a) Three-point charges are placed at the following points on the x-axis: +2 µC at x = 0 cm.-3 µC at x = 40 cm, and -5 µC at x = 120 cm. Find the electrical force, F on the - 3 μC charge. [3 marks]

The null hypothesis states that the distribution of fatal injuries for riders not wearing a helmet follows the distribution for all riders, while the alternative hypothesis states that it does not follow the distribution for all riders. We can use the level of significance, which is typically set at 0.05, to test this hypothesis.

In hypothesis testing, the null hypothesis is the hypothesis that is being tested, while the alternative hypothesis is the hypothesis that is being considered if the null hypothesis is rejected. The null hypothesis in this case is that the distribution of fatal injuries for riders not wearing a helmet follows the distribution for all riders, while the alternative hypothesis is that it does not follow the distribution for all riders. To test this hypothesis, we can use the level of significance, which is typically set at 0.05. This means that we would reject the null hypothesis if the probability of getting the observed result under the null hypothesis is less than 0.05.

Therefore, the null hypothesis states that the distribution of fatal injuries for riders not wearing a helmet follows the distribution for all riders, while the alternative hypothesis states that it does not follow the distribution for all riders. We can use the level of significance, which is typically set at 0.05, to test this hypothesis.

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The object's velocity is √23 m/s when its potential energy is 23 J. The velocity of an object can be calculated by the equation [tex]KE=1/2mv²[/tex], where KE is the kinetic energy, m is the mass of the object, and v is the velocity of the object. Therefore, we can use this equation to find the velocity of an object when its potential energy is 23 J.

In order to solve this problem, we must first find the mass of the object. We know that potential energy is given by the equation [tex]PE=mgh[/tex], where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.

Since we are not given the height of the object, we cannot directly calculate its mass. However, we can use another equation to find the mass.

The equation is [tex]PE= 1/2mv²+ mgh[/tex], where PE is the potential energy, m is the mass of the object, v is the velocity of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.

Since we know the potential energy and the height of the object is 0, we can simplify the equation to [tex]PE=1/2mv².[/tex]

Solving for m, we get [tex]m=2PE/v²[/tex].

Substituting the given values, we have m=2(23)/v²=46/v².

Now that we have the mass, we can use the equation [tex]KE=1/2mv²[/tex] to find the velocity.

Since the potential energy of the object is equal to the kinetic energy, we have PE=KE=1/2mv².

Substituting the values we have, we get 23=1/2(46/v²)v².

Simplifying this equation, we get v²=46/2=23.

Therefore, v=√23. Hence, the object's velocity is √23 m/s when its potential energy is 23 J.

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The car traveled a distance of 90 m in that time. The correct option is C.

To find the distance traveled by the car, we can use the equation of motion: distance = initial velocity × time + (1/2) × acceleration × time². In this case, the initial velocity is 18.0 m/s, the time is 5.0 s, and the car comes to rest, which means the final velocity is 0 m/s. Since the acceleration is constant, we can use the equation to calculate the distance traveled.

Plugging in the values, we have:

distance = (18.0 m/s) × (5.0 s) + (1/2) × 0 × (5.0 s)²

distance = 90 m + 0 m

distance = 90 m

Therefore, the car traveled a distance of 90 m in 5.0 s. Option C is the correct answer.

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The statement "Technician B says the voltage regulator controls the strength of the rotor's magnetic field" is true.

A voltage regulator is an electrical regulator that changes the amount of voltage in an electrical circuit. Voltage regulators can be designed to handle varying amounts of input voltage and output voltage.

Technician B states that the voltage regulator is responsible for regulating the strength of the rotor's magnetic field. This is true because the rotor's magnetic field strength is determined by the voltage that is applied to it. If the voltage regulator fails, the magnetic field strength will decrease and the motor's performance will suffer.Therefore, technician B's statement is correct.

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As per the details given here, the current in the 2 Ω resistor is 3 A.

The potential difference across both resistors is the same since the 2Ω and 4Ω resistors are parallel.

In order to determine the total resistance of the parallel combination, we can apply the equivalent resistance formula for parallel resistors as follows:

[tex]\frac{1}{R_{re}} =\frac{1}{R_1} +\frac{1}{R_2}[/tex]

[tex]\frac{1}{R_{eq}} =\frac{1}{2}+ \frac{1}{4} \\\\\frac{1}{R_{eq}} = \frac{3}{4} \\\\R_{eq}=\frac{4}{3}[/tex]

Using ohm's law,

I = V/R

[tex]V_2=\frac{R_1}{R_1+R_2} (V_{total})[/tex]

[tex]V_2=\frac{2}{4/3} (12)[/tex]

So,

I = 6/2 = 3A.

Thus, the current in the 2 Ω resistor is 3 A.

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A) The magnification produced by the objective lens is -1/2. B) The angular magnification produced by just the eyepiece lens is approximately -0.107.

Lateral magnification is given by the ratio of the size of the image (I) to the size of the object (O). Since the image is inverted, this ratio is negative. So, the lateral magnification of the objective lens is given by:M = -I/O The objective lens has a focal length of f1 = 0.15 cm and is adjusted for relaxed viewing, meaning that the final image is at an infinite distance from the eye.

As a result, the distance between the objective lens and the eyepiece lens, d = 20 cm, is equal to the focal length of the eyepiece lens. Assume that the distance between the object and the objective lens is equal to the focal length of the objective lens, f1 = 0.15 cm.

Then, the distance between the objective lens and the image produced by the objective lens, d1 = f1(1 + M1), is also equal to 20 cm.Substituting the given values into the formula for the magnification produced by the objective lens:M1 = -d1/f1 = -(f1(1 + M1))/f1M1 = -1/2

Angular magnification is given by the ratio of the angle subtended by the image (θ') to the angle subtended by the object (θ). Since the image is magnified and inverted, this ratio is negative. So, the angular magnification of the eyepiece lens is given by:A = -θ'/θ

The final image produced by the objective lens is a real and inverted image, which is then used as the object for the eyepiece lens. Assume that the distance between the eyepiece lens and the final image is equal to the focal length of the eyepiece lens, f2 = 1.4 cm.

Then, the distance between the object (real and inverted) and the eyepiece lens is given by:d2 = f2Substituting the given values into the formula for the angular magnification produced by the eyepiece lens:A = -f1/f2A = -(0.15 cm)/(1.4 cm)A ≈ -0.107 Therefore, the angular magnification produced by just the eyepiece lens is approximately -0.107.

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The angular magnification of the telescope is -40.

To calculate the angular magnification (M) of a telescope, you can use the formula:

M = (-) (focal length of the objective) / (focal length of the eyepiece)

Given that the focal length of the objective (f_obj) is 100 cm and the focal length of the eyepiece (f_eye) is 2.50 cm, we can substitute these values into the formula:

M = (-) (100 cm) / (2.50 cm)

M = -40

The angular magnification of the telescope is -40. Note that the negative sign indicates that the image formed is inverted.

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At very low driving frequencies, an RLC circuit's capacitor behaves as an open circuit, while the inductor behaves as a short circuit.

In a low driving frequency, an RLC circuit operates differently than it does at a higher frequency. When the driving frequency is close to zero, it implies the frequency is very low, and therefore the impedance of the capacitor is high, making it appear like an open circuit.

The impedance of the inductor is low, making it appear as a short circuit, due to the flow of current through an inductor that generates a magnetic field, and the magnetic field opposes any changes in current flow, the inductor stores energy in its magnetic field, which is why it is considered as a short circuit at very low driving frequencies. In conclusion, at very low driving frequencies, an RLC circuit's capacitor behaves as an open circuit, while the inductor behaves as a short circuit.

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To determine the position of the shadow at a specific time, we can use the concept of angular velocity and the relationship between angular displacement and linear displacement.

Given:

Radius of the circle (r) = 3.39 m

Angular speed (ω) = 8.00 rad/s

Initial x-coordinate of the shadow (x) = 2.00 m The ball rotates counterclockwise, which means the shadow moves to the right initially. We can use the equation: x = r * cos(θ) At t = 0, the angular displacement (θ) is 0, and the x-coordinate of the shadow is 2.00 m. We can solve for θ using the inverse cosine function:

θ = cos^(-1)(x/r)

θ = cos^(-1)(2.00 m / 3.39 m)

Calculating the value of θ: θ ≈ 55.40 degrees. Since the ball rotates counterclockwise at a constant angular speed, we can determine the angular displacement at any given time using the equation: θ = ω * tmNow, let's find the angular displacement at t = 0. We substitute the values:θ = 8.00 rad/s * 0 s θ = 0 rad. Therefore, the shadow is initially at an angular displacement of 55.40 degrees, and the angular displacement remains 0 at t = 0.

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The force acting on a particle has a magnitude (a) The x-component of the force is 139.5 N. (b) The y-component of the force is 86.3 N.

To determine the x- and y-components of the force, we can use trigonometry. The given force has a magnitude of 162 N and is directed 32.4° above the positive x-axis.

(a) The x-component of the force is given by the equation:

x-component = force * cos(angle)

Plugging in the values:

x-component = 162 N * cos(32.4°) ≈ 139.5 N

(b) The y-component of the force is given by the equation:

y-component = force * sin(angle)

Plugging in the values:

y-component = 162 N * sin(32.4°) ≈ 86.3 N

Therefore, the x-component of the force is approximately 139.5 N and the y-component of the force is approximately 86.3 N.

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The possible index of refraction of the glass is 1.52 (Option B).

This means that light is not transmitted from the glass to air at an angle of 55° if the glass has an index of refraction around 1.52.

According to Snell's law, the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two media:

n₁ * sin(θ₁) = n₂ * sin(θ₂)

In this case, the angle of incidence (θ₁) is 55°, and the index of refraction of air (n₂) is 1.00. We need to determine the index of refraction of the glass (n₁).

Let's substitute the given values into Snell's law and solve for n₁:

n₁ * sin(55°) = 1.00 * sin(θ₂)

Since no light is transmitted in air, it means that the angle of refraction (θ₂) is 90°. Therefore, sin(θ₂) = 1.

n₁ * sin(55°) = 1.00 * 1

n₁ = 1 / sin(55°)

n₁ ≈ 1.52

Based on the calculation, the possible index of refraction of the glass is approximately 1.52 (Option B). This means that light is not transmitted from the glass to air at an angle of 55° if the glass has an index of refraction around 1.52.

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1 - The equilibrium temperature reached by the gold and copper can be determined using the principle of energy conservation.

The heat gained by one object is equal to the heat lost by the other object. The equation for heat transfer is:

m_gold * c_gold * (T_final - T_gold_initial) = -m_copper * c_copper * (T_final - T_copper_initial)

Substituting the given values, we can solve for the equilibrium temperature (T_final).

2 - The change in entropy (ΔS) of the copper block can be calculated using the relationship ΔS = S_final - S_initial = C * ln(T_final / T_initial), where C is the heat capacity at constant volume. Since there is no change in volume, we have AS = S * 4dU, where dU represents the change in internal energy. For no change in volume, dU is zero. Therefore, the entropy change of the copper block is zero (ΔS = 0 J/K).

3 - The total change in entropy (ΔS_total) of the gold and copper system can be calculated by summing the individual entropy changes:

ΔS_total = ΔS_gold + ΔS_copper

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The electric field at a point due to a charged body is defined as the amount of force experienced by a unit positive charge placed at that point. The magnitude of the electric field on the x-axis at x = -8 m is 4.68×10⁴ N/C. Option (D) is correct 16 N/C.

The magnitude of the electric field due to a point charge, q, at a distance, r, from the charge is given by:E = (1/4πε₀)q/r²where ε₀ is the permittivity of free space.In this case, we know that the charge is -6.00 µC, the distance from the charge to the point where we want to find the electric field is -8 m.To find the electric field on the x-axis at x = -8 m, we can use the formula:E = (1/4πε₀)q/r² where r = 8m, q = -6.00µC.Substituting the values of r, q and ε₀ into the above equation, we get:E = (1/4πε₀)(-6.00 µC)/8²E = (-2.70×10⁶)/8²ε₀ = 8.854×10⁻¹² F/mE = -4.68×10⁴ N/CSo, the magnitude of the electric field on the x-axis at x = -8 m is 4.68×10⁴ N/C which is option (d).Hence, the correct option is d) 16 N/C.

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A pendulum is an object that hangs from a fixed point and is allowed to swing freely under the influence of gravity. It consists of a weight called a bob that is suspended from a fixed point by a string. In the laboratory, a student studies a pendulum by graphing the angle θ that the string makes with the vertical as a function of time t, observing the period of the pendulum, and measuring its length.

The period of a pendulum is the time it takes for one complete cycle of motion. The period of a pendulum is influenced by the length of the string, as well as the acceleration due to gravity. A longer string will have a longer period than a shorter one because it has a larger arc to travel through, while a shorter string will have a shorter period. The acceleration due to gravity, on the other hand, is constant, so it will not affect the period of a pendulum. The angle θ that the string makes with the vertical is also influenced by the length of the string. The longer the string, the less it will swing, and the smaller the angle θ will be.

The shorter the string, the more it will swing, and the larger the angle θ will be. Graphing the angle θ as a function of time t will reveal that the pendulum follows a periodic pattern. The amplitude of the angle θ will decrease over time due to the resistance of the air, resulting in a damping effect on the pendulum.

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The final mass of the food item less than the original mass se of the loss of water.

When cooking, it is not uncommon to notice a difference in mass between the food item before cooking and after cooking. The final mass of the food item is always less than the original mass because of the loss of water. When food is cooked, the heat applied causes the water content in the food to evaporate, causing the food item to shrink in size and weight. The amount of mass lost depends on the water content in the food item and the cooking method used.

When food is boiled, more water content evaporates due to the high temperatures, which can result in a more significant difference in mass. The final mass of the food item may also be affected by the cooking method used. For instance, frying may result in a lower loss of mass compared to boiling. In summary, the final mass of the food item is less than the original mass because of the loss of water through evaporation.

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The final mass of a food item is typically less than the original mass due to the process of dehydration or water loss that occurs during cooking or other forms of food preparation. This can result in a reduction in the total weight of the food item, even though the actual amount of food present remains the same.

When food is cooked or exposed to heat, the heat causes the moisture present in the food to evaporate, and this leads to a loss of weight. As a result, the final mass of the food item is less than the original mass.

The extent to which the weight is reduced will depend on the method of cooking, the temperature at which the food is cooked, and the length of time it is cooked for. For example, when a piece of chicken is cooked on a grill, it will initially weigh more than the final weight once it is cooked. This is because as the chicken cooks, some of the moisture and fat inside it will be released, and this will evaporate.

Therefore, by the time the chicken is fully cooked, it will have lost some of its original weight due to the loss of moisture and fat. Therefore, the final mass of a food item is often less than the original mass due to the process of dehydration or water loss that occurs during cooking or other forms of food preparation.

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The lowest pressure that can be achieved using the best available vacuum techniques is about 10−12n/m2.

Vacuum technology is used in a wide range of scientific and industrial applications. The vacuum is obtained using a range of methods, including mechanical pumps, turbomolecular pumps, and diffusion pumps, to name a few. Vacuum systems are used in many fields, including high-energy physics, surface science, and semiconductor manufacturing, among others.

In vacuum technology, the pressure is usually measured in pascal, torr, or millibar. The lowest pressure that can be achieved using the best available vacuum techniques is about 10−12n/m². This pressure is known as the ultra-high vacuum (UHV), which is used for a variety of applications, including surface analysis, material science, and vacuum deposition.

The UHV systems are expensive and require a high level of expertise to operate because they are extremely sensitive to contamination. As a result, UHV is used only when an uncontaminated environment is critical for the process being conducted.

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In the given sentence, "one, by the band u2, is a song that I find very inspirational," add quotation marks around one, hence option A is correct.

Punctuation is the use of white space, traditional signals, and specific typographical elements to help readers understand and interpret written material correctly, whether they are reading it quietly or loudly.

In many writing systems, quote marks are punctuation symbols that are used in pairs to demarcate a quotation, direct speech, or a phrase.

Thus, the correct sentence is: "One," by the band u2, is a song that I find very inspirational.

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Part a) The frequency that the moth detects is 86.33 kHz. Part b) The frequency that the bat detects in the returning echo from the moth is 100.01 kHz.

Ultrasonic waves are sound waves with frequencies greater than can be heard by humans. Bats find their way and search for food by emitting and detecting reflections of ultrasonic waves. A bat emits ultrasound at frequency f be = 93.67 kHz while flying with a velocity = 12.00 î as it chases a moth that flies with velocity V m = 6.00 î. Part a) The frequency that the moth detects can be calculated using the Doppler effect formula: f_ m = f_ be(1 + V_ b/V_ w) / (1 + V_ m/V_ w )f_ be = 93.67 kHz V_ b = 12 î V_ w = 343 m/s V_ m = 6 î Putting all the values in the above formula: f_ m = 86.33 kHz The frequency that the moth detects is 86.33 kHz. Part b) The frequency that the bat detects in the returning echo from the moth can be calculated using the Doppler effect formula: f_ b = f_ be(1 + V_ b/V_ w) / (1 - V_ m/V_ w)f_ be = 93.67 kHz V_ b = 12 î V_ w = 343 m/s V_ m = 6 î Putting all the values in the above formula: f_ b = 100.01 kHz  The frequency that the bat detects in the returning echo from the moth is 100.01 kHz.

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During time interval t2, the voltage is decreasing from the maximum value to 0, as the magnetic flux linkage with the coil is reduced.

When an isolated magnetic North pole is discovered and dropped through the set-up (magnet through a coil experiment), there is a change in magnetic flux linkage within the coil. Therefore, the induced electromotive force (EMF) will cause a voltage pattern to form.The Faraday's Law of Electromagnetic Induction states that when there is a change in magnetic flux linkage within a coil, an EMF is induced in the coil.

In this scenario, the magnetic flux linkage increases as the magnetic North pole enters the coil and decreases when it exits the coil, which will result in a change in the direction of the induced EMF within the coil.The voltage pattern obtained from the experiment depends on the rate at which the magnetic North pole is dropped through the set-up and the number of turns of the coil.

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The assigned movie, as a representative art form, immerses viewers in a temporal and spatial experience, inviting them to reflect on the essence of being and their own existence, aligning with Heidegger's emphasis on ontological exploration through art.

The art piece assigned is a movie. It is considered a good representative piece of the art form due to its ability to immerse the viewer in a temporal and spatial experience.

Drawing on Heideggerian concepts, the movie reveals the essence of being through its portrayal of human existence and the unfolding of time.

The film creates a world that invites the viewer to engage with their own understanding of existence and meaning.

It prompts reflection on the nature of being and encourages a deeper exploration of one's own existence in relation to the world.

Through its narrative and cinematic techniques, the movie provides a platform for existential questioning and philosophical contemplation, aligning with Heidegger's emphasis on the ontological aspects of art.

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The linear speed required for an Earth satellite to be in a circular orbit at an altitude of 107 km above Earth's surface is approximately 7.85 km/s. The period of revolution for this satellite is around 1 hour and 34 minutes.

To explain this further, let's consider the concept of circular motion and gravitational force. When an object is in a circular orbit, it experiences centripetal force directed towards the center of the circle. In the case of a satellite orbiting the Earth, this force is provided by the gravitational pull of the Earth.

The centripetal force (F) can be calculated using the equation F = m * a, where m is the mass of the satellite and a is the acceleration towards the center of the circle. In this case, the acceleration is provided by gravity, which can be represented as g (approximately 9.8 m/s²).

Since the satellite is in a circular orbit, the centripetal force is equal to the gravitational force between the satellite and the Earth, given by the equation F = G * (m * M) / r², where G is the gravitational constant, M is the mass of the Earth, and r is the distance between the satellite and the center of the Earth.

By equating these two equations, we can solve for the speed of the satellite. The centripetal force can be rewritten as m * v² / r, where v is the linear speed of the satellite. Setting these two equal and solving for v, we get v = √(G * M / r).

Plugging in the values for G (6.67430 x 10⁻¹¹ m³ kg⁻¹ s⁻²), M (5.97219 x 10²⁴ kg), and r (107 km + 6371 km, since the altitude is given above Earth's surface), we can calculate the linear speed, which comes out to approximately 7.85 km/s.

To find the period of revolution, we can use the formula T = 2πr / v, where T is the period and π is a mathematical constant. Plugging in the values, we find the period to be approximately 1 hour and 34 minutes.

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The image of the belt buckle seen in the mirror is 0.01 times the height of the actual object, and it is inverted.

The image formed in a mirror depends on the position of the object from the mirror and its size and orientation with respect to the mirror. The distance of an object from a mirror is known as its image distance and is denoted by v, while the distance of an object from a mirror is known as its object distance and is denoted by u. The focal length of the mirror is denoted by f. In this case, the image distance v, object distance u, and the radius of curvature R are equal to the focal length f. The object distance u is 2.4 mm from the mirror, while the distance between the belt buckle and the eyes is 0.74 mm. Hence, the image distance v = f = R = 2.4 mm. The magnification of the image formed is given by the ratio of the height of the image to the height of the object. Since the belt buckle is below the eyes, the image is inverted and the height of the image is -0.74 mm. The height of the object is the distance between the eyes and the belt buckle, which is 74 mm. Hence, the magnification is given by:-0.74/74 = -0.01. The negative sign indicates that the image is inverted.

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To determine how long the ball is in the air, we need to consider its motion as it goes up and then comes back down so when a ball is thrown straight up into air at 49m/s. For 10s ball is in the air.

The correct answer is option B.

To determine how long the ball is in the air, we need to consider its motion as it goes up and then comes back down. We can calculate the time it takes for the ball to reach its highest point and then double that time to find the total time in the air.

Given:

Initial velocity (u) = 49 m/s

a) To find the time for the ball to reach its highest point, we can use the formula:

v = u + gt

Where:

v is the final velocity,

u is the initial velocity,

g is the acceleration due to gravity (approximately -9.8 m/s²),

t is the time.

At the highest point, the ball's final velocity is 0 m/s. Substituting the given values, we have:

0 = 49 m/s + (-9.8 m/s²)[tex]t_highest[/tex]

Solving for [tex]t_highest[/tex], we get:

[tex]t_highest[/tex] = 49 m/s / 9.8 m/s² ≈ 5 s

The time for the ball to reach its highest point is approximately 5 seconds.

b) To find the total time in the air for 8 seconds, we simply double the time to reach the highest point

Total time = 2 *[tex]t_highest[/tex] = 2 * 5 s = 10 s

c) To find the total time in the air for 10 seconds, we again double the time to reach the highest point:

Total time = 2 * [tex]t_highest[/tex] = 2 * 5 s = 10 s

d) To find the total time in the air for 7 seconds, we compare it to the time to reach the highest point:

7 s <[tex]t_highest[/tex]

Therefore, the ball is not in the air for 7 seconds.

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Velocity of piece C (v₃) is 0 m/s, meaning it comes to a stop after the block shatters. Thus, the direction of the smallest piece of ice (C) immediately after the shattering is stationary (not moving).

(a) To calculate the speed and direction of the smallest piece of ice (C) immediately after the block has shattered, we can apply the conservation of linear momentum.

Given:

Mass of piece A (m₁) = 0.90 kg

Mass of piece B (m₂) = 0.80 kg

Mass of piece C (m₃) = 0.10 kg

Speed of piece A (v₁) = -0.60 m/s (negative x-direction)

Speed of piece B (v₂) = 0.40 m/s (positive y-direction)

The total momentum before the block shatters is equal to the total momentum after the shattering. The momentum is given by:

Initial momentum = (mass of A × velocity of A) + (mass of B × velocity of B) + (mass of C × velocity of C)

Since piece C is the smallest piece, its mass (m₃) is the smallest. Let the speed of piece C be v₃. The momentum after the shattering is given by:

Final momentum = (mass of A × velocity of A) + (mass of B × velocity of B) + (mass of C × velocity of C)

According to the conservation of linear momentum, the initial momentum and final momentum are equal:

Initial momentum = Final momentum

Solving for v₃:

(mass of A × velocity of A) + (mass of B × velocity of B) + (mass of C × velocity of C) = (mass of A × velocity of A) + (mass of B × velocity of B) + (mass of C × v₃)

(0.90 kg × -0.60 m/s) + (0.80 kg × 0.40 m/s) + (0.10 kg × v₃) = (0.90 kg × -0.60 m/s) + (0.80 kg × 0.40 m/s) + (0.10 kg × v₃)

Simplifying the equation, we find:

0.10 kg × v₃ = 0

This implies that the velocity of piece C (v₃) is 0 m/s, meaning it comes to a stop after the block shatters. Thus, the direction of the smallest piece of ice (C) immediately after the shattering is stationary (not moving).

(b) (i) The resulting change in gravitational potential energy will be negative. When an object moves closer to a gravitational field, its gravitational potential energy decreases, resulting in a negative change.

(ii) To calculate the change in gravitational potential energy, we can use the formula:

Change in gravitational potential energy = - G * (mass of the sphere) * (mass of the particle) / (final distance - initial distance)

Given:

Mass of the sphere = 4.5 × 10^7 kg

Mass of the particle = 5.0 × 10^-3 kg

Initial distance = 12 m

Final distance = 160 m

Gravitational constant (G) = 6.67 × 10^-11 N m²/kg²

Change in gravitational potential energy = - (6.67 × 10^-11 N m²/kg²) * (4.5 × 10^7 kg) * (5.0 × 10^-3 kg) / (160 m - 12 m)

Calculating the change in gravitational potential energy will give us the numerical value.

(iii) The change in gravitational potential energy does not depend on the path taken by the particle. It only depends on the initial and final positions and the masses involved. Therefore, whether the particle moves in a straight line or follows a complicated path

, the change in gravitational potential energy remains the same.

(iv) Substituting the values into the formula from part (ii):

Change in gravitational potential energy = - (6.67 × 10^-11 N m²/kg²) * (4.5 × 10^7 kg) * (5.0 × 10^-3 kg) / (160 m - 12 m)

Calculating this expression will give us the numerical value of the difference in gravitational potential between the particle's initial position (12 m from the centre of the sphere) and its final position (160 m from the centre of the sphere).

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The equation of the resultant wave is y = 0.4sin(2π0.2x)cos(2π30t), where the amplitude is 0.4 and the frequencies are 0.2 cycles per unit length (x) and 30 cycles per unit time (t).

For the equation of the resultant wave, we need to add the displacements of the two waves.

The instantaneous displacements of the two waves are given by:

y1 = 0.2sin(2π0.2x + 2π30t)

y2 = 0.2sin(2π0.2x - 2π30t)

We can use the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) to simplify the equation. Applying this identity, we get:

y1 + y2 = 0.2sin(2π0.2x + 2π30t) + 0.2sin(2π0.2x - 2π30t)

   = 0.2sin(2π0.2x)cos(2π30t) + 0.2cos(2π0.2x)sin(2π30t) + 0.2sin(2π0.2x)cos(2π30t) - 0.2cos(2π0.2x)sin(2π30t)

       = 0.4sin(2π0.2x)cos(2π30t)

Therefore, the equation of the resultant wave is

y = 0.4sin(2π0.2x)cos(2π30t).

This equation represents a wave with a displacement that varies sinusoidally in both space (x) and time (t).

The amplitude of the wave is 0.4, and the frequency of the wave in space is 0.2 cycles per unit length (x), while the frequency in time is 30 cycles per unit time (t).

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The maximum kinetic energy of the ejected electron is approximately [tex]\(4.51 \times 10^{-19}\, \text{J}\)[/tex], and its maximum speed is approximately [tex]\(5.79 \times 10^5\, \text{m/s}\)[/tex]. the critical frequency below which no electrons are ejected from sodium is approximately [tex]\(9.27 \times 10^{14}\, \text{Hz}\)[/tex].

The kinetic energy of electrons emitted when yellow light of [tex]\(\lambda = 600\)[/tex] nm is incident on sodium is approximately [tex]\(2.15 \times 10^{-19}\, \text{J}\)[/tex].

a) To calculate the maximum kinetic energy [tex](\(E_{\text{max}}\))[/tex] and speed [tex](\(v_{\text{max}}\))[/tex] of an electron ejected from a sodium surface in the photoelectric effect, we can use the following formulas:

[tex]\[E_{\text{max}} = h \cdot \nu - \phi\]\\\\\v_{\text{max}} = \sqrt{\frac{{2E_{\text{max}}}}{{m_e}}}\][/tex]

where:

[tex]\(h\) is Planck's constant (\(6.62607015 \times 10^{-34}\, \text{J}\cdot\text{s}\))[/tex],

[tex]\(\nu\)[/tex] is the frequency of the incident light [tex](\(\frac{c}{\lambda}\), where \(c\)[/tex] is the speed of light and [tex]\(\lambda\)[/tex] is the wavelength of the light),

[tex]\(\phi\)[/tex] is the work function of sodium (in electron volts, eV),

[tex]\(m_e\)[/tex] is the mass of an electron [tex](\(9.10938356 \times 10^{-31}\, \text{kg}\))[/tex].

Given:

Wavelength of the incident light [tex](\(\lambda\))[/tex] = 410 nm [tex](\(410 \times 10^{-9}\, \text{m}\))[/tex],

Work function of sodium [tex](\(\phi\))[/tex] = 2.28 eV [tex](\(2.28 \times 1.602176634 \times 10^{-19}\, \text{J}\))[/tex].

First, calculate the frequency of the incident light:

[tex]\[\nu = \frac{c}{\lambda} = \frac{3 \times 10^8\, \text{m/s}}{410 \times 10^{-9}\, \text{m}} \\\\= 7.317 \times 10^{14}\, \text{Hz}\][/tex]

Now substitute the values into the equations to calculate [tex]\(E_{\text{max}}\) and \(v_{\text{max}}\)[/tex]:

[tex]\[E_{\text{max}} = (6.62607015 \times 10^{-34}\, \text{J}\cdot\text{s}) \cdot (7.317 \times 10^{14}\, \text{Hz}) - (2.28 \times 1.602176634 \times 10^{-19}\, \text{J})\]\\\v_{\text{max}} = \sqrt{\frac{{2 \cdot E_{\text{max}}}}{{9.10938356 \times 10^{-31}\, \text{kg}}}}\][/tex]

After evaluating the equations, we find:

[tex]\(E_{\text{max}} \approx 4.51 \times 10^{-19}\, \text{J}\)[/tex]

[tex]\(v_{\text{max}} \approx 5.79 \times 10^5\, \text{m/s}\)[/tex]

Therefore, the maximum kinetic energy of the ejected electron is approximately [tex]\(4.51 \times 10^{-19}\, \text{J}\)[/tex], and its maximum speed is approximately [tex]\(5.79 \times 10^5\, \text{m/s}\)[/tex].

b) The critical frequency [tex](\(\nu_{\text{c}}\))[/tex] is the threshold frequency below which no electrons are ejected. It can be calculated using the formula:

[tex]\[\nu_{\text{c}} = \frac{{\phi}}{{h}}\][/tex]

Substituting the values into the formula:

[tex]\[\nu_{\text{c}} = \frac{{2.28 \times 1.602176634 \times 10^{-19}\, \text{J}}}{{6.62607015 \times 10^{-34}\, \text{J}\cdot\text{s}}}\][/tex]

After evaluating the equation, we find:

[tex]\(\nu_{\text{c}} \approx 9.27 \times 10^{14}\, \text{Hz}\)[/tex]

Therefore, the critical frequency below which no electrons are ejected from sodium is approximately [tex]\(9.27 \times 10^{14}\, \text{Hz}\)[/tex].

c) To calculate the kinetic energy of electrons emitted when yellow light of [tex]\(\lambda = 600\)[/tex] nm is incident on sodium, we can use the same formula as in part (a):

[tex]\[E_{\text{max}} = h \cdot \nu - \phi\][/tex]

Given:

Wavelength of the yellow light [tex](\(\lambda\))[/tex] = 600 nm [tex](\(600 \times 10^{-9}\, \text{m}\))[/tex]

Calculate the frequency of the yellow light:

[tex]\[\nu = \frac{c}{\lambda} = \frac{3 \times 10^8\, \text{m/s}}{600 \times 10^{-9}\, \text{m}} \\\\= 5 \times 10^{14}\, \text{Hz}\][/tex]

Substitute the values into the equation to calculate [tex]\(E_{\text{max}}\)[/tex]:

[tex]\[E_{\text{max}} = (6.62607015 \times 10^{-34}\, \text{J}\cdot\text{s}) \cdot (5 \times 10^{14}\, \text{Hz}) - (2.28 \times 1.602176634 \times 10^{-19}\, \text{J})\][/tex]

After evaluating the equation, we find:

[tex]\(E_{\text{max}} \approx 2.15 \times 10^{-19}\, \text{J}\)\\[/tex]

Therefore, the kinetic energy of electrons emitted when yellow light of [tex]\(\lambda = 600\)[/tex] nm is incident on sodium is approximately [tex]\(2.15 \times 10^{-19}\, \text{J}\)[/tex].

d).

graph is in the image attached

KE is kinetic energy

f is frequency

Wo is work function and h is slope of the graph

fo is critical frequency

slope of the graph will represent plank's constant

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An object must travel at a speed of 0.14 times the speed of light for its total energy to be 1% more than its rest energy. To have its total energy 99% more than its rest energy, the object must travel at a speed of 0.8654 times the speed of light.

To determine how fast must an object travel for its total energy to be 1% more than its rest energy and 99% more than its rest energy, we use the formula for relativistic kinetic energy K = (γ - 1)mc² where γ = 1/√(1 - v²/c²). The object must travel at a speed of 0.14 times the speed of light for its total energy to be 1% more than its rest energy.

Similarly, the object must travel at a speed of 0.8654 times the speed of light for its total energy to be 99% more than its rest energy. The speed at which an object must travel to achieve relativistic speeds becomes closer and closer to the speed of light as the object's total energy approaches infinity. At the speed of light, an object's total energy would be infinite.

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(a) At t=0, the switch is closed for the first time. Hence, the capacitor will start to charge from 0 to the full voltage of the battery over time. The time constant τ is given by:τ = RC = 61.9 Ω × 4.00 mF = 0.247 s When the switch is closed, the capacitor acts like an open circuit (i.e., does not allow current to flow) for a very short time until it charges up. Hence, we can consider the circuit without the capacitor for t < 0 and then add the capacitor at t = 0.

According to Kirchhoff's loop rule, the voltage around the loop should be zero: iR + vC = 0 Here, i is the current in the circuit at time t, R is the resistance, vC i the voltage across the capacitor terminals. iR = i × R = (9.42 V)/(61.9 Ω) = 0.152 A Voltage across the capacitor at t = 0 is given by: vC = V0(1 - e-t/τ) = 9.42 V(1 - e-0/0.247) = 9.42 V(1 - 1) = 0 V Substituting the values in the loop rule: iR + vC = 0 we get: iR = - vC => i = - vC/R = - 0/61.9 = 0 Also, the current in the circuit at t = 0 is 0 A.(b) One time constant after the switch is closed (t = τ)Let's consider the circuit diagram again as shown below: The voltage across the capacitor terminals is given by: vC = V0(1 - e-t/τ) = 9.42 V(1 - e-τ/τ) = 9.42 V(1 - e-1) = 3.53 V According to Kirchhoff's loop rule, the voltage around the loop should be zero: iR + vC = 0Substituting the values in the loop rule: iR + vC = 0 we get: iR = - vC => i = - vC /R = - 3.53 V/61.9 Ω = - 0.057 Also, the current in the circuit at t = τ is - 0.057 A.

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a) The current flowing through the resistor at t = 0 is 0.152 A.

b) The current flowing through the resistor one time constant after the switch is closed is 0.099 A.

Given Data; Resistor, R = 61.9 Ω, Capacitance, C = 4.00 mF = 4.00 x 10^⁻3 FEMF of battery, ε = 9.42 V.

(a) Current through the resistor at t = 0, when the switch is closed. We know that initially (i.e., for t < 0), the capacitor was uncharged. Therefore, there is no charge on the capacitor before closing the switch. When the switch is closed, the capacitor starts charging, and the current flows in the circuit. Hence, current starts flowing through the circuit instantaneously. The current is maximum at t = 0.

According to Kirchhoff's Loop Rule, we have: ε = V_R + V_C, where V_R is the potential difference across the resistor, and V_C is the potential difference across the capacitor at any time t. Since the capacitor is uncharged before closing the switch, there is no potential difference across the capacitor at t = 0.Now, applying Kirchhoff's Loop Rule, we get:ε = V_R + V_Cε = IR + (q / C) ...(1) where, I is the current in the circuit at any time t and q is the charge on the capacitor at time t=0.At t = 0, the capacitor is uncharged, so q = 0. Substituting the given values in equation (1), we get;9.42 = I x 61.9I = 0.152 A. Therefore, the current flowing through the resistor at t = 0 is 0.152 A.

(b) Current through the resistor at t = t_r = R x C = 61.9 x 4.00 x 10^⁻3 = 0.2476 s. One-time constant (t_r) after the switch is closed, the charge on the capacitor will be (1 - 1/e) times the maximum charge (q_max) on the capacitor. Hence, the potential difference across the capacitor at t = t_r is given by: V_C = q / C = q_max (1 - e^(-t/t_r)) / C. Substituting q_max = ε x C in the above equation, we get: V_C = ε (1 - e^(-t/t_r)). Therefore, the potential difference across the resistor is given by: V_R = ε - V_CV_R = ε - ε (1 - e^(-t/t_r))V_R = ε e^(-t/t_r) Substituting the value of V_R in the equation (1), we get;ε = IR + V_Cε = IR + ε (1 - e^(-t/t_r)) / CI = (ε / R) (1 - e^(-t/t_r)) Substituting the given values, we get; I = (9.42 / 61.9) (1 - e^(-0.2476 / 0.2476))I = 0.099 A. Therefore, the current flowing through the resistor one time constant after the switch is closed is 0.099 A.

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The total momentum of the system never changes when two or more objects collide. The correct option is B.

When two or more objects collide in a closed system, the total momentum of the system remains constant. Momentum is defined as the product of an object's mass and its velocity, given by the equation p = mv, where p is the momentum, m is the mass, and v is the velocity.

In a closed system, the total momentum before the collision is equal to the total momentum after the collision, provided that no external forces act on the system. This principle is known as the law of conservation of momentum.

Kinetic energy, on the other hand, is not conserved during collisions. Kinetic energy is given by the equation KE = (1/2)mv², where KE is the kinetic energy, m is the mass, and v is the velocity. During a collision, some of the kinetic energy may be converted into other forms, such as heat, sound, or deformation.

Therefore, while the kinetic energy of each object and the total kinetic energy of the system may change during a collision, the total momentum of the system remains constant. Hence, the correct answer is B, that the total momentum of the system never changes when two or more objects collide.

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(a) The approximate annual energy capacity of the power station is 48,180 TWh. (b) The energy conversion efficiency is 8.3%. (c) The amount of coal supply needed is 24,090,000,000 tonnes.

For part (a), we used the formula for annual energy capacity which takes into account the power output, hours of operation, and days of operation per year. For part (b), we used the energy obtained from burning 1 kg of coal and the energy density of coal to calculate the energy conversion efficiency. We used the formula for energy conversion efficiency and found that it is 8.3%.

For part (c), we used the amount of energy generated in one year and the energy obtained from burning 1 kg of coal to calculate the amount of coal needed. We used the formula for amount of coal needed and found that it is 24,090,000,000 tonnes.

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