Young, et al., Sears and Zemansky’s University Physics: with Modern Physics 13th Ed) When a batted baseball moves with air drag, does it travel (a) a greater horizontal distance while climbing up to its maximum height, (b) a greater horizontal distance while descending from its maximum height, or (c) the same horizontal distance for both? Explain your answer.

Three objects A, B, and C are moving as shown below: we need to find the x and y components of the net momentum of the particles if we define the system to consist of:(a) A and C(b) B and C(c) All three objects.

So, let's calculate the momentum of each object:P = m × v Where,P = Momentum of an objectm = Mass of an objectv = Velocity of an object(a) Momentum of objects A and C can be calculated as follows:Momentum of object A = m1v1 = 4 kg × 4 m/s = 16 kg m/s Momentum of object C = m2v2 = 3 kg × (-5 m/s) = -15 kg m/s Net momentum of objects A and C = m1v1 + m2v2 = 16 kg m/s - 15 kg m/s = 1 kg m/sTherefore, the x and y components of the net momentum of the particles (A and C) are 1 kg m/s and 0 kg m/s, respectively.(b) Momentum of objects B and C can be calculated as follows: Momentum of object B = m1v1 = 2 kg × 5 m/s = 10 kg m/s Momentum of object C = m2v2 = 3 kg × (-5 m/s) = -15 kg m/s Net momentum of objects B and C = m1v1 + m2v2 = 10 kg m/s - 15 kg m/s = -5 kg m/s. Therefore, the x and y components of the net momentum of the particles (B and C) are -5 kg m/s and 0 kg m/s, respectively.(c) Net momentum of all the three objects can be calculated as follows: Net momentum = (m1v1 + m2v2 + m3v3)Where,m1 = 4 kg, v1 = 4 m/sm2 = 3 kg, v2 = -5 m/sm3 = 2 kg, v3 = 5 m/sNet momentum = (4 × 4) + (3 × (-5)) + (2 × 5) = 16 - 15 + 10 = 11 kg m/s. Therefore, the x and y components of the net momentum of all the three objects are 11 kg m/s and 0 kg m/s, respectively.

The x and y components of the net momentum of the particles are as follows:(a) For objects A and C: 1 kg m/s and 0 kg m/s, respectively.(b) For objects B and C: -5 kg m/s and 0 kg m/s, respectively.(c) For all the three objects: 11 kg m/s and 0 kg m/s, respectively.The x and y components of the net momentum of the particles for each system of objects A and C, B and C, and all three objects are calculated as 1 kg m/s and 0 kg m/s, -5 kg m/s and 0 kg m/s, and 11 kg m/s and 0 kg m/s, respectively

Therefore, the x and y components of the net momentum of the particles for different systems of objects are different as we have seen above.

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The sound intensity is approximately 158,490,000,000.

The sound intensity can be calculated using the formula:

[tex]\[ I = I_0 \times 10^{(dB/10)} \][/tex]

Where, I is the sound intensity, I_0 is the threshold of hearing, and dB is the decibel level above the threshold.

In this case, the decibel level above the threshold is given as

[tex]\( 112 \, dB \)[/tex] and the threshold of human hearing is [tex]\( I_0 = 1.00 \)[/tex].

Substituting the values into the formula, we have:

[tex]\[ I = 1.00 \times 10^{(112/10)} \][/tex]

To calculate this, we divide 112 by 10 to get 11.2. Therefore, the sound intensity is:

[tex]\[ I = 1.00 \times 10^{11.2} \][/tex]

Now, we can simplify this equation by using the fact that [tex]\( 10^{11.2} \)[/tex] is equal to [tex]\( 10^{10} \times 10^{1.2} \)[/tex].

The value of [tex]\( 10^{10} \)[/tex] is [tex]\( 10,000,000,000 \) (10 billion),[/tex] and the value of [tex]\( 10^{1.2} \)[/tex] is approximately [tex]\( 15.849 \).[/tex]

Multiplying these values, we get:

[tex]\[ I = 1.00 \times 10,000,000,000 \times 15.849 \][/tex]

Therefore, the sound intensity is:

[tex]\[ I \approx 158,490,000,000 \][/tex]

So, the sound intensity is 158,490,000,000.

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Answer:

The correct answer is: It would correspond to an infinite instantaneous acceleration.

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The formula to calculate the heat transfer coefficient (h) using the Nusselt number is Nu = 0.023 * Re^0.8 * Pr^0.3

To determine the heat transfer coefficient for both cases of heating, we can use the concepts of forced convection and apply appropriate correlations.

Case 1: Condensing Steam on Outer Surface (Uniform Surface Temperature)

In this case, the heat transfer coefficient can be determined using the Nusselt number correlation for forced convection over a tube, such as the Dittus-Boelter equation. The Dittus-Boelter equation is valid for fully developed turbulent flow and can be expressed as:

Nu = 0.023 * Re^0.8 * Pr^0.4

where Nu is the Nusselt number, Re is the Reynolds number, and Pr is the Prandtl number.

The Reynolds number can be calculated using the mean velocity (V) and tube diameter (D) as:

Re = (ρ * V * D) / μ

where ρ is the air density and μ is the air dynamic viscosity.

The Prandtl number can be evaluated at 400 K using the air properties.

Once we have the Nusselt number, we can determine the heat transfer coefficient (h) using the equation:

Nu = h * (D / λ)

where λ is the thermal conductivity of air.

Case 2: Electric Resistance Heating (Uniform Surface Heat Flux)

In this case, we can use the Dittus-Boelter equation modified for uniform heat flux, given as:

Nu = 0.023 * Re^0.8 * Pr^0.3

The subsequent steps are the same as in Case 1 to calculate the heat transfer coefficient (h) using the Nusselt number.

By evaluating the heat transfer coefficients for both cases, we can compare the effectiveness of condensing steam and electric resistance heating for transferring heat to the flowing air in the tube.

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The distance at which the sound waves have an intensity of 0.02 W/m² is 400m.

a) Sound intensity can be defined as the amount of sound energy that passes through a unit area per unit time. It is measured in watts per square meter (W/m²).

The sound intensity, I, can be determined using the following equation: Where P is the power of the sound source and r is the distance from the sound source to the receiver. We can substitute the given values to get: Thus, the intensity at 8m away from the source is 1.5625 W/m².

b) The distance, r, can be determined using the following equation: Substituting the given values: Therefore, the distance at which the sound waves have an intensity of 0.02 W/m² is 400m.

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a. To find the total resistance of the parallel combination, we use the formula:

1/RTotal = 1/R1 + 1/R2 + 1/R3

Substituting the given values:

1/RTotal = 1/35 + 1/55 + 1/85

Simplifying this equation gives us:

1/RTotal = 0.02857 + 0.01818 + 0.01176

Adding these values, we get:

1/RTotal = 0.05851

Taking the reciprocal of both sides, we find:

RTotal = 1/0.05851

RTotal ≈ 17.09 Ω

b. To find the current through each resistor, we use Ohm's law:

I = V/R

For each resistor, we have:

I1 = V/R1 = 35/35 ≈ 1 A
I2 = V/R2 = 35/55 ≈ 0.636 A
I3 = V/R3 = 35/85 ≈ 0.412 A

So the current through each resistor is approximately:

I1 ≈ 1 A
I2 ≈ 0.636 A
I3 ≈ 0.412 A

19. a. To find the total resistance, we first find the equivalent resistance of R1 and R2 in series:

Req = R1 + R2 = 15 + 9 = 24 Ω

Next, we find the total resistance (RTotal) by adding the equivalent resistance (Req) and R3 in parallel:

1/RTotal = 1/Req + 1/R3 = 1/24 + 1/8 = 1/24 + 3/24 = 4/24

Simplifying this equation gives us:

1/RTotal = 4/24

Taking the reciprocal of both sides, we find:

RTotal = 24/4 = 6 Ω

b. To find the current in R3, we use Ohm's law:

I = V/R

Substituting the given values, we have:

I = 6/8 = 0.75 A

So the current in R3 is 0.75 A.

c. To find the potential difference across R2, we use Ohm's law again:

V = I * R

Substituting the given values, we have:

V = 0.75 * 9 = 6.75 V

So the potential difference across R2 is 6.75 V.

20. a. To find the total resistance, we first find the equivalent resistance (Req) of the two 10Ω resistors in parallel:

1/Req = 1/10 + 1/10 = 2/10

Simplifying this equation gives us:

1/Req = 2/10

Taking the reciprocal of both sides, we find:

Req = 10/2 = 5 Ω

Next, we add the resistance of the 15Ω resistor in series:

RTotal = Req + 15 = 5 + 15 = 20 Ω

b. To find the circuit's current, we use Ohm's law:

I = V/RTotal = 120/20 = 6 A

So the circuit's current is 6 A.

c. To find the current in one of the 10Ω resistors, we note that the current in each parallel branch is the same, which is equal to the circuit's current:

I = 6 A

So the current in one of the 10Ω resistors is 6 A.

d. To find the potential difference across the 15Ω resistor, we use Ohm's law:

V = I * R = 6 * 15 = 90 V

So the potential difference across the 15Ω resistor is 90 V.

In summary:

a. The total resistance is 17.09 Ω.
b. The current through each resistor is approximately: I1 ≈ 1 A, I2 ≈ 0.636 A, I3 ≈ 0.412 A.
19. a. The total resistance is 6 Ω.
b. The current in R3 is 0.75 A.
c. The potential difference across R2 is 6.75 V.
20. a. The total resistance is 20 Ω.
b. The circuit's current is 6 A.
c. The current in one of the 10Ω resistors is 6 A.
d. The potential difference across the 15Ω resistor is 90 V.

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The beat frequency heard by the bystander is approximately 21.90 Hz.

To calculate the beat frequency heard by the bystander, we need to consider the Doppler effect caused by the moving car. The beat frequency is the difference between the frequencies of the two cars' horns as heard by the bystander.

Given:

The observed frequency of the moving car's horn (fm) as heard by the bystander can be calculated using the formula:

fm = fs * (v / (v ± vm))

where

Let's calculate the beat frequency for the scenario where the moving car is approaching the bystander:

fm = 395 Hz * (343 m/s / (343 m/s + 19.6 m/s))

Simplifying the equation:

fm = 395 Hz * (343 m/s / 362.6 m/s)

fm ≈ 373.10 Hz

The beat frequency (fb) is the difference between the frequency of the parked car's horn and the observed frequency of the moving car's horn:

fb = fs - fm

fb = 395 Hz - 373.10 Hz

fb ≈ 21.90 Hz

Therefore, the beat frequency heard by the bystander is approximately 21.90 Hz.

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The coefficient of friction between the ladder and the ground must be less than or equal to √3 / 2.

To determine the coefficient of friction between the ladder and the ground, we need to consider the forces acting on the ladder.

Start by drawing a free body diagram of the ladder. -

The weight of the ladder acts vertically downwards, with a magnitude of mg, where m is the mass of the ladder and g is the acceleration due to gravity. The normal force acts perpendicular to the ground and balances the weight of the ladder. The friction force acts parallel to the ground and opposes the motion of the ladder.

The ladder will slip when the friction force reaches its maximum value, which is given by the equation:

F_max = μN

Where, F_max is the maximum friction force, μ is the coefficient of friction, and N is the normal force.

Determine the normal force acting on the ladder. - Since the ladder is at rest in the vertical direction, the normal force is equal to the weight of the ladder, which is mg.

Substitute the known values into the equation for the maximum friction force.

F_max = μN

F_max = μmg.

The maximum friction force must be equal to or less than the force required to make the ladder slip, which is the weight component parallel to the ground. The weight component parallel to the ground is given by: F_parallel = mg sinθ,

Where, θ is the angle the ladder makes with the ground.

Set up the inequality:

F_max ≤ F_parallel.

Substitute the expressions for F_max and F_parallel:

μmg ≤ mg sinθ.

Cancel out the mass, m, on both sides of the inequality:

μg ≤ g sinθ.

Cancel out the acceleration due to gravity, g, on both sides of the inequality:

μ ≤ sinθ.

From the given information, we know that the ladder will slip when θ < 60º. Substitute θ = 60º into the inequality:

μ ≤ sin(60º).

Simplify the expression:

μ ≤ √3 / 2.

Therefore, It is necessary for the ladder's coefficient of friction with the ground to be less than or equal to√3 / 2.

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Complete question is,

An 8.0 kg ladder that is 4.0 m tall leans against a building as shown below. The ground is rough and provides some friction against the ladder while the building’s side is smooth and provides no friction.

The ladder creates an angle, θ, with the ground. The ladder will slip whenever θ < 60º. Using this information, determine the coefficient of friction between the ladder and the ground.

Therefore, the potential difference that stopped the proton is approximately equal to 14,300 V.

Given data Initial speed of proton = 456,000 m/sFinal speed of proton = 0 m/s Charge of proton = 1.6 × 10⁻¹⁹ CVoltage difference or potential difference is given by

V = KE/qV = q(Et/m) / qV = Et/m where, V = Potential difference

E = Electric fieldt = time takenm = mass of the protonq = Charge of the proton K = Kinetic energy Initially, the kinetic energy of the proton is given by;KE = 1/2mv²KE = (1/2)(1.6 × 10⁻¹⁹)(456,000)²KE = 2.304 × 10⁻¹¹ J

Opposing this kinetic energy of proton, the electric field provides work done on the proton, which is equal to the kinetic energy. Let the potential difference created due to this electric field be V.

So, V = Et/mV = KE/qV = (1/2mv²) / qV = [1/(2q)] mv²V = [1/(2(1.6 × 10⁻¹⁹)] (1.6 × 10⁻²⁷)(456,000)²V = 14,300 V ≈ 1.4 × 10⁴ V Therefore, the potential difference that stopped the proton is approximately equal to 14,300 V.

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The tension in the wire after the tuning peg has been turned through one complete revolution is approximately 96.80 N (rounded to four significant figures).

Density of steel wire (ρ) = 8040 kg/m³

Length of the wire (L) = 0.84 m

Diameter of the wire (d) = 0.93 mm

Diameter of the tuning peg (D) = 3.78 mm

Initial tension in the wire (T_initial) = 18.1 N

To find the tension in the wire after one complete revolution, we need to calculate the new tension (T) in the wire.

First, let's calculate the new length of the wire after one complete revolution:

The length of the wire wrapped around the tuning peg is equal to the circumference of the peg:

L_new = 2πr = 2π(0.5D) = πD

Next, let's calculate the cross-sectional area of the wire:

The cross-sectional area of a wire is given by:

A = πr², where r is the radius of the wire.

The radius of the wire can be calculated as half of its diameter:

r = 0.5d

Now, let's calculate the mass of the wire:

The mass of the wire is given by:

m = ρVL, where V is the volume of the wire.

The volume of the wire can be calculated as the product of its cross-sectional area and length:

V = AL

Now, let's substitute the values and calculate the mass of the wire.

Next, let's calculate the new tension in the wire by taking moments about the point where the wire is fixed:

T_new × r - (m × g) × (L_new - L) = 0, where g is the acceleration due to gravity.

Finally, let's solve the equation for T_new to find the tension in the wire after one complete revolution.

Now, let's calculate the values step by step using the given values:

Diameter of the wire (d) = 0.93 mm = 0.00093 m

Radius of the wire (r) = 0.5d = 0.000465 m

Diameter of the tuning peg (D) = 3.78 mm = 0.00378 m

New length of the wire (L_new) = πD = π(0.00378) = 0.011887 m

Cross-sectional area of the wire (A) = πr² = π(0.000465)² = 0.0000006797 m²

Volume of the wire (V) = AL = (0.0000006797 m²)(0.84 m) = 0.000000571 m³

Mass of the wire (m) = ρV = (8040 kg/m³)(0.000000571 m³) = 0.00459 kg

Acceleration due to gravity (g) = 9.8 m/s²

Now, let's calculate the new tension (T_new):

T_new × r - (m × g) × (L_new - L) = 0

T_new × 0.000465 - (0.00459 kg × 9.8 m/s²) × (0.011887 m - 0.84 m) = 0

T_new × 0.000465 - 0.0450228 N = 0

T_new × 0.000465 = 0.0450228 N

T_new = 0.0450228 N / 0.000465

T_new = 96.7978495 N

Therefore, the tension in the wire after the tuning peg has been turned through one complete revolution is approximately 96.80 N (rounded to four significant figures).

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To design, simulate, build, and test an amplifier that satisfies the given requirements and constraints, you can follow these steps:

1. Determine the amplifier specifications: Start by identifying the desired characteristics of the amplifier, such as the gain, bandwidth, input impedance, and output impedance. This information will guide the design process.

2. Select an appropriate amplifier configuration: There are various amplifier configurations to choose from, such as common-emitter, common-collector, and common-base for transistor amplifiers. The choice depends on the specific requirements and constraints.

3. Calculate the component values: Once the amplifier configuration is chosen, you need to calculate the values of the components (resistors, capacitors, and transistors) based on the desired specifications. This can be done using circuit analysis techniques and amplifier design equations.

4. Simulate the circuit using OrCad: Use OrCad or any other circuit simulation software to simulate the designed amplifier circuit. This step allows you to validate the design and observe its performance before building the physical circuit.

5. Build the amplifier circuit: After the simulation results are satisfactory, gather the required components and build the amplifier circuit on a breadboard or a PCB. Ensure the correct placement and connections of each component.

6. Test the amplifier: Connect the amplifier circuit to the load (a 100-Ω resistor in this case) and apply an input signal. Use an oscilloscope or a multimeter to measure the amplifier's output voltage and current. Verify if the amplifier meets the specified requirements and constraints.

7. Fine-tune the design if necessary: If the amplifier does not meet the desired specifications, analyze the results and identify areas for improvement. You may need to adjust component values or try a different configuration to optimize the amplifier's performance.

Remember, the process of designing and building an amplifier requires careful consideration of the specifications, constraints, and circuit analysis techniques. Simulation and testing are crucial steps to ensure the amplifier functions as intended.

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The angular position of the m=1 line (in degree) for the given diffraction grating with 4000 lines per centimeter is illuminated with yellow sodium light of wavelength 589 nm is 23.12°.

What is diffraction grating?

A diffraction grating is an optical component consisting of a substrate with parallel grooves or slits. A diffraction grating separates a light source into its constituent wavelengths and spreads them into an angular spectrum by diffracting light.:

Calculation for the angular position of the m=1 line:

The formula for calculating the angular position of the m=1 line in degree is given by:

θ = msinθ.dλ

θ = sin⁻¹(mλ/d)

where, m = 1λ = 589 nm (the wavelength of yellow sodium light)

d = 1/4000 cm (the distance between adjacent slits)

We can convert this distance to meter as: d= 1/4000 × 10^-2 = 2.5 × 10⁻⁵

msinθ = (1 × 589 × 10^-9) / (2.5 × 10^-5) = 0.02344θ = sin⁻¹(0.02344)θ = 23.12°

Therefore, the angular position of the m=1 line (in degree) is 23.12°.

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To determine which choice is better, we need to calculate the present value (PV) of the costs associated with each lightbulb.

Let's start with the incandescent bulb. The cost of the bulb is $0.50, and it lasts for three years. Each year, it uses $0.75 of electricity. The discount rate is 12%.

To calculate the present value of the electricity cost for each year, we can use the formula

PV = C / (1 + r)^t,

where C is the cost, r is the discount rate, and t is the number of years. Plugging in the values, we have:

PV = $0.75 / (1 + 0.12)^1 + $0.75 / (1 + 0.12)^2 + $0.75 / (1 + 0.12)^3

Calculating this gives us a present value of $1.89 for the electricity cost.

Next, let's calculate the present value of the LED bulb. The cost of the bulb is $4, and it lasts for eleven years. Each year, it uses $0.20 of electricity.

Using the same formula as before, we can calculate the present value of the electricity cost for each year:

PV = $0.20 / (1 + 0.12)^1 + $0.20 / (1 + 0.12)^2 + ... + $0.20 / (1 + 0.12)^11

Calculating this gives us a present value of $1.18 for the electricity cost.

Now, we can compare the present values of the costs for each lightbulb. The incandescent bulb has a present value of $1.89, while the LED bulb has a present value of $1.18.

Since the present value of the LED bulb is lower, it is the better choice. The LED bulb is more cost-effective in the long run, even though it has a higher upfront cost. It lasts longer and uses less electricity, resulting in lower overall costs.

In conclusion, the LED lightbulb is the better choice based on the calculations of present value. It offers long-term savings due to its longer lifespan and lower electricity usage.

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The equilibrium position for a test charge q, with Q1 = 6C and Q2 = -4C, at a distance r = 1m, is located at x = 0.5m.

To find the equilibrium position for a test charge q along the x-axis in the presence of charges Q1 and Q2, we can use the principle of electrostatic equilibrium. In this case, with Q1 = 6C and Q2 = -4C, we can determine the equilibrium position.

The condition for electrostatic equilibrium is that the net force on the test charge q should be zero. This means that the forces exerted on the test charge by Q1 and Q2 should balance each other.

Since Q1 is positive and Q2 is negative, the forces they exert on the test charge q will be in opposite directions. The magnitude of the force between two charges is given by Coulomb's law:

F = k * |Q1 * Q2| / r^2

where k is the electrostatic constant and r is the distance between the charges.

To achieve equilibrium, the magnitudes of the forces exerted by Q1 and Q2 on the test charge q should be equal. This means:

k * |Q1 * Q2| / r^2 = k * |Q1 * q| / (x^2)

Simplifying this equation, we have:

|Q2 / r^2| = |q / x^2|

Given that Q2 = -4C and Q1 = 6C, the equation becomes:

|-4 / r^2| = |q / x^2|

Since the charges and distances are positive values, we can ignore the absolute value signs.

4 / r^2 = q / x^2

Solving for x, we have:

x^2 = (q * r^2) / 4

x = sqrt((q * r^2) / 4)

Plugging in the values Q1 = 6C, Q2 = -4C, and r = 1m, and assuming a test charge q of any positive value, we can calculate the equilibrium position.

For example, if q = 1C, the equilibrium position is:

x = sqrt((1C * (1m)^2) / 4)

x = sqrt(1/4) = 0.5m

Therefore, the equilibrium position for a test charge q of 1C is located at x = 0.5m, assuming Q1 = 6C, Q2 = -4C, and r = 1m.

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Twin-lead cable with #26 AWG copper conductors and centers separated by 1 cm, the characteristic impedance is 0 ohms, and the propagation velocity is 3 * 10^8 m/s.

To calculate the distributed parameters at 1 MHz for a twin-lead cable made up of #26 AWG copper conductors with centers separated by 1 cm, we need to determine the characteristic impedance and propagation velocity.

1. First, let's calculate the characteristic impedance (Z0) using the formula:

  Z0 = (138 * log10((D + H) / D)) / sqrt(Er)

  where:
  - D is the distance between the centers of the conductors (1 cm = 0.01 m)
  - H is the height of each conductor (which we assume to be negligible for twin-lead cables)
  - Er is the relative permittivity of air (approximately 1)

  Plugging in the values, we get:
  Z0 = (138 * log10((0.01 + 0) / 0.01)) / sqrt(1)
     = (138 * log10(1)) / 1
     = (138 * 0) / 1
     = 0

  Therefore, the characteristic impedance is 0 ohms.

2. Next, let's calculate the propagation velocity (Vp) using the formula:

  Vp = c / sqrt(Er)

  where:
  - c is the speed of light (approximately 3 * 10^8 m/s)
  - Er is the relative permittivity of air (approximately 1)

  Plugging in the values, we get:
  Vp = (3 * 10^8) / sqrt(1)
     = (3 * 10^8) / 1
     = 3 * 10^8 m/s

  Therefore, the propagation velocity is 3 * 10^8 m/s.

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To calculate the distributed parameters of a twin-lead cable at 1 MHz, we need to determine the characteristic impedance and propagation velocity.

First, let's calculate the distributed capacitance (C) and inductance (L) of the twin-lead cable.

1. Distributed capacitance (C):
  - The distributed capacitance is determined by the spacing between the conductors and the permittivity of the medium (air in this case).
  - The formula for distributed capacitance per unit length (C'') is given by:
    [tex]C'' = (8.854 x 10^-12 F/m) / (2π x frequency) x ln(b/a)[/tex],
    where a is the radius of one conductor and b is the separation between the centers of the conductors.
  - In this case, we are given that the conductors are made of #26 AWG copper, so we can calculate the radius using the wire gauge chart.
  - The radius for #26 AWG copper is approximately 0.127 mm or [tex]1.27 x 10^-4 m[/tex].
  - Plugging in the values, we get:
   [tex] C'' = (8.854 x 10^-12 F/m) / (2π x 1 MHz) x ln(0.01 m / (2 x 1.27 x 10^-4 m))[/tex].

2. Distributed inductance (L):
  - The distributed inductance is primarily determined by the dimensions and shape of the conductors.
  - For a twin-lead cable, the formula for distributed inductance per unit length (L'') is given by:
    L'' = (μ₀ / (2π)) x ln(b/a),
    where μ₀ is the permeability of free space, a is the radius of one conductor, and b is the separation between the centers of the conductors.
  - Plugging in the values, we get:
    [tex]L'' = (μ₀ / (2π)) x ln(0.01 m / (2 x 1.27 x 10^-4 m))[/tex],
    where [tex]μ₀ = 4π x 10^-7 H/m[/tex].

Next, let's determine the characteristic impedance (Z₀) and propagation velocity (v) of the cable.

3. Characteristic impedance (Z₀):
  - The characteristic impedance is given by the formula:
    [tex]Z₀ = √(L'' / C'')[/tex],
  - Plugging in the values of L'' and C'', we get:
    [tex]Z₀ = √((μ₀ / (2π)) x ln(0.01 m / (2 x 1.27 x 10^-4 m))) / ((8.854 x 10^-12 F/m) / (2π x 1 MHz) x ln(0.01 m / (2 x 1.27 x 10^-4 m)))[/tex].

4. Propagation velocity (v):
  - The propagation velocity is given by the formula:
    [tex]v = 1 / √(L'' x C'')[/tex],
  - Plugging in the values of L'' and C'', we get:
    [tex]v = 1 / √((μ₀ / (2π)) x ln(0.01 m / (2 x 1.27 x 10^-4 m))) / ((8.854 x 10^-12 F/m) / (2π x 1 MHz) x ln(0.01 m / (2 x 1.27 x 10^-4 m)))[/tex].

By calculating the above formulas, we can find the distributed parameters (C and L) at 1 MHz for the twin-lead cable made up of #26 AWG copper conductors with centers separated by 1 cm. Additionally, we can determine the characteristic impedance (Z₀) and propagation velocity (v).

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An oscillator's amplitude decreases by 4.83% in ten cycles due to dissipative forces. We need to determine the percentage by which the oscillator's energy decreases in ten cycles.

We must first comprehend how the amplitude and energy of an oscillator are related to one another and the impact of dissipative forces on it. A harmonic oscillator is an object that oscillates back and forth about its equilibrium position when it is disturbed. The displacement of an oscillator from its equilibrium position is referred to as its amplitude. Energy is defined as the capacity to perform work, and it is conserved in an isolated system with no external forces acting on it. Dissipative forces are forces that remove energy from a system over time, causing the amplitude of an oscillator to decrease. The loss of amplitude results in a decrease in the oscillator's energy as well.

Now, let's solve the problem. The amplitude of the oscillator decreases by 4.83 percent in 10 cycles. If we know the amplitude, we can calculate the energy using the formula for the total mechanical energy of an oscillator.E = 1/2 kA²Where,

E = Total mechanical energy of oscillator

k = Spring constant

A = Amplitude

We know that the amplitude decreased by 4.83 percent, which means that the amplitude after 10 cycles is 0.9517 times the original amplitude.

Thus, the energy after 10 cycles will be

E' = 1/2 k (0.9517A)² = 0.9043 E

Thus, the oscillator's energy decreases by 9.57% in ten cycles, which is our final answer.

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When the number of turns is increased from 7 to 24, the magnetic field increases by approximately 180.45 Gauss

If the slope of the magnetic field versus number of turns of a coil is 10.65 G/turn, and the number of turns is increased from 7 to 24, the magnetic field will increase by a certain value.

To calculate this increase, we can multiply the slope by the change in the number of turns. The result will give us the increase in magnetic field in units of Gauss.

The increase in the magnetic field can be calculated by multiplying the slope of the magnetic field versus number of turns by the change in the number of turns. In this case, the slope is given as 10.65 G/turn, and the change in the number of turns is 24 - 7 = 17 turns.

Therefore, the increase in the magnetic field is equal to (10.65 G/turn) × 17 turns = 180.45 G. Rounding this value to two decimal places, the magnetic field increases by approximately 180.45 G.
The slope of the magnetic field versus number of turns represents the rate at which the magnetic field changes with respect to the number of turns.

In this case, the slope is given as 10.65 G/turn, which means that for each additional turn added to the coil, the magnetic field increases by 10.65 Gauss.

To calculate the increase in the magnetic field when the number of turns is increased from 7 to 24, we can multiply the slope by the change in the number of turns.

The change in turns is obtained by subtracting the initial number of turns (7) from the final number of turns (24). Multiplying the slope by the change in turns gives us the increase in the magnetic field, which is 180.45 Gauss in this case.

Therefore, when the number of turns is increased from 7 to 24, the magnetic field increases by approximately 180.45 Gauss.

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a)  The charge stored on capacitor C₂ is approximately 8.94 μC. b)  The energy stored in capacitor C₁ is approximately 15.61 μJ, and the energy stored in capacitor C₂ is approximately 6.32 μJ.

(a) To find the charge stored on each capacitor, we can use the formula:

Q = C * V

where Q is the charge stored, C is the capacitance, and V is the voltage applied.

Given:

C₁ = 14.0 μF

C₂ = 6.00 μF

V = 1.49 V

For capacitor C₁:

Q₁ = C₁ * V = (14.0 μF) * (1.49 V) = 20.86 μC

For capacitor C₂:

Q₂ = C₂ * V = (6.00 μF) * (1.49 V) = 8.94 μC

Therefore, the charge stored on capacitor C₁ is approximately 20.86 μC, and the charge stored on capacitor C₂ is approximately 8.94 μC.

(b) The energy stored in a capacitor can be calculated using the formula:

E = (1/2) * C * V²

Given the capacitance and voltage values, we can determine the energy stored in each capacitor.

For capacitor C₁:

E₁ = (1/2) * C₁ * V² = (1/2) * (14.0 μF) * (1.49 V)² = 15.61 μJ

For capacitor C₂:

E₂ = (1/2) * C₂ * V² = (1/2) * (6.00 μF) * (1.49 V)² = 6.32 μJ

Therefore, the energy stored in capacitor C₁ is approximately 15.61 μJ, and the energy stored in capacitor C₂ is approximately 6.32 μJ.

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The magnitude of the electric field between the plates is approximately 126050 N/C.  The magnitude of the force on an electron between the plates is approximately 2.02 × 10^(-14) N.  The amount of work that must be done on the electron to move it to the negative plate is approximately 5.78 × 10^(-17) J.

(a) The magnitude of the electric field between the plates can be calculated using the formula: Electric Field = Voltage / Distance. Given that the potential difference is 600 V and the distance between the plates is 4.76 mm (which is 0.00476 m), we can substitute these values into the formula:

Electric field = 600 V / 0.00476 m

Electric Field ≈ 126050 N/C

Therefore, the magnitude of the electric field between the plates is approximately 126050 N/C.

(b) The magnitude of the force on an electron between the plates can be calculated using the formula: Force = Charge × Electric Field. Since the electron has a charge of -1.6 × 10^(-19) C and the electric field is given as 126050 N/C, we can substitute these values into the formula:

Force = (-1.6 × 10^(-19) C) × (126050 N/C)

Force ≈ -2.02 × 10^(-14) N

The negative sign indicates that the force is acting in the opposite direction of the electric field.

Therefore, the magnitude of the force on an electron between the plates is approximately 2.02 × 10^(-14) N.

(c) The work done on the electron to move it to the negative plate can be calculated using the formula: Work = Force × Distance. Given that the force is approximately -2.02 × 10^(-14) N and the initial distance from the positive plate is 2.86 mm (which is 0.00286 m), we can substitute these values into the formula:

Work = (-2.02 × 10^(-14) N) × (0.00286 m)

Work ≈ -5.78 × 10^(-17) J

The negative sign indicates that work is done against the electric field.

Therefore, the amount of work that must be done on the electron to move it to the negative plate is approximately 5.78 × 10^(-17) J.

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The maximum speed is 7.5 m/s. The maximum speed for a lorry of mass 4500 kg to be able to clear the hill safely is  5.0 m/s. The frictional force at the top of the hill when the car is traveling at the maximum speed is 2564.07 N.

The following is the solution to the problem:

A 1500 kg car drives over a small hill with radius, r=150 m at a constant speed, v.

i. To determine the maximum speed for the car to clear the hill safely. The gravitational force on the car can be resolved into two components i.e., the normal force, N and the force pulling the car down the hill which is a component of the weight of the car, W. This component can be determined as follows;

W = m × g

Where m is the mass of the car and g is the gravitational acceleration which is equal to 9.81 m/s² at the surface of the Earth.

W = 1500 × 9.81 = 14715 N

The component of the weight of the car pulling it down the hill = W × sin θ  where θ is the angle of the hill with respect to the horizontal, which is given by: sin θ = opposite / hypotenuse = h / r where h is the height of the hill. Rearranging for h, we get; h = r × sin θ  Now we can find the force pulling the car down the hill;

F = W × sin θ F = 14715 × sin 5° = 1281.67 N

The maximum force of static friction on the car to prevent it from slipping down the hill is given by;

fstatic = µstatic × N where µstatic is the coefficient of static friction between the tyres and the road and N is the normal force. N can be determined as follows;

N = m × g - F  = 1500 × 9.81 - 1281.67 = 12820.33 N

The maximum speed can now be determined as follows;

Centripetal force = frictional forcef = mv²/rµstatic × N = mv²/rv² = µstatic × N × r / m

Now we can substitute the values given to determine the maximum speed that the car can safely go over the hill.

v = √(µstatic × N × r / m)v = √(0.2 × 12820.33 × 150 / 1500)v = 7.5 m/s

ii. The maximum speed for a lorry of mass 4500 kg to be able to clear the hill safely.

The same procedure can be used as above to find the maximum speed of the lorry by just substituting the value of m with 4500 kg which gives;

v = √(0.2 × 12820.33 × 150 / 4500)v = 5.0 m/s

iii. The frictional force at the top of the hill when the car is traveling at the maximum speed can be determined as follows;

fstatic = µstatic × N

fstatic = 0.2 × 12820.33fstatic = 2564.07 N

Therefore the frictional force at the top of the hill when the car is traveling at the maximum speed is 2564.07 N.

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The grape was in the air for 0.23 seconds (rounded to the nearest hundredth of a second).

When an object is thrown upwards or downwards, the direction of velocity and acceleration are opposite. The direction of the velocity of the object is upward and the direction of acceleration of the object is downward. The direction of velocity and acceleration is opposite and hence the acceleration will be negative in such cases.

As we know the velocity of the grape at the topmost point will be zero because at that point it stops moving upwards and starts moving downwards. The distance traveled by the grape during its upward and downward journey will be the same and the acceleration will be constant. Thus, the time taken to go up and to come down will be the same, i.e., t

Up = tDown.Let's find the time for the grape to go up. The initial velocity, u = 1.11 m/s. The final velocity, v = 0 (At the highest point the grape comes to rest)Acceleration, a = -9.8 m/s² (The acceleration is negative as it is directed in the opposite direction to the velocity of the grape)We know, v = u + at0 = 1.11 - 9.8tUp=-1.11/9.8=0.11327s (approx)

The time taken for the grape to come down will be the same as the time it took to go up.Total time taken by the grape in the air is= tUp + tDown= 0.11327 + 0.11327= 0.2265s (approx)The grape was in the air for 0.23 seconds (rounded to the nearest hundredth of a second).

Hence, the required answer is 0.23 seconds.

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Therefore, the tension of the rope connecting the two blocks is 12 N. Hence the correct option is d) 4.0 N.

Given The masses of the two blocks on a flat level board are 2.0 kg and 4.0 kg. A taut rope connects the blocks and they slide on the board without friction. Something is pulling 6.0 N to the right on a rope attached to the right side of the 4.0 kg block, causing the blocks to slide to the right faster and faster.

According to the problem, two blocks are connected by a rope and are moving together as if they were one object. Therefore, the force acting on the two blocks is F=6 N.

The tension T of the rope connecting the two blocks can be calculated as follows:

T - F = ma where m is the mass of the combined blocks and a is their acceleration.To find the acceleration a, we need to find the net force acting on the system. There is only one force acting on the system i.e 6 N. Now using the equation F = ma, we get:

6 = (2+4)a6 = 6aa = 1m/s²

Now substituting the value of a, we get:

T - 6 = (2+4)1T = 12 NTherefore, the tension of the rope connecting the two blocks is 12 N. Hence the correct option is d) 4.0 N.

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The power delivered by the net force is 64 Watts.

The object of 2 kg is moving in one dimension with a velocity given by

V(t)=1/2 t² at t=4s.

We need to calculate the power delivered by the net force.

Power can be calculated by multiplying the force by velocity. That is

P = F x V

Since we know the velocity, we can find the force applied to the object.

The velocity is given by

V(t)=1/2 t² at t=4s.

Hence the velocity of the object can be found by substituting

t = 4 s.

So

V(4) = 1/2 (4)²= 1/2 * 16= 8 m/s

Now, the force applied can be calculated using the formula,

F = m*a

where

a is the acceleration of the object

m is the mass of the object

Here, the acceleration is given by the time derivative of the velocity which is

a(t) = V'(t) = t

Therefore, the acceleration at t = 4 s is

a(4) = 4 m/s²

The force is,

F = m*a= 2 kg * 4 m/s²= 8 N
Now, the power delivered by the net force is,

P = F * V= 8 N * 8 m/s= 64 Watts

Hence, the power delivered by the net force is 64 Watts.

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It will take a force of 25729 N to accelerate a 669.31 kg sports car from 0 to 60 mph in 4.44 seconds.

When a vehicle speeds up, the vehicle's acceleration is affected by the amount of force it generates. The force required to speed up a 669.31 kg sports vehicle from 0 to 60 mph in 4.44 seconds may be determined using the following formula:

[tex]\[force = mass \times acceleration\][/tex]

Force = (669.31 kg) \times (60 mph / 2.23694 m/s) / (4.44 s)

Force = 25729 N

It will take a force of 25729 N to accelerate a 669.31 kg sports car from 0 to 60 mph in 4.44 seconds.

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The direction of a skydiver's mass is always straight down towards the center of the Earth due to the force of gravity, regardless of its initial velocity or motion.

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Electric field at radial distances

(a) r=0 cm, (b) r=a/2.00 (c) r=a, (d) r=1.50a, (e) r=2.30a, and (f) r=3.50a :

Part a: At r = 0 cm, the electric field is zero as no point charge is present.

Part b: At r = a/2.00, the electric field inside the sphere (r < a) is calculated as below.

[tex]E = (1/4πε0​) q1​​/a^3 × r[/tex]

where,ε0​ is the permittivity of free space, q1​ is the charge on the sphere= 3.59

[tex]fC = 3.59 × 10^-15 C.[/tex]

Putting the values, we getE = 1.04 × 10^9 N/C[tex]1.04 × 10^9 N/C[/tex]

Part c:

At r = a, the electric field is just outside the sphere (r > a) and is calculated using the same formula as for Part b.

[tex]E = (1/4πε0​) q1​/r^2[/tex]

where,ε0​ is the permittivity of free space, q1​ is the charge on the sphere= [tex]3.59 fC = 3.59 × 10^-15 C.[/tex]

Putting the values, we getE =[tex]2.77 × 10^8 N/C[/tex]

Part d:

At r = 1.50a, the electric field is calculated using Gauss's law where we take a  Gaussian surface around the entire system.

Electric flux = q1​ + q2

​We know that q2​ =[tex]-q1​q1​ = 3.59 fC = 3.59 × 10^-15 C q2​ = -3.59 fC = -3.59 × 10^-15 C[/tex]

Electric flux = 0Electric flux through the Gaussian surface is zero since there is no charge enclosed by it.

electric field at r = 1.50a is zero.

Part e: At r = 2.30a, the electric field is just outside the shell (b < r < c).

Thus the electric field can be calculated as below:

[tex]E = (1/4πε0​) q2​/r^2[/tex]

where,ε0​ is the permittivity of free space, q2​ is the charge on the shell= [tex]-3.59 fC = -3.59 × 10^-15 C.[/tex]

Putting the values, we getE = 1.89 × 10^8 N/C

Part f: At r = 3.50a, the electric field is outside the shell (r > c) and is calcule formula as for Part e.

calcuated using the samE =

[tex](1/4πε0​) q2​/r^2[/tex]

where,ε0​ is the permittivity of free space, q2​ is the charge on the shell= [tex]-3.59 fC = -3.59 × 10^-15[/tex]

C. Putting the values

we getE = 4.67 × 10^7 N/C

Net charge on the (g) inner and (h) outer surface of the shell:

Net charge on the inner surface of the shell =[tex]q1​ = 3.59 fC = 3.59 × 10^-15 C[/tex]

Net charge on the outer surface of the shell = [tex]q2​ - q1​ = -2q1​ = -7.18 fC = -7.18 × 10^-15 C[/tex]

The net charge on the inner surface of the shell is 3.59 × 10^-15 C and the net charge on the outer surface of the shell is -7.18 × 10^-15 C.

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The maximum distance the remote control can extend beyond the edge of the table and still not tip over when pressing the power button is 7.05 cm (approx).

As per data,

A = 0.122 kg, L= 22.0 cm, t = 0.0151 m, and F= 0.355 N

In order for the remote control not to tip over when pressing the power button, we need to check if the center of gravity of the remote control lies within the table's edges.

If the center of gravity lies outside the table's edges, the remote-control tips over. Let us find the center of mass or center of gravity of the remote control.

For a uniform distribution of mass, we can find the center of mass as follows:

Since the mass is uniformly distributed along the entire length of the remote control, its center of gravity lies at the midpoint of the entire length of the remote.

Hence, we can find the distance from the center of gravity to the edge by:

Thus, the maximum distance the remote control can extend beyond the edge of the table and still not tip over when pressing the power button is 7.05 cm (approx).

Therefore, the power button is pressed, the remote control can extend 7.05 cm (approximately) beyond the edge of the table without toppling.

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Sound level at that point = 100.98 dB.

a. The tension in the string is calculated as follows:

$$T=\frac{m}{L}\left(\frac{v}{\lambda}\right)^2$$

where L is the length of the string,

m is the mass per unit length,

v is the speed of the wave, and

λ is the wavelength.

Mass per unit length, m=7.48×10−4 kg/m

                                    v=0.85 m/s,

                                   λ=2A

                                     =0.245×2

                                    =0.49m.

Let's substitute these values in the formula:

T = (7.48 × 10⁻⁴ kg/m) × [(0.85 m/s) / (0.49 m)]²

  = 13.2 N

Tension in the string is 13.2 N.

b. The frequency at which the crests pass a given point in space is given by:

$$f = \frac{v}{\lambda}$$v

     = 0.85 m/s

 λ = 2A

    = 0.49m

We can calculate the frequency as follows:

f = v / λ

 = 0.85 m/s / 0.49m

 = 1.73 Hz

The frequency at which the crests pass a given point in space is 1.73 Hz.

c. The distance between two adjacent crests on the wave is given by:

$$\lambda = \frac{v}{f}$$v = 0.85 m/s

f = 1.73 Hz

We can calculate the wavelength as follows:

λ = v / f

  = 0.85 m/s / 1.73 Hz

  = 0.49 m

The distance between two adjacent crests on the wave is 0.49 m.

d. The spot moves one wavelength over one period of the wave. Therefore, its displacement is equal to one wavelength (λ) given by,

λ = v / f

λ = 0.85 m/s / 1.73 Hz

  = 0.49 m

So, the distance the blue spot travels in one period is 0.49 meters. Phase difference ϕ in the sound waves from the two sources at that point is given by:

$$\phi = \frac{2\pi d}{\lambda}$$

where d is the distance between the point and the sources.

d1 = 6.28 m

d2 = 7.28 m

λ = v / f

 = 343 / 107

 = 3.206 m

The phase difference is, ϕ = (2 × π × (d2 - d1)) / λ

                                            = (2 × π × (7.28 m - 6.28 m)) / 3.206 m

                                            = 1.96 rad

The displacement amplitude of the resultant wave is given by,

Ar = 2A

   = 2 × 5.91 × 10⁻⁷ m

   = 1.18 × 10⁻⁶ m

The sound intensity at the point is given by,

I = Ar²ρv/2I

 = (1.18 × 10⁻⁶ m)² × 1.22 kg/m³ × 343 m/s / 2

 = 2.90 × 10⁻¹¹ W/m²

The sound level at that point is given by,

L = 10 log 10 (I/I₀)

where I₀ is the reference intensity and is equal to 1 × 10⁻¹² W/m²

L = 10 log 10 (2.90 × 10⁻¹¹/1 × 10⁻¹²)

  = 100.98 dB

a) Tension = 13.2 N

b) Frequency = 1.73 Hz

c) Distance between two adjacent crests on the wave = 0.49 m

d) Distance that blue spot travels in one period is 0.49 meters

i) Phase difference, ϕ = 1.96 rad

ii) Displacement amplitude of the resultant wave, Ar = 1.18 × 10⁻⁶ m

iii) Sound intensity at that point = 2.90 × 10⁻¹¹ W/m²

iv) Sound level at that point = 100.98 dB.

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The speed of the ship relative to the Earth is 6.08 m/s.

a) In what direction (in degrees west of north) would the ship have to travel in order to have a velocity straight north relative to the Earth, assuming its speed relative to the water remains 6.85 m/s?

b) What would its speed (in m/s) be relative to the Earth?

                                                   1.48 m/s × cos 36.5 degrees = 1.226 m/s

                                                   1.48 m/s × sin 36.5 degrees = 0.876 m/s

                                                   6.85 m/s × sin 53.5 degrees = 5.5 m/s

                                                    6.85 m/s × cos 53.5 degrees = 4.6 m/s

Now we can use the Pythagorean theorem to find the magnitude of the velocity of the ship relative to the Earth as follows:

                                                 √(5.5 m/s + 1.226 m/s)² + (4.6 m/s + 0.876 m/s)²

                                                                                   = √36.91 m²/s²

                                                                                  = 6.08 m/s

Therefore, the speed of the ship relative to the Earth is 6.08 m/s.

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1. A juggler tosses up a ball and catches it at the same level from which it was thrown. It leaves his hand with a speed of 12.0 m/s.

Given, The initial velocity of the ball, u = 12.0 m/final velocity of the ball,

v = -12.0 m/s (At the highest point, the velocity is zero)Acceleration acting on the ball,

a = -9.8 m/s2 (Acceleration due to gravity, acting downwards)Distance travelled by the ball,

s = ?Time taken for the ball to reach the maximum height,

t = ?Using the third equation of motion,

s = ut + 0.5at²Here, the ball is thrown upwards from a height 'h'. Therefore, the initial height of the

ball = hWhen the ball reaches the maximum height, the vertical velocity of the ball becomes zero.i.e., v = 0Using the first equation of motion,

v = u + at0

= u - 9.8t⇒

t = u/9.8Time taken for the ball to reach the maximum height,

t = 12/9.8

= 1.22 s.

Since the rocket is in free fall after it runs out of fuel, the time taken to reach the ground is given by:t = sqrt(2s/g)Here, s = 63 m (Maximum height reached by the rocket) and

g = 9.8 m/s² (Acceleration due to gravity)

t = sqrt(2 × 63 / 9.8)

= 4.05 sThus, the total time spent in the air is 4.05 s.

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