Which of the following is NOT another name for the Big and Little Dippers?
A. Cart B. Drinking gourd. C. Many little eyes. D. Plow.

To solve this problem, we can apply the principle of conservation of momentum. To find the resulting speed of the block, we need to determine the velocity of the block after the collision.

we can write the equation for conservation of momentum in the x-direction as:

(m_bullet * v_bullet_initial) + (m_block * v_block_initial) = (m_bullet * v_bullet_final) + (m_block * v_block_final)

where:

m_bullet = mass of the bullet = 4.60 g = 0.0046 kg

v_bullet_initial = initial velocity of the bullet = 632 m/s

m_block = mass of the block = 710 g = 0.710 kg

v_bullet_final = final velocity of the bullet = 436 m/s

Substituting the known values into the equation and solving for v_block_final, we get:

(0.0046 kg * 632 m/s) + (0.710 kg * 0 m/s) = (0.0046 kg * 436 m/s) + (0.710 kg * v_block_final)

0.0029072 kg·m/s = 0.0020056 kg·m/s + (0.710 kg * v_block_final)

0.0009016 kg·m/s = 0.710 kg * v_block_final

v_block_final = 0.0009016 kg·m/s / 0.710 kg

v_block_final ≈ 0.00127 m/s

(b) The speed of the bullet-block center of mass can be calculated using the conservation of momentum equation in the x-direction:

(m_bullet * v_bullet_initial) + (m_block * v_block_initial) = (m_bullet + m_block) * v_center_of_mass

we have:

(0.0046 kg * 632 m/s) + (0.710 kg * 0 m/s) = (0.0046 kg + 0.710 kg) * v_center_of_mass

2.9152 kg·m/s = 0.00531 kg * v_center_of_mass

v_center_of_mass = 2.9152 kg·m/s / 0.00531 kg

v_center_of_mass ≈ 549.055 m/s

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When elements and compounds move from one phase to another, heat is present.

Phase changes can happen when the temperature or pressure changes. Temperature affects the phase of matter. The temperature at which a substance changes from a solid to a liquid to a gas varies depending on the pressure.

The temperature at which water boils, for example, changes based on elevation. It takes more energy to break down bonds when the substance's temperature rises, causing the substance to change phases. Heat is used up by a substance when it changes from a solid to a liquid or from a liquid to a gas.

Therefore, Heat is created by a substance when it changes from a gas to a liquid or from a liquid to a solid.

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The gravitational force that Saturn exerts on Rhea is 3.546 × 10^17 Newtons.

To calculate the gravitational force that Saturn exerts on Rhea, we can use the formula for gravitational force:

F = G * (m1 * m2) / r^2

Where:

F is the gravitational force

G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2)

m1 is the mass of Saturn

m2 is the mass of Rhea

r is the distance between Saturn and Rhea

Given:

m1 (mass of Saturn) = 5.68 × 10^26 kg

m2 (mass of Rhea) = 2.31 × 10^21 kg

r (distance between Saturn and Rhea) = 527,108 km = 527,108,000 m

a) Calculating the gravitational force:

F = G * (m1 * m2) / r^2

F = (6.67430 × 10^-11 N m^2 / kg^2) * (5.68 × 10^26 kg * 2.31 × 10^21 kg) / (527,108,000 m)^2

Calculating this expression:

F ≈ 3.546 × 10^17 N

Therefore, the gravitational force that Saturn exerts on Rhea is approximately 3.546 × 10^17 Newtons.

b) To find a point between Saturn and Rhea where a spacecraft does not experience any gravitational pull, we need to consider the gravitational force equation.

Since gravitational force depends on the masses of the objects and their distance, there is no point between Saturn and Rhea where a spacecraft would be completely free from gravitational pull.

The gravitational force between two objects decreases with distance, but it never becomes zero unless the distance becomes infinitely large.

So, in the vicinity of Saturn and Rhea, there will always be a gravitational force acting on any object present.

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The false statement among the options is B. The statement "du is independent of pressure as it is only a function of T and p" is incorrect.

In thermodynamics, the differential of internal energy (du) is given by the expression:

du = TdS - pdV

This equation shows that du depends not only on temperature (T) and pressure (p) but also on entropy (S) and volume (V). The du term represents the infinitesimal change in internal energy of a system.

The first term, TdS, accounts for the heat transfer into the system, where T is the temperature and dS is the infinitesimal change in entropy. The second term, -pdV, represents the work done by the system against external pressure, where p is the pressure and dV is the infinitesimal change in volume.

Therefore, du is not independent of pressure. The presence of the -pdV term in the equation clearly indicates that pressure has an impact on the change in internal energy.

While it is true that du can be expressed as a function of T and p alone (assuming constant entropy and volume), it does not imply that du is independent of pressure in general. The specific conditions and constraints of a system determine the dependence of du on various variables.

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The total resistance of the network is 3 Ohms. The current flowing through the battery in the circuit is 10 Amperes.

To find the total resistance of the network, we can use the formula for resistors in series:

R_total = R_1 + R_2

R_1 = 2 Ohms

R_2 = 1 Ohm

Substituting the given values into the formula:

R_total = 2 Ohms + 1 Ohm

R_total = 3 Ohms

Therefore, the total resistance of the network is 3 Ohms.

To find the current flowing through the battery in the circuit, we can use Ohm's Law:

I = V / R

I is the current

V is the voltage (emf) of the battery

R is the total resistance of the network

V = 30 V

R = 3 Ohms

Substituting the given values into the formula:

I = 30 V / 3 Ohms

I = 10 Amperes

Therefore, the current flowing through the battery in the circuit is 10 Amperes.

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The given question is based on true or false statements. Below mentioned are the answers for the given statements:

a) True

b) True

c) False

d) True

e) True

f) True

g) True

h) False

i) True

j) False

The given question is asking to identify the given statements which are true or false. All the statements are related to fluid mechanics and aerodynamics. Some of the important definitions are defined below:

Static pressure: The pressure of fluid when it is at rest is called static pressure.

Stagnation pressure: The pressure of a fluid when it is forced to stop moving is called stagnation pressure.

Isentropic: A process in which entropy remains constant is called isentropic.

Expansion wave: The wave generated when a supersonic flow slows down to a subsonic flow is called an expansion wave.

Wedge angle: The angle made by the forward edge of the wedge with the horizontal axis is called wedge angle. Wave angle: The angle between the direction of incoming flow and the line representing the wave's direction is called wave angle.

Critical Mach number: The Mach number at which the flow over the wing reaches supersonic velocity is called critical Mach number. The answers to the given statements are:

a) The static pressure is the pressure measured by a sensor moving at the same velocity as the fluid velocity. True

b) In a large, pressurized air tank, the stagnation pressure is larger than the static pressure at the same point. True

c) The flow across a normal shock wave is isentropic. False

d) Density p is constant across the expansion wave since it is an isentropic process. True

e) For a wedge of given deflection angle, wave angle of an attached oblique shock decreases as the Mach number decreases. True

f) A thinner airfoil will generally have a higher critical Mach number Mer compared to a thicker airfoil. True

g) Area ruling is a process in which the wing area of the airplane is changed to reduce supersonic drag. True

h) Supercritical airfoils achieve better performance by increasing Mer. False

i) An optimal shape for a re-entry vehicle moving at hypersonic Mach numbers is a sharp conical shape. True

j) Convective heating becomes less important than radiative heating as re-entry velocity increases. False

Hence, the correct answers for the given statements are True, True, False, True, True, True, True, False, True, and False.

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The given sequence of characters "(in) e-amaness fied r-artuanere xmr" seems to be gibberish or a random combination of characters.

It doesn't seem to form any meaningful word or phrase when unscrambled. However, if we apply some techniques to unscramble it, we might get some results. Some of the methods that we can use to unscramble a word or phrase include:Rearranging the letters in a word/phrase.

Rotating the letters 180 degrees and 90 degrees.Playing around with anagrams or combinations of letters and numbers.The given sequence of characters might be an encrypted message or code that requires decryption to make sense of it. It might also be a random combination of characters without any meaning or significance.

Therefore, without any additional context or information, it is impossible to determine what the sequence of characters means or how to unscramble it.

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Option 1 is correct. The gravity on a planet with a mass of 3 times that of Earth and a radius 3 times that of Earth would be 1/9th of Earth's gravity.

The force of gravity on a planet is determined by its mass and radius. According to Newton's law of universal gravitation, the force of gravity (F) between two objects is given by the equation [tex]F = (G * m1 * m_2) / r^2[/tex], where G is the gravitational constant, [tex]m_1[/tex] and [tex]m_2[/tex] are the masses of the two objects, and r is the distance between their centres.

In this case, we are comparing the gravity of a planet with a mass ([tex]m_2[/tex]) of 3 times that of Earth ([tex]M_\oplus[/tex]) and a radius (r) of 3 times that of Earth. Since the radius is directly proportional to the distance between the centres of the two objects, the value of [tex]r^2[/tex] would be [tex]3^2 = 9[/tex] times larger than Earth's radius.

As a result, the force of gravity on this planet would be [tex]1/9th (1/3^2)[/tex] of Earth's gravity, which is the first option given. Therefore, the correct answer is 1/9 × Earth's gravity.

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The net electric field vector at point A, due to a positive charge Q₁ = 30 μC and a negative charge Q₂ = -20 μC, can be determined using vector addition.

To find the net electric field vector at point A, we need to consider the electric fields produced by each charge individually and then combine them using vector addition. The electric field at a point in space due to a point charge is given by the equation:

E = k * (Q / r²) * u

Where:

- E is the electric field vector

- k is the electrostatic constant (k = 9 x 10^9 N m²/C²)

- Q is the charge of the source

- r is the distance from the source charge to the point of interest

- u is the unit vector pointing from the source charge to the point of interest

Step 1: Electric field due to Q₁

The electric field at point A due to Q₁ can be calculated using the above equation. The magnitude of the electric field is given by:

E₁ = k * (Q₁ / r₁²)

Step 2: Electric field due to Q₂

Similarly, the electric field at point A due to Q₂ can be calculated as:

E₂ = k * (Q₂ / r₂²)

Step 3: Net electric field at point A

To find the net electric field at point A, we need to add the electric field vectors due to each charge. Since the electric field is a vector quantity, we need to consider both magnitude and direction.

To add two vectors, we can break them down into their x and y components. Assuming the x-axis points to the right and the y-axis points upward, we can calculate the x and y components of each electric field vector. Let's denote the x-component of a vector V as Vₓ and the y-component as Vᵧ.

The x-component of the net electric field at point A (Eₐₓ) is the sum of the x-components of the electric field vectors due to each charge:

Eₐₓ = E₁ₓ + E₂ₓ

Similarly, the y-component of the net electric field at point A (Eₐᵧ) is the sum of the y-components of the electric field vectors due to each charge:

Eₐᵧ = E₁ᵧ + E₂ᵧ

Finally, the magnitude and direction of the net electric field at point A can be calculated using the x and y components:

|Eₐ| = √(Eₐₓ² + Eₐᵧ²)

θ = atan(Eₐᵧ / Eₐₓ)

By calculating the x and y components and using the above equations, we can determine the net electric field vector at point A due to the given charges Q₁ and Q₂.

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The magnitude of the acceleration of the alpha particles is 3.5 × 10¹⁴ m/s².The charge on an alpha particle is 3.2 × 10⁻¹⁹ C. The distance between them is 2.3 × 10⁻¹² N.

(a) The electric force acting between two alpha particles is given as:F = k(q1q2)/r² where q1 and q2 are the charges of alpha particles, r is the separation between them, and k is Coulomb's constant.

The alpha particle consists of 2 protons, each having a charge of +1.6 × 10⁻¹⁹ C.

Therefore, the charge on an alpha particle is 2 × 1.6 × 10⁻¹⁹ C = 3.2 × 10⁻¹⁹ C.

The distance between them is 3.6 × 10⁻¹⁵ m.F = (9 × 10⁹ Nm²/C²) × [(3.2 × 10⁻¹⁹ C)²]/(3.6 × 10⁻¹⁵ m)²F = 2.3 × 10⁻¹² N

(b) The force between the two alpha particles causes an acceleration in them.

We can use the second law of motion to find the acceleration.a = F/m where m is the mass of one alpha particle.

The mass of an alpha particle is 4.0026 u = 6.65 × 10⁻²⁷ kg.a = (2.3 × 10⁻¹² N)/(6.65 × 10⁻²⁷ kg)a = 3.5 × 10¹⁴ m/s².

Therefore, the magnitude of the acceleration of the alpha particles is 3.5 × 10¹⁴ m/s².

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The inventor who co-created the film Fred Ott's Sneeze, which was one of the first American movies was Thomas Edison. So option B is correct.

Thomas Edison, along with his team at the Edison Manufacturing Company, co-created the film titled "Fred Ott's Sneeze" in 1894. It is considered one of the earliest American motion pictures. The film features Fred Ott, an employee of Edison, sneezing and was a short, silent film that lasted just a few seconds. Thomas Edison was a prolific inventor and played a crucial role in the early development of motion pictures and filmmaking technology.Therefore option B is correct.

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The dimensional expressions for the quantities on the right-hand side of the given equations are ML²T⁰, ML²T⁻¹, and MLT⁻², corresponding to different physical quantities involved in the equations.

Physical quantities are m, r, v, a, and t with dimensions [m] = M, [r] = L, [v] = LT⁻¹, [a] = LT⁻², and [t] = T. The dimensional expression of the quantity on the right-hand side of each equation is given below:

L0 = mvr

where [L0] = L1[L] = [M]a[L]b[T]c = MaLbTc

By equating the dimensions on both sides, we get

LHS = RHS.

LHS = L0 = L¹

RHS

mvr = [M][L][LT⁻¹] = MaL²T⁻¹

Comparing the exponents of M, L, and T on both sides, we get

M : 1 = aL : 2 = bT : -1 + 1 = c⇒ a = 1, b = 2, and c = 0.

So, the dimensional expression of the quantity on the right-hand side of L0 = mvr is MaL²T⁰ = ML²T⁰W = mar

where [W] = [F][d] = MLT⁻²LT = ML²T⁻¹

By equating the dimensions on both sides, we get

LHS = RHS.

LHS = W = ML²T⁻¹

RHS

mar = [M][LT⁻²][L] = ML²T⁻¹

Comparing the exponents of M, L, and T on both sides, we get

M : 1 = 1

L : 2 = 1

T : -1 - 2 = -3⇒ the dimensional expression of the quantity on the right-hand side of W = mar is ML²T⁻¹.

K = -rma

By equating the dimensions on both sides, we get

LHS = RHS.

LHS = K = [M][L²][T⁻²]

RHS

-rma = -[L][M][T⁻²] = MLT⁻²

Comparing the exponents of M, L, and T on both sides, we get

M : 1 = 1

L : 2 = -1

T : -2 = -2⇒ the dimensional expression of the quantity on the right-hand side of K = -rma is MLT⁻².

Hence, the dimensional expression of the quantity on the right-hand side of each equation is

ML²T⁰, ML²T⁻¹, and MLT⁻².

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a) The required slit separation to achieve the fourth-order constructive interference at an angle of 6.0° with monochromatic light of wavelength λ=5.90×10⁻⁷m is approximately 9.83×10⁻⁶m.

b) With a different light source having a wavelength λ=6.50×10⁻⁷m, the angle at which third-order constructive interference will occur using the same slits is approximately 7.13°.

a) In a double-slit experiment, the condition for constructive interference is given by the equation: d × sin(θ) = m × λ,

where d is the slit separation, θ is the angle of the interference pattern, m is the order of the interference, and λ is the wavelength of the light.

Given that the fourth-order constructive interference occurs at an angle of 6.0° (converted to radians: 6.0° × π/180 ≈ 0.105 radians) and the wavelength is λ=5.90×10⁻⁷m, we can rearrange the equation to solve for the slit separation:

d = (m × λ) / sin(θ),

d = (4 × 5.90×10⁻⁷m) / sin(0.105),

d ≈ 9.83×10⁻⁶m.

b) Using the same slits but with a different light source having a wavelength λ=6.50×10⁻⁷m, we can determine the angle at which third-order constructive interference occurs. Rearranging the equation as before:

θ = arcsin((m × λ) / d),

θ = arcsin((3 × 6.50×10⁻⁷m) / 9.83×10⁻⁶m),

θ ≈ 7.13°.

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The answer is that the orbital period of the object in Earth years is approximately 25.9 years. Given that an object orbits the sun at an average distance of 17 AU, we need to determine its orbital period in Earth years.

The period of revolution or time taken for an object to complete one revolution around the sun is given by Kepler's third law of planetary motion. Kepler's third law of planetary motion states that the square of the time period of revolution of a planet is proportional to the cube of its average distance from the Sun.

Mathematically, the expression for Kepler's third law can be written as: T² ∝ r³ where T is the period of revolution of the planet and r is the average distance of the planet from the sun.

According to Kepler's third law, the square of the time period of revolution of the object is proportional to the cube of its average distance from the Sun. That is: T² ∝ r³ Therefore, we can write:T² = k × r³where k is a constant.

The above equation can be rearranged as:T² = (r³) / k

On substituting the values of T and r, we have:T^2=17*(AU^3)/k

The value of k can be determined if we know the orbital period of Earth. The average distance of Earth from the Sun is 1 AU. The time period of revolution of Earth is 1 year. Substituting these values into the equation, we get:

1^2= 1*(AU^3)/k

Simplifying the above expression, we get: k = 1

On substituting the value of k in the equation and solving for T, we have: T= √(17*AU^3) ≈25.9

Therefore, the orbital period of the object in Earth years is approximately 25.9 years (rounded to one decimal place).

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These  diagrams represent the motion of the jet ski as described in the problem, starting from rest, accelerating to a constant speed, and then maintaining that speed.

Motion Diagram:

The motion diagram shows the position of the jet ski at different time intervals. Since the jet ski starts from rest, we can represent it as follows:

Constant Speed

The "o" represents the starting position of the jet ski, and the arrow indicates the direction of motion. As time progresses, the jet ski moves to the right.

Position vs. Time Graph:

Since the jet ski starts from rest and then continues at a constant speed, the position vs. time graph would be a straight line with a positive slope (representing constant velocity). The graph would look like this:

markdown

Velocity vs. Time Graph:

The velocity vs. time graph would show the change in velocity as a function of time. Since the jet ski starts from rest and then maintains a constant speed, the graph would be a step function. It would show an instant increase in velocity from zero to a constant value and then remain constant. The graph would look like this:

markdown

Acceleration vs. Time Graph:

Since the jet ski starts from rest and then maintains a constant speed, the acceleration vs. time graph would be zero throughout. It would be a horizontal line at zero acceleration. The graph would look like this:

markdown

Acceleration

These diagrams represent the motion of the jet ski as described in the problem, starting from rest, accelerating to a constant speed, and then maintaining that speed.

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The actuating force required by the gripper to hold the vertically positioned part with a mass of 2kg, given a coefficient of friction of 0.52 and a safety factor of 2, is 41.6 N.

To calculate the actuating force, we first need to determine the force due to gravity acting on the part. The weight of the part can be calculated as the mass (m) multiplied by the acceleration due to gravity (g). In this case, the weight of the part is 2kg × 10m/s^2 = 20N.

Next, we need to consider the friction force between the gripper fingers and the part. The friction force can be calculated as the product of the coefficient of friction (μ) and the normal force. The normal force is equal to the weight of the part in this vertically positioned scenario, which is 20N. Thus, the friction force is 0.52 × 20N = 10.4N.

To hold the part safely, the gripper must exert a force greater than the sum of the weight and the friction force. Considering the safety factor of 2, the required actuating force is 2 × (20N + 10.4N) = 62.8N. However, since the gripper has two fingers, the force exerted by each finger is half of the total actuating force. Therefore, each finger needs to exert a force of 31.4N.

In summary, the actuating force required by the gripper to hold the vertically positioned part with a mass of 2kg, a coefficient of friction of 0.52, and a safety factor of 2 is 41.6N. (Gripper force calculation with friction coefficient and safety factor)

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(a) As measured by an observer also in the ship, the heartbeat rate of the astronaut will be lower than 69 beats/min.

(b) As measured by an observer at rest on Earth, the heartbeat rate of the astronaut will still be 69 beats/min.

(a) According to time dilation in special relativity, time appears to pass more slowly for an object that is moving relative to an observer. In this case, when the astronaut is traveling in a spaceship at 0.86c (86% of the speed of light), the observer in the ship will measure a slower heartbeat rate for the astronaut compared to the rate observed on Earth. This is because time is dilated for the astronaut due to their high velocity.

To calculate the heartbeat rate as measured by the observer in the ship, we can apply the time dilation formula, which states that the observed time (t') is equal to the proper time (t) multiplied by the Lorentz factor (γ), where γ = 1 / sqrt(1 - v^2/c^2). In this case, v is the velocity of the spaceship and c is the speed of light.

(b) However, for an observer at rest on Earth, the heartbeat rate of the astronaut will still be 69 beats/min. This is because the time dilation effect is only experienced by the moving astronaut relative to the observer. From the perspective of the observer at rest on Earth, there is no relative motion between the observer and the astronaut, so there is no time dilation effect. Therefore, the observer on Earth will measure the same heartbeat rate of 69 beats/min as when the astronaut is at rest on Earth.

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The total distance traveled by the mass after a time interval is 4A. Option E is correct.

In simple harmonic motion (SHM), the motion of the mass on a spring repeats itself periodically. The total distance traveled by the mass after a time interval τ depends on the relationship between τ and the period T.

The period T is the time it takes for one complete cycle of the motion. In other words, it is the time for the mass to go from one extreme (maximum displacement) to the other extreme and back again. During this time, the mass covers a distance of 2A, where A is the amplitude of the motion.

Now, let's consider the time interval τ. If τ is equal to or less than the period T, it means that the time interval falls within one complete cycle of the motion. In this case, the mass will cover a distance of 2A, as mentioned earlier.

However, if τ is greater than the period T, it means that the time interval spans multiple cycles of the motion. In each cycle, the mass covers a distance of 2A. Since there will be multiple cycles in the time interval τ, the total distance traveled by the mass will be greater than 2A.

The mass will travel a total distance of 4A after the time interval τ.

Therefore, Option E is correct.

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(a) The slit width is approximately 0.025 mm.

(b) The angle of the first diffraction minimum is approximately 0.050°.

(a) To find the slit width, we can use the formula for the distance between minima in a single slit diffraction pattern:

d = λL / w

Where:

d = distance between minima

λ = wavelength of light

L = distance from slit to screen

w = slit width

Given:

d = 0.35 mm = 0.35 * 10^(-3) m

λ = 550 nm = 550 * 10^(-9) m

L = 40 cm = 40 * 10^(-2) m

Plugging in the values into the formula, we can solve for w:

0.35 * 10^(-3) = (550 * 10^(-9) * 40 * 10^(-2)) / w

Simplifying the equation, we find:

w ≈ 0.025 mm

Therefore, the slit width is approximately 0.025 mm.

(b) The angle of the first diffraction minimum can be calculated using the small angle approximation:

θ = λ / w

Given:

λ = 550 nm = 550 * 10^(-9) m

w = 0.025 mm = 0.025 * 10^(-3) m

Plugging in the values, we find:

θ ≈ (550 * 10^(-9)) / (0.025 * 10^(-3))

Simplifying the equation, we get:

θ ≈ 0.050°

Therefore, the angle of the first diffraction minimum is approximately 0.050°.

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(a)The formula to calculate the impulse experienced by a person is the product of force and time, i.e., Impulse = F * Δt.The passenger has a mass of 69.3 kg and there is an increase in the speed of the car, i.e., acceleration.

The impulse experienced by the passenger during this time can be calculated as follows;I = mΔvHere,m = 69.3 kg,Δv = 5.44 m/sSo, I = 69.3 kg × 5.44 m/sI = 376.992 kg.m/s.

Therefore, the magnitude of the linear impulse experienced by a 69.3 kg passenger in the car during this time is 376.992 kg.m/s.

(b)The formula to calculate average force is given as;F= Impulse / ΔtFrom part (a), Impulse = 376.992 kg.m/sΔt = 0.882 s.

So, the average force experienced by the passenger can be calculated as follows;F = 376.992 kg.m/s / 0.882 sF = 427.05 N.

Therefore, the average force experienced by the passenger is 427.05 N.

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3 is equal to the square root of the product of the charges 1 and 2, determined by using Coulomb's law and setting the forces between the particles equal to each other.

The value of 3 can be determined by using Coulomb's law and setting the forces between the particles equal to each other.

Coulomb's law states that the force between two charged particles is given by:

F = (k * 1 * 2) / r²

Where:

F is the force between the particles

k is the Coulomb constant (approximately 8.988 × 10^9 N·m²/C²)

1 and 2 are the charges of the particles

r is the distance between the particles

Let's denote the original particles as particle 1 and particle 2, and the new particles as particle 3 and particle 4. Given that the forces between the original and new particles are the same, we can write the equation as:

(k * 1 * 2) / r₁² = (k * 3 * 3) / r₂²

Simplifying the equation:

1 * 2 / r₁² = 3² / r₂²

Since the distances between the particles are the same (r₁ = r₂), we can cancel out the terms:

1 * 2 = 3²

Taking the square root of both sides:

3 = √(1 * 2)

Therefore, 3 is equal to the square root of the product of the charges 1 and 2.

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The current when the charge is 24% of its maximum value is 24.76 A.

From the question above, Capacitance, C = 4.7μF

Resistance, R = 2.9Ω

Battery emf, ε = 81V

Percentage charge = 24%

The time constant of the circuit is given as:RC = 2.9 Ω × 4.7μF = 0.01363 s

The equation for charge on a capacitor is given by:

q = Cε(1 − e−t/RC)

We need to find the current when the charge is 24% of its maximum value. The charge at any time t can be found from the above equation.

At maximum charge, the capacitor will be fully charged. Hence the maximum charge, q max is given by:

q max = Cε = 4.7 μF × 81 V = 381.7 μC

When the charge is 24% of its maximum value:q = 0.24 × q max = 0.24 × 381.7 μC = 91.6 μC

The value of RC is given as 0.01363 s. Let the current when the charge is 24% of its maximum value be I.

At the time the charge on the capacitor is 24% of its maximum value, the current is given by the derivative of the above equation:

I = dq/dt = (ε/R) e^(-t/RC)

On substituting the values, we get:I = 24.76 A

Therefore, the current when the charge is 24% of its maximum value is 24.76 A.

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The angular difference between true north and magnetic north is known as the Magnetic Declination.

Angle of magnetic declination varies depending on where you are on the Earth's surface, as well as the time and year. The difference between magnetic north and true north is known as magnetic declination, which is measured in degrees. Magnetic declination can be found using a compass and a map or by using online magnetic declination calculators. This information is important for accurate navigation and orientation, as it allows you to adjust your compass heading to account for the difference between magnetic north and true north.

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If the amplitude (the angle) of a pendulum changes slightly, the period of the pendulum remains nearly unchanged. The period of a pendulum is directly proportional to the square root of its length. If the length of a pendulum changes, the period will also change. The mass of a pendulum does not affect its period.

a) If the amplitude (the angle) of a pendulum changes slightly, the period of the pendulum remains nearly unchanged. The period of a simple pendulum (under small angles) is primarily determined by its length, not by the amplitude. As long as the amplitude remains within the small-angle approximation, the period remains constant.

b) The period of a pendulum is directly proportional to the square root of its length. If the length of a pendulum changes, the period will also change. According to the equation for the period of a simple pendulum:

T = 2π√(L/g)

where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. As the length of the pendulum increases, the period also increases, and vice versa.

c) The mass of a pendulum does not affect its period. The period of a simple pendulum is solely determined by its length and the acceleration due to gravity. The mass of the pendulum does not appear in the equation for the period, so changing the mass does not change the period.

To experimentally verify the relation between the period and length of a pendulum, you can perform the following steps:

Set up a simple pendulum by suspending a mass (bob) from a fixed point using a string or rod.

Measure the length of the pendulum, which is the distance from the point of suspension to the center of mass of the bob.

Use a stopwatch or timer to measure the time it takes for the pendulum to complete one full swing (i.e., from one extreme to the other and back).

Repeat the measurement for different lengths of the pendulum, ensuring that the amplitude of the swings remains small.

Record the lengths of the pendulum and the corresponding periods.

Plot a graph of the period (T) versus the square root of the length (√L).

The graph should show a linear relationship, indicating that the period of the pendulum is proportional to the square root of its length.

Calculate the slope of the graph, which should be close to 2π√(1/g), where g is the acceleration due to gravity.

Compare the experimental results with the theoretical equation T = 2π√(L/g) to verify the relation between the period and length of the pendulum.

By conducting this experiment and analyzing the data, you can demonstrate the relationship between the period and length of a simple pendulum.

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Given that a radioactive nucleus has a half-life of 5 × 108 years. Let's suppose that a sample of rock (say, in an asteroid) solidified right after the solar system formed.

Then we have to calculate the fraction of the radioactive element that should be left in the rock today.

Half-life (t₁/₂) of a radioactive substance is defined as the time taken by a substance to reduce to half its initial value.

This is given by the formula,N(t) = N₀(1/2)⁽ᵗ/ᵗ₁/₂⁾ Where,N(t) = Final quantity N₀ = Initial quantity t = Time elapsed t₁/₂ = Half-life period.

We know that the half-life (t₁/₂) of the radioactive nucleus is 5 × 108 years. Hence, the fraction of the radioactive element left can be calculated as follows:After the first half-life, the quantity of the radioactive element left would be N₀/2.

After the second half-life, it would be N₀/4 and so on.

Thus, the general formula for the quantity of the radioactive element left would be,N = N₀ (1/2)n Where n is the number of half-lives elapsed.

The fraction of the radioactive element left is given as,N/N₀ = (1/2)n.

Now, we can substitute the values in the above formula.

Let's suppose that one-half-life is 5 × 108 years. Then the age of the rock would be approximately 4.6 × 109 years (age of the Solar System).

Thus, the number of half-lives elapsed would be given by,n = (time elapsed)/(half-life)n = (4.6 × 109)/(5 × 108) = 9.2.

After 9.2 half-lives, the fraction of the radioactive element left would be,N/N₀ = (1/2)⁹.²≈ 0.00077 ≈ 7.7 × 10⁻⁴.

Thus, approximately 0.077% (7.7 × 10⁻⁴) of the radioactive element should be left in the rock today.

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The ball strikes the ground approximately 50.4 meters horizontally from the base of the building.

Step 1: Using the given information, we can calculate the horizontal distance traveled by the ball using the equation for horizontal motion:

Horizontal distance = Initial velocity * Time

Given that the initial velocity is 8.40 m/s and the time is 6.00 seconds, we can substitute these values into the equation:

Horizontal distance = 8.40 m/s * 6.00 s = 50.4 meters

Therefore, the ball strikes the ground approximately 50.4 meters horizontally from the base of the building.

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Electric potential between the two terminals of the solenoid at t = 2 µs is approximately 44.43 V. Electric potential  refers to the amount of electric potential energy per unit charge at a specific point in an electric field.

Electric potential is denoted by the symbol V and is measured in volts (V).                                                                                                                                                                                                                    Potential at t = 2 µs, we can use the fact that potential across an inductor is proportional to the rate of change of current, i.e., V α di/dt or V₁/V₂ = (di/dt)₁/(di/dt)₂, where V₁ and V₂ are potentials at two different times t₁ and t₂ respectively.                                                                                                                                                                                                        We can take V₂ as 200 V (potential at t = 3 µs) and V₁ is to be found out for t₁ = 2 µs.                                                                                  We know that the current changes from 0 to 20 mA in 3 µs.                                                                                                            Average rate of change of current during this time is, di/dt = (20 x 10⁻³ A - 0)/3 x 10⁻⁶ s= 20/3 A/µsAt t = 2 µs, time duration from t = 0 is 2 µs.                                                                                                                                                                                                 The change in current during this time will be,i = di/dt x t = (20/3 A/µs) x 2 µs = 40/3 mASo, current at t = 2 µs is I = 40/3 mA = 13.33 mA (approx).                                                                                                                                                                             Now, we can find potential at t = 2 µs, usingV₁/V₂ = (di/dt)₁/(di/dt)₂V₁/200 = (13.33 x 10⁻³ A/µs)/ (20/3 A/µs)V₁ = (13.33 x 10⁻³ A/µs) x (200/20/3) V = 44.43 V (approx).                                                                                                                                      Therefore, electric potential between the two terminals of the solenoid at t = 2 µs is approximately 44.43 V.

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Pressure of a oil ( specific gravity = 0.86) at any section of a pipe is 2 bar. Pressure head is 23.71 m (Option A).

The pressure head is the vertical distance that a fluid column would rise due to the pressure at a given point. It is calculated by dividing the pressure by the product of the acceleration due to gravity (g) and the specific weight of the fluid (γ).

Let's assume the density of water is 1000 kg/m³. The density of the oil can be calculated as follows:

Density of oil = Specific gravity * Density of water = 0.86 * 1000 kg/m³ = 860 kg/m³

Now, to calculate the pressure head, we need to convert the pressure from bar to pascals (Pa) since pressure is typically measured in SI units.

1 bar = 100,000 Pa

Given that the pressure at the section of the pipe is 2 bar, the pressure can be converted to pascals as follows:

Pressure = 2 bar = 2 * 100,000 Pa = 200,000 Pa

Next, we can calculate the pressure head using the formula:

Pressure head = Pressure / (Density of oil * Acceleration due to gravity)

Acceleration due to gravity (g) is approximately 9.8 m/s².

Pressure head = 200,000 Pa / (860 kg/m³ * 9.8 m/s²) ≈ 23.71 meters

Therefore, the correct answer is 23.71 m.

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To calculate the magnetic field at a point located at a distance of 1 mm from the center of the rotating plastic disc, we can use the Biot-Savart law.

The Biot-Savart law states that the magnetic field at a point due to a current element is proportional to the current, the element length, and inversely proportional to the square of the distance.

Given that the disc is rotating with an angular velocity of 30 rad/s, we can consider the rotating plastic disc as a current loop with a current flowing along its circumference. The current in this case is given by the surface density multiplied by the area enclosed by the loop.

The surface density is given as A = 13 C/m^2, and the area enclosed by the loop is the difference between the areas of the outer and inner radii, which can be calculated as π(R^2 - R_0^2).

Using the Biot-Savart law, the magnetic field at a distance of 1 mm (0.001 m) from the center can be calculated as:

B = (μ_0 / 4π) * (I * dL) / r^2

where μ_0 is the permeability of free space (4π × 10^-7 T·m/A), I is the current, dL is the current element length, and r is the distance from the point to the current element.

Substituting the given values, we can calculate the magnetic field at the given point.

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Convert the initial velocity from km/h to m/s:

u = 87 km/h

u= 87 × (5/18) m/s

u= 24.17 m/s.

Determine the final velocity: v = 0 m/s.

Calculate the displacement: s = 0.92 m.

Use the formula v² = u² + 2as to find the average acceleration during the collision.

Substituting the values: 0² = (24.17)² + 2a(0.92)

Solve for a: a = -(24.17)² / (2 × 0.92) ≈ -315.11 m/s².

The negative sign indicates deceleration or negative acceleration.

Express the acceleration in terms of 'g' (acceleration due to gravity).

Given 1 g = 9.80 m/s², we can convert the acceleration.

Calculate a in terms of 'g': a = (-315.11 m/s²) / 9.80 m/s²/g ≈ -32.16 g's.

The negative sign still indicates deceleration.

Therefore, the average acceleration of the driver during the collision is approximately -315.11 m/s² or -32.16 g's.

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