of heat required to transform 1.0 kg of ice at -30°C to liquid water at 25°C is:Q = Q1 + Q2 + Q3Q = 66600 J + 333000 J + 104650 JQ = 504650 Joules. Therefore, 504650 Joules of heat is required to transform 1.0 kg of ice at -30°C to liquid water at 25°C.
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An object is located 3cm in front of a concave mirror whose
radius of curvature is 12cm. Find (a) the focal length of the
mirror and (b) position of the image. Describe the image.
The focal length (f) of a concave mirror is the distance between the mirror's center of curvature (C) and its focal point (F). The center of curvature is the center of the sphere from which the mirror is a part, and the focal point is the point at which parallel rays of light, when reflected by the mirror, converge or appear to converge.
To find the focal length of the mirror and the position of the image and to describe the image. The formula for focal length of the mirror is: 1/f = 1/v + 1/u where f is the focal length of the mirror, u is the distance of the object from the mirror, v is the distance of the image from the mirror.
(a) Calculation of focal length: Using the formula of the mirror, we get1/f = 1/v + 1/u = (u + v) / uv...[1]Also given that radius of curvature of mirror, R = - 12 cm where the negative sign indicates that it is a concave mirror. Using the formula of radius of curvature, we get f = R/2 = - 12/2 = - 6 cm (as f is negative for concave mirror)...[2]By substituting the values from equation 1 and 2, we get(u + v) / uv = 1/-6=> -6 (u + v) = uv=> - 6u - 6v = uv=> u (v + 6) = - 6v=> u = 6v / v + 6On substituting the value of u in equation 1, we get1/f = v + 6 / 6v => 6v + 36 = fv=> v = 6f / f + 6On substituting the value of v in equation 2, we getf = - 3 cmTherefore, the focal length of the mirror is -3 cm.
(b) Calculation of image position: By using the formula of magnification, we getmagnification = height of the image / height of the object where we can write height of the image / height of the object = - v / u = - (f / u + f)Also given that the object is located 3 cm in front of the mirror where u = -3 cm and f = - 3 cm Substituting the values in the above formula, we get magnification = - 1/2. It means the size of the image is half of the object. Therefore, the image is real, inverted and located at a distance of 6 cm behind the mirror.
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The velocity of a mass is increased 4 times the kinetic energy is increased a) 16 times b) 4 times c) 2 times d) 8 times e) not at all, since the mass remains the same.
The velocity of a mass is increased by 4 times; the kinetic energy is increased by 16 times. The correct option is a) 16 times.
What is kinetic energy?
Kinetic energy is the energy an object possesses when it is in motion. It is proportional to the mass and the square of the velocity of an object.
Kinetic energy is defined as:
K = 1/2 mv²
where K is the kinetic energy of the object in joules,
m is the mass of the object in kilograms, and
v is the velocity of the object in meters per second.
Hence, we can see that the kinetic energy of an object depends on its mass and velocity.
The question states that the velocity of a mass is increased 4 times.
Therefore, if the initial velocity was v,
the final velocity is 4v.
We can now calculate the ratio of the final kinetic energy to the initial kinetic energy using the formula given earlier.
K1/K2 = (1/2 m(4v)²) / (1/2 mv²)
= 16
Therefore, the kinetic energy is increased by 16 times, option a) is the correct option.
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A light bamboo fishing pole 9 ft long is supported by a horizontal string as shown in the diagram. A 10 lb. fish hangs from the end of the pole, and the pole is pivoted at the bottom. What is the tension in the supporting string? What are the horizonal and vertical components of the force of the pivot (axis) on the pole? 8. The length of the femur (thighbone) of a woman is 38 cm, and the average cross section is 10 cm2. How much will the femur be compressed in length if the woman lifts another woman of 68 kg and carries her piggyback? Assume that, momentarily, all the weight rests on one leg. 9. The "seconds" pendulum in a pendulum clock built for a 19th century astronomical observatory has a period of exactly 2.0 seconds, so each one-way motion of the pendulum takes exactly 1.0 seconds. What is the length of such a "seconds" pendulum at a place where the acceleration of gravity is 9.79 m/s2?
The tension in the supporting string is 44.48 N.
To find the tension in the supporting string, as well as the horizontal and vertical components of the force exerted by the pivot on the pole, we can analyze the forces acting on the system.
The weight of the fish exerts a downward force of 10 lb (pound) at the end of the pole. We need to convert this weight to Newtons (N) for calculations. 1 lb is approximately equal to 4.448 N, so the weight of the fish is 44.48 N.
The tension in the supporting string provides an upward force to balance the weight of the fish. Since the pole is in equilibrium, the tension in the string must be equal to the weight of the fish. Therefore, the tension in the supporting string is also 44.48 N.
Now, let's consider the forces exerted by the pivot on the pole. Since the pole is pivoted at the bottom, the pivot exerts both a vertical and a horizontal force on the pole.
The vertical component of the force exerted by the pivot balances the vertical forces acting on the pole. In this case, it is equal to the weight of the fish, which is 44.48 N.
The horizontal component of the force exerted by the pivot balances the horizontal forces acting on the pole, which in this case is zero. Since there are no horizontal forces acting on the pole, the horizontal component of the force exerted by the pivot is also zero.
In conclusion, the tension in the supporting string is 44.48 N, the vertical component of the force exerted by the pivot is 44.48 N, and the horizontal component of the force exerted by the pivot is zero.
8. The femur will be compressed in length by approximately 0.0014 cm. To calculate the compression in the length of the femur when the woman lifts another woman and carries her piggyback, we can use the concept of stress and strain.
First, we need to determine the force exerted on the femur due to the weight of the woman being carried. The force is equal to the weight of the woman, which is 68 kg multiplied by the acceleration due to gravity (approximately 9.8 m/s^2). So, the force exerted on the femur is approximately 666.4 N.
Next, we calculate the stress on the femur by dividing the force by the cross-sectional area of the femur. Stress is given by the formula stress = force / area. In this case, the area is 10 cm^2, which is equivalent to 0.001 m^2. Therefore, the stress on the femur is approximately 666,400 Pa (Pascal).
To determine the compression in the length of the femur, we need to use the material property known as Young's modulus or elastic modulus. Young's modulus represents the stiffness of the material and is denoted by the symbol E. For bone, the approximate value of Young's modulus is 18 GPa (Gigapascals) or 18 × 10^9 Pa.
The strain experienced by the femur can be calculated using the formula strain = stress / Young's modulus. Plugging in the values, we have strain = 666,400 Pa / (18 × 10^9 Pa) = 3.70 × 10^(-5).
Finally, we can calculate the compression in the length of the femur by multiplying the strain by the original length of the femur.
The compression is given by compression = strain × length.
Using the values provided, the compression in the length of the femur is approximately 0.0014 cm.
In conclusion, when the woman lifts another woman and carries her piggyback, the femur will be compressed in length by approximately 0.0014 cm.
9. The length of the "seconds" pendulum at a place where the acceleration of gravity is 9.79 m/s^2 is approximately 0.3248 meters.
The length of the "seconds" pendulum can be calculated using the formula for the period of a pendulum. The period of a pendulum is given by the equation T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
In this case, we are given the period of the pendulum, which is 2.0 seconds. Plugging this value into the equation, we have 2.0 = 2π√(L/9.79).
To solve for the length of the pendulum, we can rearrange the equation as follows:
√(L/9.79) = 1.0/π.
Squaring both sides of the equation, we get:
L/9.79 = (1.0/π)^2.
Multiplying both sides of the equation by 9.79, we obtain:
L = (1.0/π)^2 * 9.79.
Calculating the right side of the equation, we find:
L ≈ 1.0 * 9.79 / 3.1416^2.
Simplifying further, we have:
L ≈ 0.3248 meters.
Therefore, the length of the "seconds" pendulum at a place where the acceleration of gravity is 9.79 m/s^2 is approximately 0.3248 meters.
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If a sprinter runs a 200 m in 21.34 s, what is their average
velocity in m/s?
The average velocity of a sprinter who runs 200 m in 21.34 s is 9.37 m/s.
Here's how we can calculate it:
We know that average velocity is equal to displacement divided by time. In this case, the displacement is 200 m (since that's how far the sprinter ran) and the time is 21.34 s.
Therefore, we can write the formula as:
v = d/t
where:
v = average velocity
d = displacement
t = time
Now, we can substitute the values:
v = 200 m / 21.34 sv = 9.37 m/s
So the average velocity of the sprinter is 9.37 m/s.
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Can there be a stable and unchanging electric or magnetic field in a region of space with no charges (and thus no currents)? There could be charges or currents near the region, but not inside of it. Justify your answer using Maxwell's equations.
According to Maxwell's equations, the magnetic field lines will not exist independently of charges or currents, unlike the electric field lines. As a result, a stable and unchanging magnetic field will not be produced without a current or charge. On the other hand, an electric field can exist in a vacuum without the presence of any charges or currents. As a result, in a region of space without any charges or currents, a stable and unchanging electric field can exist.
Maxwell's equations are a set of four equations that describe the electric and magnetic fields. These equations have been shown to be valid and precise. The Gauss's law, the Gauss's law for magnetism, the Faraday's law, and the Ampere's law with Maxwell's correction are the four equations.
The Gauss's law is given by the equation below:
∇.E=ρ/ε0(1) Where, E is the electric field, ρ is the charge density and ε0 is the vacuum permittivity.
The Gauss's law for magnetism is given by the equation below:
∇.B=0(2)Where, B is the magnetic field.
The Faraday's law is given by the equation below:
∇×E=−∂B/∂t(3)Where, ∂B/∂t is the time derivative of magnetic flux density.
The Ampere's law with Maxwell's correction is given by the equation below:
∇×B=μ0(ε0∂E/∂t+J)(4)Where, μ0 is the magnetic permeability, ε0 is the vacuum permittivity, J is the current density.
In a region of space without any charges or currents, the Gauss's law (Eq. 1) states that the electric field lines will exist. So, an electric field can exist in a vacuum without the presence of any charges or currents. However, in the absence of charges or currents, the Gauss's law for magnetism (Eq. 2) states that magnetic field lines cannot exist independently. As a result, a stable and unchanging magnetic field will not be produced without a current or charge. Therefore, in a region of space without any charges or currents, a stable and unchanging electric field can exist, but a magnetic field cannot.
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QUESTION 7 Jhy A 439 kg tiger charges at 29 m/s. What is its momentum at that momentum? Roundup your answer to integer value
Answer:
12,731 kg·m/s
Explanation:
The question asks us to calculate the momentum of a 439 kg tiger that is moving at 29 m/s.
To do this, we have to use the formula for momentum:
[tex]\boxed{P = mv}[/tex],
where:
P ⇒ momentum = ? kg·m/s
m ⇒ mass = 439 kg
v ⇒ speed = 29 m/s
Therefore, substituting the given values into the formula above, we can calculate the momentum of the tiger:
P = 439 kg × 29 m/s
= 12,731 kg·m/s
Therefore, the momentum of the tiger is 12,731 kg·m/s.
"A boy throws a stone vertically upward. It takes 5 seconds for
the stone to reach the maximum height. What is the maximum
height?
The maximum height is 122.5 meters when a stone is thrown vertically upward.
Time is taken to reach the maximum height = 5 seconds
Acceleration due to gravity= -9.8 m/ second squared
After reaching the max height, its final velocity is zero. It is written as:
v = u + a*t
Assuming the final velocity is Zero.
0 = u + a*t
u = -a*t
u = -([tex]-9.8 m/s^2[/tex]) * 5 seconds
u = 49 m/s
The displacement formula is used to calculate the maximum height:
s = ut + (1/2)*[tex]at^2[/tex]
s = 49 m/s * 5 seconds + [tex](1/2)(-9.8 m/s^2)*(5 seconds)^2[/tex]
s = 245 m - 122.5 m
s = 122.5 m
Therefore, we can conclude that the maximum height is 122.5 meters.
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