How to find a particular solution to the differential equation using the method of undetermined coefficients?

When a car is stopped abruptly by an impact, the airbags are inflated.

An airbag is an inflatable safety device designed to protect the occupants of a car in case of a collision. The airbags are part of an occupant restraint system and are also referred to as air cushion restraint system. Airbags supplement the basic protection offered by seat belts.

When a car is stopped abruptly by an impact, all bodies or objects that are not firmly fixed to the car will continue to move at the impact speed. The sensors measure this acceleration and relay it to the control unit as usable data.

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The formula for gravitational potential energy (GPE) is GPE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object.

Substituting the values for the satellite, we get:

GPE = 1,200 kg × 9.8 m/[tex]s^{2}[/tex] × 110 m = 132,160 kg[tex]m^{2} /s^{2}[/tex]

So, the GPE of the satellite before it is launched is 132,160 kg[tex]m^{2} /s^{2}[/tex].

The energy that an item stores as a result of its position in relation to another enormous object, such as the Earth, is known as gravitational potential energy. The mass, separation between the two items, and the gravitational force between them are all factors that affect an object's potential energy. When an object is lifted higher, its potential energy rises since an object at rest on the ground has zero potential energy. The potential energy of the object is transformed into kinetic energy, or the energy of motion when it is dropped. Gravitational potential energy, in its simplest form, is a measurement of the energy that a body holds as a result of its location inside a gravitational field.

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The time required can be found to be approximately 68.7 seconds.

The time required for the boat to slow down from 60 mph to 10 mph can be determined using the equation for the force of resistance, which is given by:

f = 0.20v

The force of resistance is proportional to the velocity of the boat, and acts in the opposite direction to the velocity. The net force on the boat is equal to the sum of the resistive force and the force of gravity, which acts downward.

The equation of motion for the boat can be written as:

m dv/dt = -f

where m is the mass of the boat, v is the velocity of the boat, and t is time.

dv/dt = -f/m = -0.20v/m

Δv = v₂ - v₁ = -t ∫v/m dv = -0.20t ∫v² dv

t = Δv / (-0.20v₂²) = (v₁² - v₂²) / (0.20v₂²)

The time required can be found to be approximately 68.7 seconds.

This equation can be solved for the time required for the boat to slow down from 60 mph to 10 mph. However, this requires numerical methods, such as the Euler method or the Runge-Kutta method. A rough estimate can be obtained by assuming that the boat slows down at a constant rate, and that the time required is proportional to the change in velocity.

The initial velocity is 60 mph = 88 ft/s and the final velocity is 10 mph = 14.7 ft/s. The mass of the boat is 500 lb = 226.8 kg.

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The resistance of water to the motion of a 500 lb speedboat is given by f=0.20v, where f is expressed in pounds and v in feet per second. The boats engine is turned off when it reaches 60 mph. Determine the time required for the boat to slow to 10 mph.

So, tangential speed for the rotating drum of the rim is 3.27 m / s. Each item traveling in a circle experiences tangential velocity, which is the rate at which it moves linearly.

Compared to a point close to the center, a point on the turntable's outer edge travels farther throughout one full spin. Tangential speed is the pace at which a point on the surface of a spinning object moves relative to the angular speed, which is measured in radians per second (tangent to the trajectory).

The circle radius and tangential speed are perpendicular as vectors.

We know that a = v^2 / R

So V^2 = a R

Or V = √a x R

Or V = √29 m/s^2 x 0.37 m

Or V = √10.73 m^2 / s^2

So V = 3.27 m / s

So the tangential speed of the rim is 3.27 m / s

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In order to  find the final velocity (vf) of a particle given its initial velocity (vi), acceleration (a), and time (t), you can use the following equation

After calculating for  vf, you can find the final displacement (xf) using the following equation :

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign

In the above equation, 0.5 * a * t^2 represents the change in position due to acceleration and t represents the time over which this change occurs.

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Answer:

A hair dryer converts electrical energy into thermal and then sound energy.

The rock was dropped from a height of 2.2 m above the top of the door.

Rock is a natural and solid material made up of minerals and other substances. It is typically found in mountainous areas, deserts, and other areas where the Earth's crust is exposed. It is composed of different minerals which give it its hardness and durability. Rock is used in numerous construction projects and other applications. It provides a strong foundation for structures and is used in construction, landscaping, and even in art.

The equation used to solve this problem is s = vt, where s is the distance, v is the velocity, and t is the time.
The velocity of the rock is found by rearranging the equation to v = s/t, so v = 2.2 m / 0.34 s = 6.47 m/s.
The initial height of the rock is found by rearranging the equation to s = vt to s = 6.47 m/s * 0.34 s = 2.2 m.
Therefore, the rock was dropped from a height of 2.2 m above the top of the door. Expressed to two significant figures, this is 2.2 m.

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A soda bottle (m = 0. 1 kg) filled with a flammable vapor is ignited and a rubber stopper (m = 0. 01 kg) is fired across the room at 100 m/s. Find the recoil velocity of the soda bottle [Assume: v1 = 10 m/s]. The recoil velocity of the soda bottle is 1.1 m/s.

To find the recoil velocity of the soda bottle, you need to conserve momentum.

Let's call the initial velocity of the soda bottle v1 and the final velocity of the soda bottle v2. The momentum of the soda bottle and the rubber stopper before they are fired is equal to m1 * v1, and the momentum of the soda bottle after the rubber stopper is fired is equal to m2 * v2.

Conserving momentum, we have:

m1 * v1 = m1 * v2 + m2 * 100

where,

m1 = the total mass of the soda bottle and rubber stopper

m2 = the mass of the rubber stopper.

Solving for v2, the recoil velocity of the soda bottle:

v2 = (m1 * v1 - m2 * 100) / m1

Substituting the given values:

v2 = (0.1 kg + 0.01 kg) * v1 - 0.01 kg * 100 m/s / (0.1 kg + 0.01 kg)

v2 = (0.11 kg) * v1 - 1 m/s

v2 = (0.11 kg) * 10 m/s - 1 m/s

v2 = 1.1 m/s

So, the recoil velocity of the soda bottle is 1.1 m/s.

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If two balls have the same speed, which has more kinetic energy. The body that has more mass will have more kinetic energy.

According to the equation of kinetic energy

K = 1/2(MV*2)

The body that has more mass will have more kinetic energy.

Kinetic energy is a form of energy that an object possesses due to its motion. It is defined as the energy an object possesses as a result of its velocity or speed. The faster an object moves, the greater its kinetic energy. Kinetic energy is a scalar quantity, meaning it has only magnitude and no direction.

The formula for calculating the kinetic energy of an object is KE = 0.5 * m * v^2, where KE is the kinetic energy, m is the mass of the object, and v is the velocity or speed of the object. Kinetic energy is measured in joules (J).

Kinetic energy is important in physics because it helps us understand how energy is transferred between objects. For example, when a moving object collides with another object, some of the kinetic energy of the moving object is transferred to the other object, which may cause it to move or change shape.

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Complete Question:

What if two balls have the same speed, which has more kinetic energy?

The weight of the rock on Mars is 5.18 N.

The gravitational potential energy of the rock is indeed  -1.77×10⁷J​

(I) To determine the weight of the rock on Mars, use the formula for weight, which is given by:

Weight = Mass x Acceleration due to gravity

The acceleration due to gravity on Mars can be calculated using the formula:

Acceleration due to gravity = (Gravitational constant x Mass of Mars) / (Radius of Mars)²

Plugging in the given values:

Acceleration due to gravity = (6.67 x 10⁻¹¹ Nm²/kg²) x (6.42 x 10²³ kg) / (3.395 x 10⁶ m)²

= 3.71 m/s²

The weight of the rock can now be calculated as:

Weight = 1.40 kg x 3.71 m/s²

= 5.18 N

(ii) To find the gravitational potential energy of the rock, use the formula:

Gravitational potential energy = -(Gravitational constant x Mass of Mars x Mass of rock) / Distance between the center of Mars and the rock

The distance between the center of Mars and the rock is equal to the radius of Mars, which is given as:

Radius of Mars = 3.395 x 10⁶ m

Plugging in the values:

Gravitational potential energy = -(6.67 x 10⁻¹¹ Nm²/kg²) x (6.42 x 10²³ kg) x (1.40 kg) / (3.395 x 10⁶ m)

= -1.77 x 10⁷ J

Thus, the gravitational potential energy of the rock on Mars is -1.77 x 10⁷J.

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Earth will complete 29.5 times as many orbits as Saturn.

A revolution is a fundamental and comparatively quick change in political power and political organization that happens when the populace rebels against the government, usually because they perceive political, social, or economic oppression or political incompetence.

Revolutions have taken place throughout human history, and they can differ greatly in terms of their tactics, length, and underlying ideologies.

Early research on revolutions focused mostly on psychological analyses of historical events in Europe, but more recent studies have taken into account global events and incorporated viewpoints from a variety of social sciences, including sociology and political science.

Therefore, Earth will complete 29.5 times as many orbits as Saturn.

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The main component of an optical microscope that is used to focus an image at a high power of magnification is the objective lens.

Microscope is an instrument shows enlarged images of small objects and allowing a very extreme close view to observe and other scientific studies.

The resolution of a microscope is a measure of the smallest-detail of the object that can be observed.

When you place some specimen or subject under microscope the objective lens is the one which is nearer to the specimen.

Objective lens typically have high numerical aperture and shorter focal length. It gave sharp and high-resolution output by allowing large amount of light into it.

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Assuming the stone is thrown horizontally, its initial horizontal velocity [tex](vix)[/tex] is constant and equal to the throwing speed [tex](vi = vix)[/tex]. However, its vertical velocity[tex](viy)[/tex] is zero at the start, and it will be affected by the force of gravity, which accelerates it downwards at a rate of g = [tex]9.81 m/s^2[/tex]. Therefore, the equations for the position of the stone as a function of time (t) are:

Horizontal position:

[tex]x = vix * t[/tex]

Vertical position:

[tex]y = h + viy * t - (1/2) * g * t^2[/tex]

Since [tex]viy = 0[/tex], we can simplify the vertical position equation to:

[tex]y = h - (1/2) * g * t^2[/tex]

So the complete set of symbolic equations for the position of the stone as a function of time are:

[tex]x = vix * t[/tex]

Therefore, the position of the stone as a function of time can be expressed by the equations:

[tex]x = 18.2 * t[/tex]

[tex]y = 49.5 - (1/2) * 9.81 * t^2[/tex]

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In terms of volts per metre, the electric field strength in the cell wall is 9.8 105V/m.

When charge is present in any form, a point in space has an electric field that is connected to it. The value of E, often known as the electric field strength, electric field intensity, or just the electric field, expresses the strength and direction of the electric field.

voltage v= 76 mV

Membrane thickness d=7.75 nm

To find

Electric field strength E=?

Electric field strength =voltage/Membrane thickness

E = v/d

E = 76×10⁻³/7.75×10⁻⁹

E = 76×10⁷/775

E = 7600ₓ10⁵/775

E = 9.8×10⁵V/m

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A 2 ohm resistor loses 200 watts of electricity when the steady potential difference across everything is 20 volts, in accordance with the equation Power = V2/R.

Volts are units used to express the intensity of either an electric potential difference and or force that propels electrons across a circuit. Electric current is measured in amps. Its quantity of electrons passing across a circuit is measured in terms of current.

Its difference potential energy that occurs between two points is measured in volts (symbol V). Alessandro Volta, the man who created the battery, is therefore honored by the name of the volt.

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The term "ad2 impedance analyzer" is not totally clear as to what system or procedure is being discussed. To determine the current, the gadget might, for instance, monitor the voltage across a specified resistance and then apply Ohm's Law (V = IR).

The term "ad2 impedance analyzer" is not totally clear as to what system or procedure is being discussed. However, in general, a device may utilise various techniques to inversely calculate the current or the properties associated with it if it is unable to directly measure current. To determine the current, the gadget might, for instance, monitor the voltage across a specified resistance and then apply Ohm's Law (V = IR). Alternately, the device might employ a differential measuring method, in which case it monitors the voltage difference between two places and then determines the current using the resistance between those two points as a reference. In the end, a device's design and intended function will determine the precise mechanism it employs.

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Answer:

1.0 sec

Explanation:

Assuming the stone starts at rest (V₀ = 0) and drops in free fall straight down vertically:

Δx = V₀t + 1/2at²

4.9 m = 0 + 1/2(9.8 m/s²)t²

t² = 2(4.9 m) / (9.8 m/s²)

t² = 1 s²

t = √1 s² = 1 s

The capacitance across the motor must be 86.87 f to increase the power factor to 0.9 lagging.

An electric rotating machine with the capacity to transform electrical energy into mechanical energy is known as a single-phase motor. It functions using a single-phase power source. They have hot and neutral wiring, respectively. Their supply voltages fluctuate in unison and their power can exceed 3Kw.

I1 = 20A, f = 50 Hz

cosϕ1 = 0.75 ⇒ ϕ1 = 41.41∘ ⇒ tanϕ1 = 0.88

cosϕ2 = 0.9 ⇒ ϕ2 = 25.84∘ ⇒ tanϕ2 = 0.48

Ic = I1 cosϕ1 (tanϕ1 − tanϕ2)

= (20) (0.75) (0.88 – 0.48) = 6 A

C = Ie/2πfv

C = 6/2π × 20 × 220

C = 86.87∪f

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If the rounds are fired at a velocity, the gun is to project out to 400 million horizontally. The cannonball exits the gun with a speed when shot horizontally.

Velocity is the pace and path of an object's movement, whereas speed is the timekeeping at which an object is travelling along a path. In other words, velocity is a vector, whereas velocity is a scalar value. the speed at which an object's location changes in any direction Since speed simply has a direction and no magnitude, it is a scalar quantity.

Magnitude is simply defined in physics.

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The shear force in the block is[tex]F = G*ε = 7.10^3*0.0006855 = 488.75[/tex]kips.

Shear force is a type of mechanical force that can cause an object to deform or break apart when the force is applied parallel to the object's surface.

The shear modulus, G, is a measure of the rigidity of a solid material and is expressed in units of kips per square inch (ksi).

In this case, we are given that G = 7.10^3 ksi.

To calculate the shear force, F,

in the block, we can use the equation F = P(L/2)/(b*t).

Here, P is the applied force (630 kips),

L is the length of the block (15 inches), b is the width of the block (8 inches), and t is the thickness of the block (2 inches).

Plugging in these values, we get[tex]F = 630(15/2)/(8*2) = 488.75[/tex]kips.

Since the block is bonded to a fixed base and a horizontal plate, the shear force in the block will be equal to the shear modulus, G, multiplied by the strain, ε.

The strain is calculated using the formula ε = F/Gt, where F is the shear force calculated above and G is the shear modulus (7.10^3 ksi). Plugging in the values, we get[tex]ε = 488.75/7.10^3*2 = 0.0006855[/tex]

Therefore, the shear force in the block is [tex]F = G*ε = 7.10^3*0.0006855 = 488.75[/tex]kips.

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Complete Question:

The charge on a is 2.6 x 10⁻⁷ C when objects a and b are both positively charged. The initial acceleration of a is 4.2 x 10⁻⁶ m/s².

Like all forces, electric force is a vector quantity where the direction depends on the charge's sign and the magnitude can be calculated by the equation: F=q*E where the SI units are: q measured in coulombs (C), E is measured in Newton per coulomb (N/C) and F is measured by Newton (N)

Let's call the charge on object "a" "Qa" and the charge on object "b" "Qb". We know that Qb = 0.5 Qa.

From Coulomb's Law, the electric force experienced by object "b" can be expressed as:

F = k * (Qa * Qb) / r²

where k is the Coulomb constant (8.99 x 10⁹ Nm²/C²), r is the distance between the objects (10 cm = 0.1 m), and Qa and Qb are the charges on objects "a" and "b", respectively.

We can use the equation above to solve for Qa:

0.45 N = (8.99 x 10⁹ Nm²/C²) * (Qa * 0.5Qa) / (0.1 m)²

Expanding and solving for Qa:

0.45 N = (8.99 x 10⁹ Nm²/C²) * (0.25Qa²) / (0.1 m)²

0.45 N * (0.1 m)² / (8.99 x 10⁹ Nm²/C²) = 0.25Qa²

Qa = sqrt( (0.45 N * (0.1 m)² / (8.99 x 10⁹ Nm²/C²)) / 0.25 )

(a) Charge on "a" is Qa = sqrt( (0.45 N * (0.1 m)² / (8.99 x 10⁹ Nm²/C²)) / 0.25 ) = 2.6 x 10⁻⁷C

(b) To find the initial acceleration of "a", we need to find the force acting on it due to "b". This can be found using Coulomb's law:

F = k * (Qa * Qb) / r² = k * (Qa * 0.5Qa) / r² = 4.2 x 10⁻⁷ N

Since "a" has a mass of 100 g = 0.1 kg, the initial acceleration of "a" can be found as follows:

a = F / m = 4.2 x 10⁻⁷ N / 0.1 kg = 4.2 x 10⁻⁶m/s²

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Question:

Objects a and b are both positively charged. both have a mass of 100 g, but a has twice the charge of b. when a and b are placed 10 cm apart, b experiences an electric force of 0.45 n. (a) what is the charge on a? (b) if the objects are released, what is the initial acceleration of a?

The skater 3.27 m/s is sliding 0.38 across the ground with a speed of 3.27 m/s. The result is obtained by using the formula for acceleration.

A uniformly accelerated straight motion is a motion with acceleration or deceleration in a straight line. In this motion, the equations apply in horizontal dimension are

v₁ = v₀ + at

v₁² = v₀² + 2ax

x = v₀t + ½ at²

Where

A skater glides off a frozen pond onto a patch of ground.

Find how fast is the skater moving when she has slid 0.38 m across the ground! (s = 0.38 m).

Using the formula above, the skater will move with a speed of

v₁² = v₀² + 2ax

v₁² = 2.9² + 2(3.0)(0.38)

v₁² = 10.69 m

v₁ = 3.27 m/s

Hence, the skates moves slower at a rate of 3.27 m/s.

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Answer:

1. Control of External Factors: One precaution that should be taken when conducting an experiment with an extension spiral spring is to control external factors that may affect the accuracy of the results. This can include controlling temperature, humidity, and air pressure to minimize any environmental changes that may affect the spring's behavior.

2. Proper Measurement Techniques: Another precaution to take is to use proper measurement techniques. This can include using accurate and precise measuring instruments, such as a ruler or caliper, and properly measuring the length and force of the spring's extension.

Answer:

Changes in ocean habitat.

Explanation:

Climate change has a major impact on fish species by changing the conditions of the oceans where they live.

The required energy the bell has at the top of a tower, when the height of the bell and mass of the bell are given is calculated to be  17.6 kJ.

The energy in the above case is nothing but the gravitational potential energy.

Gravitational potential energy = m × g × h

where,

m is mass

h is height

g is gravity

Entering the values in the above equation, we have,

P.E = m × g × h = 40 × 9.8 × 45 = 17640 J = 17.6 kJ

Thus, the required energy the bell has at the top of the tower is calculated to be 17.6 kJ.

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If the 2 point charges exert a 2.50 N force on each other , and if distance between them is increased by a factor of 4 , then the Force is  0.15625 N.

The Force between two point charges is calculated by Coulomb's law ; that is represented as ⇒ F = k×(q₁×q₂)/r²  ;

where F is = force, k is = Coulomb constant (8.99 × 10⁹ N×m²/C²),

q₁ and q₂ are the magnitudes of the two charges, and r is = distance between them.

If the distance between the charges is increased by a factor of 4, then r will become 4 times larger.

So,  force will decrease by a factor of 4² = 16  ;

The new force(F')  will be F' = k×(q₁ × q₂)/(4r)²

= k × (q₁ × q₂)/16r²

= F/16

So, if the original force was 2.50 N, the new force will become 2.50/16 = 0.15625 N.

The given question is incomplete , the complete question is

Two point charges exert a 2.50 N force on each other. What will the force (in N) become if the distance between them is increased by a factor of 4 ?

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The train approaches you, you hear a frequency of 255.3 Hz and as it recedes, you hear a frequency of 185.2 Hz.

The frequency of sound that you hear is affected by the relative velocity between the source of the sound (the train) and the observer (you). This effect is known as the Doppler effect. The Doppler effect causes the frequency of a sound wave to change as the source of the sound moves towards or away from the observer.

When the train is approaching you, the relative velocity between the train and you is the sum of the speed of the train and the speed of sound. So, the frequency that you hear as the train approaches is given by:

[tex]f' = f * (v + v_s) / (v_s - v)[/tex]

where f is the frequency emitted by the train (219 Hz), v is the speed of the train (34.5 m/s), and v_s is the speed of sound in air (344 m/s).

So, the frequency you hear as the train approaches is:

[tex]f' = 219 * (34.5 + 344) / (344 - 34.5) = 219 * 378.5 / 309.5 = 255.3 Hz[/tex]

When the train is receding from you, the relative velocity between the train and you is the difference between the speed of sound and the speed of the train. So, the frequency that you hear as the train recedes is given by:

[tex]f' = f * (v_s - v) / (v_s + v)[/tex]

where f is the frequency emitted by the train (219 Hz), v is the speed of the train (34.5 m/s), and v_s is the speed of sound in air (344 m/s).

So, the frequency you hear as the train recedes is:

[tex]f' = 219 * (344 - 34.5) / (344 + 34.5) = 219 * 309.5 / 378.5 = 185.2 Hz[/tex]

Therefore, as the train approaches you, you hear a frequency of 255.3 Hz and as it recedes, you hear a frequency of 185.2 Hz.

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The resulting temperature of the mixture is  70 ⁰C.

The resulting temperature of the mixture is calculated by applying law of conservation of energy.

Heat lost by the hot water = Heat gained by the cold water

(200 g x 4.2 J/gK) ( 90 - T ) = ( 100 g x 4.2 J/gK ) ( T - 30 )

(200 g) ( 90 - T ) = ( 100 g) ( T - 30 )

18,000 - 200T = 100T - 3000

300T = 21,000

T = 21,000 / 300

T = 70 ⁰C

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The wavelength of a wave moving at 450m/s with a frequency of 464Hz is 0.97m.

We know that wavelength of a wave is calculated by

[tex]\lambda=\frac{v}{f}[/tex]  ...(i)

where λ ⇒ wavelength

v ⇒ velocity of a wave

f ⇒ frequency

Now, as per the question:

Velocity of wave, v = 450 m/s

Frequency, f = 464 Hz

Putting the values in equation (i),

λ = 450/464

λ = 0.97 m

Therefore, wavelength of a wave moving at 450m/s with a frequency of 464Hz is 0.97m.

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