The resultant velocity of the plane, relative to the ground (groundspeed) is approximately 315.82 km/hr which is calculated using a trigonometric approach.
To find the groundspeed of the plane, we need to calculate the resultant velocity by considering the vector addition of the plane's airspeed and the wind velocity.
First, we decompose the airspeed into its components. The southward component of the airspeed can be found by multiplying the airspeed (662 km/hr) by the sine of the angle between the direction of the airspeed and the south direction ([tex]28.0^0[/tex]). This gives us a southward airspeed component of approximately 309.81 km/hr.
Next, we decompose the wind velocity into its components. The westward component of the wind velocity is obtained by multiplying the wind velocity (77.5 km/hr) by the cosine of the angle between the wind direction and the east direction ([tex]180^0 - 75^0 = 105^0[/tex]). This gives us a westward wind component of approximately 31.59 km/hr.
Now, we can find the resultant velocity by adding the components. The groundspeed is the magnitude of the resultant velocity and can be calculated using the Pythagorean theorem. The groundspeed is approximately 315.82 km/hr.
To summarize, the resultant velocity of the plane, relative to the ground, is approximately 315.82 km/hr. This is obtained by considering the vector addition of the plane's airspeed and the wind velocity.
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A 58 Ni ion of charge 1 proton and mass 9.62x10-26kg is accelerated trough a potential difference of 3kV and deflected in a magnetic field of 0.12T. The velocity vector is perpendicular to the direction of magnetic field. a. [5] Find the radius of curvature of the ion's orbit. b. [4] A proton, accelerated to the same velocity as the 58Ni, also enters the same magnetic field. Is the radius of curvature of proton is going to be bigger/smaller/the same? Justify your answer.
(a) The radius of curvature of a 58Ni ion's orbit can be found using the given information of its charge, mass, potential difference, and magnetic field strength.
(b) The radius of curvature for a proton accelerated to the same velocity and entering the same magnetic field will be smaller than that of the 58Ni ion due to the proton's smaller mass.
(a) The radius of curvature (r) of the 58Ni ion's orbit can be determined using the equation r = (mv) / (qB), By substituting the given values and solving the equation, the radius of curvature can be calculated.
(b) For the proton, since it has a smaller mass compared to the 58Ni ion, its radius of curvature will be smaller. This can be justified by considering the equation r = (mv) / (qB). Therefore, the radius of curvature for the proton will be smaller than that of the 58Ni ion.
In conclusion, the radius of curvature for the 58Ni ion's orbit can be calculated using the given information, and the radius of curvature for the proton will be smaller due to its smaller mass.
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In a typical electron microscope, the momentum of each electron is about 1.3 x 10⁻²² kg-m/s. What is the de Broglie wavelength of the electrons?
m
In a typical electron microscope, the momentum of each electron is about 1.3 x 10⁻²² kg-m/s. The de Broglie wavelength of the electrons is approximately 5.097 x 10^-12 meters.
To calculate the de Broglie wavelength of electrons, we can use the de Broglie wavelength equation:
λ = h / p
where:
λ is the de Broglie wavelength,
h is the Planck's constant (approximately 6.626 x 10^-34 J·s),
p is the momentum of the electron.
Given:
p = 1.3 x 10^-22 kg·m/s
Substituting the values into the equation:
λ = (6.626 x 10^-34 J·s) / (1.3 x 10^-22 kg·m/s)
Simplifying the equation:
λ = (6.626 x 10^-34 J·s) / (1.3 x 10^-22 kg·m/s)
λ ≈ 5.097 x 10^-12 meters
Therefore, the de Broglie wavelength of the electrons is approximately 5.097 x 10^-12 meters.
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two blocks hang vertically, and are connected by a maskes since which is koped over a massiss, frictionless pulley as shown. One block Stimes as much mass as the other, the magnitude of acceleration of the smaller back is
Two blocks of different masses are connected by a massless, frictionless pulley. The smaller block experiences an acceleration, and its magnitude is one-third of the acceleration due to gravity (g).
In the given scenario, let's assume the mass of the smaller block is denoted as [tex]m_1[/tex], and the mass of the larger block is [tex]2m_1[/tex] since it is stated that one block times as much mass as the other. The system is connected by a massless, frictionless pulley, implying that the tension in the string remains the same on both sides.
Considering the forces acting on the smaller block, we have the tension force (T) acting upwards and the weight force (mg) acting downwards. As the block experiences acceleration, the net force acting on it can be determined using Newton's second law: net force = mass * acceleration. Therefore, we have [tex]T - mg = m_1a[/tex], where a represents the acceleration of the smaller block.
Since the mass of the larger block is [tex]2m_1[/tex], the weight force acting on it is [tex]2m_1g[/tex]. As the pulley is frictionless, the tension in the string remains constant. Hence, we can set up an equation for the larger block as well: [tex]2m_1g - T = 2m_1a[/tex].
To find the magnitude of acceleration for the smaller block, we can eliminate T from the above two equations. Adding the equations together, we get: [tex]T - mg + 2m_1g - T = m_1a + 2m_1a[/tex]. Simplifying this expression gives: g = 3a. Therefore, the magnitude of acceleration for the smaller block is one-third of the acceleration due to gravity (g).
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If a 6.87x10-6 C charge is placed at the origin, with coordinates -- (0.0). What is the magnitude of the electric field at a point located at coordinates (18,97 Note: use epsilon value of 8.85 10-12 F/m
The magnitude of the electric field at the point (18,97) due to a 6.87x10-6 C charge placed at the origin (0,0) is approximately [tex]5.57*10^3[/tex].
To calculate the magnitude of the electric field at the given point, we can use the formula for electric field intensity:
[tex]E = k * q / r^2[/tex]
Where:
E is the electric field intensity,
k is the electrostatic constant [tex](k = 8.99*10^9 Nm^2/C^2),[/tex]
q is the charge [tex](6.87*10^-^6 C)[/tex], and
r is the distance between the charge and the point of interest.
In this case, the distance between the charge at the origin and the point (18,97) is calculated using the distance formula:
[tex]r = \sqrt((x2 - x1)^2 + (y2 - y1)^2)\\= \sqrt((18 - 0)^2 + (97 - 0)^2)\\= \sqrt(324 + 9409)\\= \sqrt(9733)\\=98.65 m[/tex]
Substituting the values into the formula, we get:
[tex]E = (8.99*10^9 Nm^2/C^2) * (6.87*10^-^6 C) / (98.65 m)^2\\= 5.57*10^3 N/C[/tex]
Therefore, the magnitude of the electric field at the point (18,97) is [tex]5.57*10^3[/tex] N/C.
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An air parcel is sinking 1 km. The temperature in the parcel increases by 10 degrees C, but the vapor pressure does not change. The vapor pressure in the parcel is 10hPa, and the saturation vapor pressure in the parcel is 20hPa. What is the relative humidity?
The relative humidity is 50%, indicating the air is holding half of the moisture it can hold at the current temperature, aiding in weather predictions.
Given that an air parcel is sinking 1 km, the temperature in the parcel increases by 10 degrees C, but the vapor pressure remains constant. The vapor pressure in the parcel is 10 hPa, and the saturation vapor pressure is 20 hPa within the parcel. To calculate the relative humidity, we use the formula: Relative Humidity = Vapor pressure / Saturation vapor pressure * 100.
Plugging in the given values, we have: Relative humidity = 10 / 20 * 100. Simplifying the equation, we find that the relative humidity is 50%.
A relative humidity of 50% indicates that the air is holding half the amount of moisture it is capable of holding at the current temperature. This measure is crucial in meteorology as it helps forecasters predict cloud formation, precipitation, and other weather phenomena.
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If a projectile is launched downwards, the value of v0y is: A. Zero B. Positive C. Negative
D. Cannot be determined from the problem.
When a projectile is launched downwards, the value of v0y is negative.
Let's define the variables: vy = vertical component of velocity.
v0y = initial vertical component of velocity. a = acceleration (due to gravity) = -9.8 m/s²
When a projectile is launched downwards, it means the angle of projection is downwards. The vertical component of velocity (v0y) will be negative. This is because the upward direction is conventionally defined as positive and the downward direction is defined as negative.
v0y = -|v0|sinθ
Here, θ is the angle of projection and |v0| is the initial velocity of the projectile. Since the angle of projection is downwards, sinθ is negative.
Therefore, v0y is negative. So the correct option is C. Negative.
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Three 560 resistors are wired in parallel with a 75 V battery. What is the current through each of the resistors? Express your answer to the nearest mA.
The current through each of the resistors is approximately 134 mA.
To find the current through each resistor in a parallel circuit, we can use Ohm's Law, which states that current (I) is equal to voltage (V) divided by resistance (R).
In a parallel circuit, the voltage across each resistor is the same as the voltage across the battery. Therefore, the current through each resistor will be determined by the individual resistance values.
Given:
Resistance of each resistor (R) = 560 Ω
Voltage (V) = 75 V
To find the current through each resistor, we use the formula:
I = V / R
Calculations:
I = 75 V / 560 Ω
I ≈ 0.134 A
To convert the current to milliamperes (mA), we multiply by 1000:
I ≈ 0.134 A * 1000
I ≈ 134 mA
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An alien spaceship, moving at constant velocity, traverses the solar system (a distance of 10.50 light-hours) in 15.75 hr as measured by an observer on Earth. Calculate the speed of the ship (as measured by an observer on Earth), and the time interval that an observer on the ship measures for the trip. A. v = 0.500c, At' = 11.7 hr B. v = 0.667c, At' = 11.7 hr C. v = 0.887c, At = 21.1 hr D. v = 0.995c, Ať = 21.1 hr E. None of the above
Correct option is B. The speed of the alien spaceship, as measured by an observer on Earth, is approximately 0.667 times the speed of light (c). The time interval that an observer on the ship measures for the trip is approximately 11.7 hours.
In order to calculate the speed of the spaceship, we can use the formula v = d/t, where v is the velocity, d is the distance, and t is the time. In this case, the distance is 10.50 light-hours and the time is 15.75 hours. Plugging in these values, we get v = 10.50 light-hours / 15.75 hours = 0.667 times c.
To find the time interval that an observer on the spaceship measures for the trip, we can use the time dilation formula t' = t / √(1 - (v^2/c^2)), where t' is the time interval as measured on the spaceship, t is the time interval as measured on Earth, v is the velocity of the spaceship, and c is the speed of light. Plugging in the values we have, t = 15.75 hours and v = 0.667 times c, we can calculate t' = 15.75 hours / √(1 - (0.667^2)) = 11.7 hours.
Therefore, the correct answer is B. The speed of the ship, as measured by an observer on Earth, is approximately 0.667c, and the time interval that an observer on the ship measures for the trip is approximately 11.7 hours.
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Which statement describes the energy transformation that occurs when a person eats a sandwich before a hike
A new Mars rover is being designed that will send signals between the
rover on Mars and a control station on Earth. The engineers working on the
rover are concerned about interference from electrical events in Earth's
atmosphere.
To address this concern, should the rover send analog or digital signals?
Choose 1 answer:
A Analog; the interference won't change an analog signal.
Analog; analog signals can be designed to minimize the effect of
interference.
B
Digital; digital signals are not affected by the interference.
Digital; digital signals can be designed to minimize the effect of
interference.
les videos
Report a problem
Answer:
B
Explanation:
The appropriate choice to address the concern of interference from Earth's atmosphere would be:
B. Digital; digital signals can be designed to minimize the effect of interference.
Digital signals are less susceptible to interference compared to analog signals. They can be encoded and designed with error correction techniques to ensure accurate transmission and reception of data, even in the presence of interference. This makes digital signals much more suitable for long range communication.Which of the following are a focus of study for the location of possible extraterrestrial life? (check all that apply)
Question 1 options:
The core of the Milky Way Galaxy
The Sun
Vulcan
Europa
Enceladus
Mars
Possible locations for extraterrestrial life being studied include: the core of the Milky Way Galaxy and its habitable planets, the Sun and its solar system with moons like Europa and Enceladus, and Mars with its potential for water and life-sustaining conditions.
The search for extraterrestrial life has fascinated humans since ancient times, and our understanding of the universe continues to expand. Scientists have narrowed down potential locations for extraterrestrial life based on factors like the presence of liquid water, organic molecules, and energy sources. Here are some of the key areas being studied:
1. The core of the Milky Way Galaxy: With millions of stars, the core of our galaxy is considered a potential hub for habitable planets. Scientists investigate this region to understand galaxy formation and the likelihood of life in other parts of the universe.
2. The Sun and its solar system: As our closest star, the Sun is crucial in the search for life within our solar system. Moons such as Europa and Enceladus, found around the outer planets, show potential for hosting life-supporting conditions. Studying these moons helps us comprehend the nature of the universe and its capacity to sustain life.
3. Mars: Known for its barren landscape, Mars has been a primary focus of research due to the possibility of water on the planet. Water is a vital ingredient for life as we know it. Investigating Mars allows us to gain insights into the conditions necessary for life and their existence elsewhere in the universe.
Vulcan, although a hypothetical planet once postulated to explain a discrepancy in Mercury's orbit, is not recognized by astronomers and is primarily featured in science fiction, particularly in Star Trek.
By exploring these locations, scientists aim to deepen our understanding of the universe and increase the chances of discovering extraterrestrial life.
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Consider two celestial objects with masses m 1
and m 2
with a separation distance between their centers r. If the separation distance r were to triple, what would happen to the magnitude of the force of attraction? It increases by a factor of 3. It decreases by a factor of 9. It decreases by a factor of 3. It remains unchanged. It decreases by a factor of 6 .
Therefore, the correct option is "It decreases by a factor of 9."So, the force of attraction between two celestial objects with masses m1 and m2 separated by a distance r decreases by a factor of 9 if the separation distance r were to triple.
According to the law of gravitation, the magnitude of the force of attraction between two celestial objects with masses m1 and m2 separated by a distance r is given byF= Gm1m2 / r2where G is the gravitational constant.If the separation distance r were to triple, the magnitude of the force of attraction between them would decrease by a factor of 9.The formula for force of attraction suggests that the force of attraction between two objects is inversely proportional to the square of the distance between them. Thus, when the distance triples, the magnitude of the force will decrease to 1/9th of the original force. Therefore, the correct option is "It decreases by a factor of 9."So, the force of attraction between two celestial objects with masses m1 and m2 separated by a distance r decreases by a factor of 9 if the separation distance r were to triple.
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Determine the output voltage for the network of Figure 2 if V₁ = 2 mV and ra= 50 kn. (5 Marks) Marking Scheme: 1. Calculation using correct Formulae 2. Simulation using any available software 6.8 k V₂ S 91 MQ HF 15 MQ ww www www Figure 2 VGTH=3V k=0.4×10-3 3.3k2
The output voltage for the given network is 2.9 V.
In the given network if V₁ = 2 mV and ra= 50 kn, the output voltage can be determined . using Kirchoff's voltage law and Ohm's law. In Kirchoff's voltage law, the sum of the voltage drops in a closed loop equals the voltage rise in the same loop. In the network, a closed loop consists of a battery and the circuit's resistance.
Thus,Vin - Ira - Vds = 0 where Vin is the voltage drop across the battery, I is the current, ra is the resistance and Vds is the voltage drop across the resistor. Rearranging the equation, we getVout = Ira which is the voltage drop across the resistance. Using Ohm's law, I=Vds/ra. Substituting Vds=VGTH−Vout and simplifying,Vout=(VGTH-Vin)*ra=3V-2mV*50kΩ=3V-100V=2.9V.Vout = 2.9 V.
Simulation can be carried out using any available software.
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Suppose a neuron membrane has a specific capacitance (i.e. capacitance per unit area) of 10³ F/m². Also suppose there are 3 x10¹ Na voltage-gated channels per m² allowing 200 Na ions to flow through each channel in 1 ms. Assuming no other changes occur during this time: a. Find the amount of charges (per area) that accumulate on the membrane 1 ms after these channels open. Your answer: Answer Coulomb/m² b. Find the change in the membrane potential (voltage) 1 ms, after these channels open. Answer in the unit of mV. Your answer: Answer mV
The amount of charges that accumulate on the membrane 1 ms after the channels open is 9.6 x 10^(-15) Coulomb/m². The change in the membrane potential 1 ms after the channels open is 9.6 x 10^(-15) mV.
To calculate the amount of charges that accumulate on the membrane 1 ms after the channels open, we need to determine the total charge passing through the channels per unit area.
The total charge passing through the channels can be calculated by multiplying the number of channels per m² (3 x 10¹) by the charge per channel (200 Na ions). This gives us a total of 6 x 10² Na ions passing through the channels per m².
Next, we need to convert the number of Na ions into Coulombs by multiplying it by the elementary charge (1.6 x 10^(-19) C) since each Na ion carries one elementary charge. Therefore, the total charge passing through the channels per m² is 9.6 x 10^(-18) C.
Since the specific capacitance of the membrane is given as 10³ F/m², we can multiply it by the total charge to get the amount of charges that accumulate on the membrane per area. This gives us a value of 9.6 x 10^(-15) C/m².
To find the change in the membrane potential (voltage), we can use the equation Q = CV, where Q is the charge, C is the capacitance per unit area, and V is the voltage. Rearranging the equation, we have V = Q/C. Plugging in the values, we get V = (9.6 x 10^(-15) C/m²) / (10³ F/m²), which simplifies to 9.6 x 10^(-18) V/m².
To convert the voltage from V/m² to mV, we multiply it by 10³, resulting in a change in membrane potential of 9.6 x 10^(-15) mV.
Therefore, the answers are:
a. The amount of charges that accumulate on the membrane 1 ms after these channels open is 9.6 x 10^(-15) Coulomb/m².
b. The change in the membrane potential 1 ms after these channels open is 9.6 x 10^(-15) mV.
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Since the investigative question has two variables, you need to focus on each one separately. Thinking only about the first part of the question, mass, what might be a hypothesis that would illustrate the relationship between mass and kinetic energy? Use the format of "if…then…because…” when writing your hypothesis.
In order to form a hypothesis that would illustrate the relationship between mass and kinetic energy, we first need to understand what kinetic energy and mass are and how they are related. Kinetic energy is the energy that an object possesses due to its motion, and is given by the formula KE = 0.5mv², where m is the mass of the object and v is its velocity. Mass, on the other hand, is a measure of the amount of matter in an object.
The relationship between mass and kinetic energy is direct, meaning that as mass increases, so does kinetic energy, provided that velocity remains constant. Similarly, if velocity increases, then kinetic energy will increase as well, provided that mass remains constant.
The hypothesis that illustrates this relationship can be stated as follows:If the mass of an object is increased, then the kinetic energy of the object will also increase, because kinetic energy is directly proportional to mass, assuming velocity remains constant.In other words, if the mass of an object is doubled, then its kinetic energy will also double, assuming that its velocity remains constant. This hypothesis can be tested through experiments that involve measuring the kinetic energy of objects with different masses, but with the same velocity.
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If the mass of an object increases, then its kinetic energy will increase proportionally because mass and kinetic energy have a linear relationship when graphed.
Before beginning a long trip on a hot day, a driver inflates an automobile tire to a gauge pressure of 1.80 atm at 300 K. At the end of the trip the gauge pressure has increased to 2.33 atm. (Caution: Gauge pressure is measured relative to the atmospheric pressure. The absolute pressure in the tire at the beginning of the trip is 2.80 atm.) Assuming the volume has remained constant, what is the temperature of the air inside the tire? (b) What percentage of the original mass of air in the tire should be released so the pressure returns to the original value? Assume the temperature remains at the value found in (a), and the volume of the tire remains constant as air is released. Also assume that the atmospheric pressure is 1.00 atm and remains constant. Hint: The percentage of the original mass is the same as the percentage of the original number of moles. The fraction of the original number of moles that should be released is equal to 1 - nƒ/n; where n, is the original (initial) number of moles and nf is the final number of moles after some of the gas has been released. Note that the volume of the gas, which remains constant throughout the problem, cancels out in the ratio, nf/n,, so that you don't need to know the volume to solve this problem.
The temperature of the air inside the tire is 363 K. To return the pressure to the original value, approximately 42.9% of the original mass of air should be released.
(a) Using the ideal gas law, we can relate the initial and final pressures and temperature: P1/T1 = P2/T2,
where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature. Rearranging the equation, we have: T2 = (P2 * T1) / P1.
T2 = (2.33 atm * 300 K) / 2.80 atm = 363 K.
(b) To find the percentage of the original mass of air that should be released to return the pressure to the original value, the relationship between pressure & the number of moles of gas. According to the ideal gas law, PV = nRT.
P1 = (nfinal / ninitial) * Pfinal.
(nfinal / ninitial) = P1 / Pfinal = 2.80 atm / 1.80 atm = 1.56.
Therefore, the percentage of the original mass of air that should be released is approximately 1 - 1.56 = 0.44, or 44%.
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A newspaper delivery boy throws a newspaper onto a balcony 0.75 m above the height of his hand when he releases the paper. Given that he throws the paper with a velocity of 15 m/s [46° above horizontal], find: a) the maximum height of the paper's trajectory (above the boy's hand) b) the velocity at maximum height c) the acceleration at maximum height d) the time it takes for the paper to reach the balcony, if it reaches the balcony as it descends
Answer: (a) The maximum height of the paper's trajectory (above the boy's hand) is 6.5 m.
(b) The velocity at maximum height is 6.57 m/s.
(c) The acceleration at maximum height is -9.8 m/s².
(d) The time it takes for the paper to reach the balcony, if it reaches the balcony as it descends, is 2.11 s.
a) To find the maximum height of the paper's trajectory (above the boy's hand), we can use the kinematic equation,
v² = u² + 2gh
where, v = 0 (at maximum height)u = uy = 11.34 m/s (initial vertical velocity), g = -9.8 m/s² (negative sign indicates deceleration in vertical direction)
Substituting the values in the above equation, 0² = (11.34)² + 2(-9.8)hh = (11.34)² / (2 × 9.8)h = 6.5 m.
Therefore, the maximum height of the paper's trajectory (above the boy's hand) is 6.5 m.
b) To find the velocity at maximum height, we can use the kinematic equation,v² = u² + 2gh
where, u = uy = 11.34 m/s (initial vertical velocity)g = -9.8 m/s² (negative sign indicates deceleration in vertical direction)h = 6.5 m (maximum height). Substituting the values in the above equation,
v² = (11.34)² + 2(-9.8)×6.5
v² = 43.15
v = √43.15
v = 6.57 m/s.
Therefore, the velocity at maximum height is 6.57 m/s.
c) At maximum height, the velocity of the paper is zero. Therefore, the acceleration at maximum height is equal to the acceleration due to gravity, i.e., -9.8 m/s² (negative sign indicates deceleration in vertical direction).
Therefore, the acceleration at maximum height is -9.8 m/s².
d) To find the time it takes for the paper to reach the balcony, if it reaches the balcony as it descends, we can use the kinematic equation,
s = ut + 0.5 at²
where, s = h = 0.75 m (height of the balcony above the hand of the delivery boy)u = ux = 10.7 m/s (horizontal velocity)g = 9.8 m/s² (acceleration due to gravity)
Substituting the values in the above equation,
0.75 = 10.7 t + 0.5 × 9.8 t²0.49 t² + 10.7 t - 0.75 = 0.
Using the quadratic formula,
t = (-10.7 ± √(10.7² + 4 × 0.49 × 0.75)) / (2 × 0.49)
t = (-10.7 ± √45.76) / 0.98t = (-10.7 ± 6.77) / 0.98t
= -4.09 or 2.11. As time cannot be negative, the time taken for the paper to reach the balcony is 2.11 s.
Therefore, the time it takes for the paper to reach the balcony, if it reaches the balcony as it descends, is 2.11 s.
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A motorcyclist is making an electric vest that, when connected to the motorcycle's 12 V battery, will warm her on cold rides. She is using 0.23−mm-diameter copper wire, and she wants a current of 4.8 A in the wire. For the steps and strategies involved in solving a similar problem, you may view a Video Tutor Solution. A 15μF capacitor initially charged to 25μC is discharged through a 1.0kΩ resistor. Part A How long does it take to reduce the capacitor's charge to 10μC ?
It takes approximately 18.97 ms for the capacitor's charge to be reduced to 10 μC.
A 15 μF capacitor initially charged to 25 μC is discharged through a 1.0 kΩ resistor. The steps and strategies involved in solving this similar problem are given below:
where q = charge on capacitor at time t, Q = initial charge on the capacitor, R = resistance, C = capacitance, and e = 2.71828 (constant)
To find the time it takes to reduce the capacitor's charge to 10 μC, substitute the given values in the above equation, q = 10 μC, Q = 25 μC, R = 1.0 kΩ, C = 15 μF.
Then solve for t.t = - RC ln(q/Q)=- (1.0 kΩ) (15 μF) ln(10 μC/25 μC)t = 18.97 ms
Therefore, it takes approximately 18.97 ms for the capacitor's charge to be reduced to 10 μC.
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(a) Interpret the term "Opto-Electronics" and Elecro-Optics" with examples. (b) From the energy band diagram, proof the smallest energy required for emission and absorption of light is |h9 = Eg. where h = Planck's constant, 9 = Frequency of the light and Eg = Band-gap
The term Opto-electronics is a branch of physics that studies the combination of optical (light) and electrical engineering. It refers to the technology that is associated with the control and conversion of light with electronic devices and systems.
Some examples of optoelectronic devices include LEDs, photodiodes, fiber optic cables, solar cells, and photovoltaic cells. Electro-optics, on the other hand, refers to the study of the properties of materials and the interaction between electromagnetic waves and matter. It is an interdisciplinary field of engineering that involves designing and developing devices and systems that can manipulate and control light using electricity. Some examples of electro-optic devices include LCDs, laser printers, and optical communication systems.
(b) The energy band diagram is a graphical representation of the distribution of electrons in the energy levels of a material. It helps to determine the energy required for the emission and absorption of light in a material. The smallest energy required for the emission and absorption of light is given by the equation |h9 = Eg, where h is Planck's constant, 9 is the frequency of the light, and Eg is the band gap of the material. The band gap is the energy difference between the valence band and the conduction band of a material.
When a photon with energy equal to the band gap is absorbed by a material, an electron in the valence band is excited to the conduction band, creating a hole in the valence band. This results in the emission of light of the same energy when the electron returns to its original position. Thus, the band gap is the minimum energy required for the emission and absorption of light in a material.
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Use your result above to calculate the incident angle θ 1
from air in entering the fiber (see notes on refraction). Use three significant digits please.
To calculate the incident angle θ1, we need additional information related to refraction, such as the refractive indices of the materials involved.
In the context of refraction, the incident angle (θ1) is the angle between the incident ray and the normal to the interface between two media. To calculate θ1, we need to know the refractive indices of the materials involved. The refractive index (n) is a property of a medium that determines how light propagates through it. The relationship between the incident angle, the refractive indices of the two media, and the angles of refraction can be described by Snell's law.
To determine the incident angle accurately, the refractive indices of both the air and the fiber are required. Once these values are known, Snell's law can be applied to calculate the incident angle.
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A long straight wire carrying a 4 A current is placed along the x-axis as shown in the figure. What is the magnitude of the magnetic field at a point P, located at y = 9 cm, due to the current in this wire?
To find the magnitude of the magnetic field at point P due to the current in the wire, we can use the formula for the magnetic field produced by a long straight wire. The magnitude of the magnetic field at point P depends on the distance from the wire and the current flowing through it.
The magnetic field produced by a long straight wire at a point P located a distance y away from the wire can be calculated using the formula B = (μ₀ * I) / (2π * y), where B is the magnetic field, μ₀ is the permeability of free space (a constant), I is the current in the wire, and y is the distance from the wire.
In this case, the current in the wire is given as 4 A and the point P is located at y = 9 cm. We can substitute these values into the formula to calculate the magnitude of the magnetic field at point P.
Remember to convert the distance from centimeters to meters before substituting it into the formula.
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A force is specified by the vector F = [(110)i + (-210)j + (-10)k] N. Calculate the angles made by F with the positive x-, y-, and z-axes.
The vector F can be written as F = 110i - 210j - 10k.
The angle made by F with the positive x-axis is given by:
θx = arctan(Fy/Fx)
where Fx is the x-component of the vector F and
Fy is the y-component of the vector F.
θx = arctan(-210/110)
θx = -62.25°
The angle made by F with the positive y-axis is given by:
θy = arctan(Fx/Fy)
where Fx is the x-component of the vector F and
Fy is the y-component of the vector F.
θy = arctan(110/-210)
θy = -28.07°
The angle made by F with the positive z-axis is given by:
θz = arctan(Fz/Fr) where Fz is the z-component of the vector F and Fr is the magnitude of the vector F.
Fr can be calculated as:
Fr = √(F²) = √(110² + (-210)² + (-10)²)Fr = 236.31 N
θz = arctan(-10/236.31)
θz = -2.42°
Hence, the angles made by F with the positive x-, y-, and z-axes are -62.25°, -28.07°, and -2.42° respectively.
Note: The angles are measured in the clockwise direction from the positive x-, y-, and z-axes.
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A radio station transmits isotropically (ie in all directions) electromagnetic radiation at a frequency of 100.8 MHz. At a certain distance from the radio station the intensity of the wave is I=0.267 W/m^2.
a) What will be the intensity of the wave at a quarter of the distance from the radio station?
b) What is the wavelength of the transmitted signal?
If the power of the antenna is 5 MW.
c) At what distance from the source will the intensity of the wave be 0.134 W/m^2?
d) and what will be the absorption pressure exerted by the wave at that distance?
e) and what will be the effective electric field (rms) exerted by the wave at that distance?
a) The intensity at a quarter of the distance is 1.068 W/[tex]m^2[/tex].
b) The wavelength of the transmitted signal is approximately 2.972 m.
c) tThe distance from the source when the intensity is 0.134 W/[tex]m^2[/tex] is approximately 1183.5 m.
d) The absorption pressure at that distance is approximately 4.47 x [tex]10^{-10}[/tex] Pa.
e) The effective electric field (rms) at that distance is approximately 1.32 x [tex]10^{-4}[/tex] V/m.
To solve the given problems, we need to use the formulas related to electromagnetic waves and their properties.
a) The intensity of a wave is inversely proportional to the square of the distance from the source.
Therefore, if the distance is reduced to a quarter, the intensity will increase by a factor of 4.
Thus, the intensity at a quarter of the distance from the radio station will be 4 times the initial intensity: I = 4 * 0.267 W/[tex]m^2[/tex] = 1.068 W/[tex]m^2[/tex].
b) The wavelength of a wave can be determined using the formula: wavelength = speed of light / frequency.
The speed of light in a vacuum is approximately 3 x 10^8 m/s.
Converting the frequency from MHz to Hz, we have f = 100.8 x 10^6 Hz.
Substituting these values into the formula, we get: wavelength = (3 x 10^8 m/s) / (100.8 x 10^6 Hz) ≈ 2.972 m.
c) To find the distance from the source when the intensity is 0.134 W/[tex]m^2[/tex], we rearrange the formula for intensity and distance: distance = √(power / (4π * intensity)).
Given that the power of the antenna is 5 MW (5 x 10^6 W) and the intensity is 0.134 W/[tex]m^2[/tex], we can calculate the distance: distance = √((5 x 10^6 W) / (4π * 0.134 W/m^2)) ≈ 1183.5 m.
d) The absorption pressure exerted by the wave can be calculated using the formula: pressure = intensity / (speed of light).
Substituting the intensity and the speed of light, we get: pressure = 0.134 W/[tex]m^2[/tex] / (3 x 10^8 m/s) ≈ 4.47 x 10^-10 Pa.
e) The effective electric field (rms) can be determined using the formula: electric field = √(2 * power / (speed of light * area)).
Given that the power is 5 MW, the speed of light is 3 x 10^8 m/s, and assuming the wave is spreading in all directions (isotropic), the area is 4π[tex]r^2[/tex], where r is the distance.
Substituting these values, we have: electric field = √(2 * (5 x 10^6 W) / (3 x 10^8 m/s * (4π * (1183.5 m)^2))) ≈ 1.32 x [tex]10^{-4}[/tex] V/m.
In summary, a) the intensity at a quarter of the distance is 1.068 W/[tex]m^2[/tex], b) the wavelength of the transmitted signal is approximately 2.972 m, c) the distance from the source when the intensity is 0.134 W/[tex]m^2[/tex] is approximately 1183.5 m, d) the absorption pressure at that distance is approximately 4.47 x [tex]10^{-10}[/tex] Pa, and e) the effective electric field (rms) at that distance is approximately 1.32 x [tex]10^{-4}[/tex] V/m.
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show the positioning of an objective lens and eye piece of the following devices with respect to their focal length . example d=fo-fe and number of lens in the device
1. simple microscopic (magnifying glass)
2. compound microscope
3. astronomical telescope
4. galilean telescope
5. prismatic binoculars
1. Simple Microscope (Magnifying Glass): Objective lens = N/A, Eyepiece = N/A (Single Lens)
2. Compound Microscope: Objective lens = Closer, Eyepiece = Farther
3. Astronomical Telescope: Objective lens = Closer, Eyepiece = Farther
4. Galilean Telescope: Objective lens = Closer, Eyepiece = Farther
5. Prismatic Binoculars: Objective lens = Closer, Eyepiece = Farther
Simple Microscope (Magnifying Glass):
In a simple microscope or magnifying glass, there is only one lens, which serves as both the objective lens and the eyepiece. The lens is convex and typically has a short focal length. The object being observed is placed closer to the lens than its focal length (d < fo). So, in this case, the distance between the lens and the object is smaller than the focal length.
Compound Microscope:
A compound microscope consists of two lenses: the objective lens and the eyepiece. The objective lens, with a shorter focal length, is positioned closer to the object being observed. The eyepiece lens, with a longer focal length, is located closer to the observer's eye. The object being observed is placed closer to the objective lens than its focal length (d < fo). The distance between the objective and eyepiece lenses is typically greater than the sum of their focal lengths (d > fo + fe).
Astronomical Telescope:
In an astronomical telescope, the objective lens is positioned closer to the object being observed, such as celestial bodies. The objective lens has a longer focal length compared to the eyepiece lens. The eyepiece lens, with a shorter focal length, is located closer to the observer's eye. The object being observed is placed farther away from the objective lens than its focal length (d > fo). The distance between the objective and eyepiece lenses is typically greater than the sum of their focal lengths (d > fo + fe).
Galilean Telescope:
A Galilean telescope has a convex objective lens and a concave eyepiece lens. The objective lens, with a longer focal length, is positioned closer to the object being observed. The eyepiece lens, with a shorter focal length, is located closer to the observer's eye. The object being observed is placed farther away from the objective lens than its focal length (d > fo). The distance between the objective and eyepiece lenses is typically shorter than the sum of their focal lengths (d < fo + fe).
Prismatic Binoculars:
Prismatic binoculars use multiple lenses and prisms to provide a magnified view. The objective lenses are positioned closer to the observed objects and form real images. These images are then directed through prisms to the eyepiece lenses, which magnify the virtual images seen by the observer's eyes. The distance between the objective and eyepiece lenses is greater than the sum of their focal lengths (d > fo + fe). Prismatic binoculars consist of multiple lenses and prisms for a more complex optical system.
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1. Write the form of the Fermi-Dirac distribution function f(E) for free electrons in a metal. 2. Show that the value of this function is one at E<< EF and zero when E >> EF. 3. Hall voltage is being measured for two identical samples. One is made of gold and other is of a semiconductor like silicon. If the values of the current and magnetic field used for the measurement are the same, which sample will give a larger Hall voltage? On what factor will the Hall voltage depend?
Answer: 1. Fermi-Dirac distribution function f(E) = 1/{exp[(E - EF) / kT] + 1}
2. 2. In a Fermi-Dirac distribution function, the value of the function is one when E<< EF and zero when E >> EF because of the following reasons:
When E << EF, the value of exp[(E - EF) / kT] is very small. When E >> EF, the value of exp[(E - EF) / kT] is very large.3. A semiconductor like silicon with a higher number density of free electrons will give a larger Hall voltage.
1. Fermi-Dirac distribution function f(E) for free electrons in a metal is expressed as shown below:
f(E) = 1/{exp[(E - EF) / kT] + 1} Where, E is the energy of an electron, EF is the Fermi energy level, k is the Boltzmann constant, and T is the absolute temperature of the metal.
2. In a Fermi-Dirac distribution function, the value of the function is one when E<< EF and zero when E >> EF because of the following reasons:
When E << EF, the value of exp[(E - EF) / kT] is very small. When E >> EF, the value of exp[(E - EF) / kT] is very large.3. A semiconductor sample such as silicon will give a larger Hall voltage when compared to a gold sample, provided that the values of the current and magnetic field used for the measurement are the same. The Hall voltage depends on the following factor: Hall voltage = (IB) / ne Where, I is the current through the sample, B is the magnetic field, n is the number density of free electrons in the material, and e is the charge of an electron. The Hall voltage is directly proportional to the number density of free electrons. Therefore, a semiconductor like silicon with a higher number density of free electrons will give a larger Hall voltage.
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A Erms = 110-V oscillator is used to provide voltage and current to a series LRC circuit. The impedance minimum value is 45.0 1, at resonance. What is the value of the impedance at double the resonance frequency?
The impedance of a series LRC circuit at double the resonance frequency is four times the impedance at resonance.
In a series LRC circuit, the impedance (Z) is given by the formula:
Z = √(R^2 + (Xl - Xc)^2)
Where R is the resistance, Xl is the inductive reactance, and Xc is the capacitive reactance. At resonance, the inductive and capacitive reactances cancel each other out, resulting in the minimum impedance value.
Given that the impedance minimum value is 45.0 Ω at resonance, we can determine the values of R, Xl, and Xc at resonance. Since the impedance minimum occurs at resonance, we have Xl = Xc.
At double the resonance frequency, the inductive and capacitive reactances will no longer cancel each other out. The inductive reactance (Xl) will increase while the capacitive reactance (Xc) will decrease. This leads to an increase in the impedance.
Since the impedance is directly proportional to the square root of the sum of squares of the resistive and reactive components, doubling the resonance frequency results in a fourfold increase in the impedance value.
Therefore, the value of the impedance at double the resonance frequency is 4 times the impedance at resonance, which is 45.0 Ω. Hence, the impedance at double the resonance frequency is 180.0 Ω.
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A heat engine containing an ideal gas is physically represented by the picture below, with its cycle described by the diagram beside it. In going from point A to point B,L increases from 15 cm to 20 cm. The engine has η=4%. A Carnot cycle operating between the same high and low temperatures as this engine would have η=40%. Determine if the gas seems to be mostly monatomic, diatomic, or polyatomic (calculations are required for credit). Problem 1 ( 30pts) A heat engine containing an ideal gas is physically represented by the picture below, with its described by the di gram beside it. In going from point A to point B,L increases from 15 20 cm. The engine has η=4%. A Carnot cycle operating between the same high an temperatures as this engine would have η=40%. Determine if the gas seems to be n monatomic, diatomic, or polyatomic (calculations are required for credit).
The gas seems to be diatomic because γ = C_p/C_v = 1 + 2/2 = 7/5, which is between 5/3 for monoatomic gas and 7/5 for diatomic gas.
At point A, the volume is V1 = π(0.15)^2 L = 0.070686 L.At point B, the volume is V2 = π(0.2)^2 L = 0.125664 L.The work done by the gas is ΔW = (P1V1 - P2V2)/(γ - 1)where γ = C_p/C_v is the specific heat ratio. In this case, the heat engine is not given a particular gas. However, a rough estimation of the specific heat ratio can be made. Monoatomic gas has γ = 5/3, diatomic gas has γ = 7/5, and polyatomic gas has γ > 7/5.The efficiency of the heat engine is η = W/Q_in = 1 - Q_out/Q_inwhere Q_in is the heat added to the engine and Q_out is the heat rejected by the engine.
By substituting the first law of thermodynamics, Q_in = ΔU + W and Q_out = -ΔU, we getη = 1 - T_L/T_Hwhere T_L and T_H are the low and high temperatures of the heat engine. Since the Carnot cycle is reversible and the efficiency of a reversible engine is η = 1 - T_L/T_H, the high and low temperatures of the heat engine are equal to those of the Carnot cycle.η_C = 1 - T_L/T_H = 0.4T_H/T_L = 2.5The efficiency of the heat engine isη_E = 0.04 = 0.4/10which implies that T_L/T_H = 9.6The high temperature of the heat engine can be determined from the ideal gas lawPV = nRTwhere n is the amount of gas and R is the gas constant. By substituting L = 0.15 m and V = πr^2L, we getP_A = nRT_A/πr^2LSubstituting r = 0.05 m, P_A = 2.4 nRT_A/L.
The temperature of the heat engine at point A can be determined from the volume.V = nRT/P and L = V/πr^2.Substituting r = 0.05 m, L = 0.15 m, and P = P_A, we getT_A = PL/0.2nR.Substituting P_A = 2.4 nRT_A/L, we getT_A = 0.6 T_AThe temperature of the heat engine at point B can be determined in a similar way.T_B = PL/0.2nRSubstituting P_B = 2.4 nRT_B/L, we getT_B = 0.6 T_B.
The temperature ratio isT_B/T_A = (PL/0.2nR)/(PL/0.15nR) = 0.75The efficiency ratio isη_E/η_C = 0.04/0.4 = 0.1The efficiency ratio can be expressed asη_E/η_C = T_L/T_H (1 - T_L/T_H)/(1 - η_E)Simplifying the equation givesT_L/T_H = (1 - η_E)/(1 - η_E/η_C) = 0.8889Since T_B/T_A = 0.75, the temperature of the heat engine at point A isT_A = T_B/0.75 = 0.8 T_BSubstituting T_A and T_L/T_H in the equation T_L/T_H = 0.8889 givesT_H = 605.2 K and T_L = 538.3 K.The gas seems to be diatomic because γ = C_p/C_v = 1 + 2/2 = 7/5, which is between 5/3 for monoatomic gas and 7/5 for diatomic gas.
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A bungee jumper with mass 52.5 kg jumps from a high bridge. After arriving at his lowest point, he oscillates up and down, reaching a low point seven more times in 43.0 s. He finally comes to rest 20.5 m below the level of the bridge. Estimate the spring stiffness constant of the bungee cord assuming SHM. μΑ ) ? Value k Units Estimate the unstretched length of the bungee cord assuming SHM
The estimated unstretched length of the bungee cord assuming simple harmonic motion (SHM) is zero.
To estimate the spring stiffness constant (k) of the bungee cord, we can use the formula for the period of a simple harmonic oscillator:
T = 2π√(m/k),
where T is the period, m is the mass of the jumper, and k is the spring stiffness constant.
Given that the jumper reaches the low point seven more times in 43.0 seconds, we can calculate the period as follows:
T = 43.0 s / 8 = 5.375 s.
Now, rearranging the equation for the period, we have:
k = (4π²m) / T².
Substituting the known values:
k = (4π² * 52.5 kg) / (5.375 s)²,
k ≈ 989.67 N/m (rounded to two decimal places).
Therefore, the estimated spring stiffness constant (k) of the bungee cord is approximately 989.67 N/m.
To estimate the unstretched length of the bungee cord, we need to determine the equilibrium position when the jumper comes to rest 20.5 m below the level of the bridge.
In simple harmonic motion (SHM), the equilibrium position corresponds to the unstretched length of the spring. At this point, the net force acting on the system is zero.
Using Hooke's Law, the force exerted by the spring is given by:
F = kx,
where F is the force, k is the spring stiffness constant, and x is the displacement from the equilibrium position.
Since the jumper comes to rest 20.5 m below the bridge, the displacement (x) is 20.5 m.
Setting F = 0 and solving for x, we have:
kx = 0,
x = 0.
This implies that the equilibrium position (unstretched length) of the bungee cord is zero, meaning that the bungee cord has no additional length when it is unstretched.
Therefore, the estimated unstretched length of the bungee cord assuming simple harmonic motion (SHM) is zero.
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The spring stiffness constant of the bungee cord is found by equating the force exerted by the spring when the bungee jumper is at his lowest point to his weight and solving for k. The unstretched length of the bungee cord can be deduced from the final resting position of the bungee jumper.
Explanation:To determine the spring stiffness constant k of the bungee cord, we need to use Hooke's Law which defines the force exerted by a spring as F = -kx, where x is the displacement of the spring from its equilibrium position.
In the case of the bungee jumper, when he is at his lowest point, the force exerted by the spring is equal to his weight, F = mg, where m is the mass of the jumper and g is the acceleration due to gravity. By equating these two forces, we get: -kx = mg. Solving for k gives k = -mg/x.
With the mass m = 52.5 kg, gravity g=9.81 m/s², and displacement (lowest point height difference) x = 20.5 m, we can calculate k to estimate the spring stiffness.
The unstretched length of the bungee cord can be estimated by observing the final resting position of the bungee jumper. If the final resting position is taken as the equilibrium position (x=0), then the length of the cord in this position would be the unstretched length.
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A light beam is traveling through an unknown substance. When it strikes a boundary between that substance and the air (Nair = 1), the angle of reflection is 29.0° and the angle of refraction is 36.0°. What is the index of refraction n of the substance? n =
The index of refraction n of the substance is 0.82.
The index of refraction of a substance can be calculated using Snell's law.
Snell's law states that: n1sinθ1 = n2sinθ2, where n1 is the refractive index of the first medium (in this case air), θ1 is the angle of incidence, n2 is the refractive index of the second medium (the unknown substance), and θ2 is the angle of refraction.
Given that a light beam is traveling through an unknown substance and when it strikes a boundary between that substance and the air (Nair = 1), the angle of reflection is 29.0° and the angle of refraction is 36.0°, we are required to find the index of refraction n of the substance.
We can use the formula: n1sinθ1 = n2sinθ2 where n1=1, θ1 = 29.0°, and θ2 = 36.0° to find the refractive index of the unknown substance.
The first step is to calculate sin θ1 and sin θ2 using a scientific calculator: sin θ1 = sin 29.0° = 0.4848 and sin θ2 = sin 36.0° = 0.5878
Substitute the given values in the formula: n1sinθ1 = n2sinθ2
Substituting the known values, we get:1 × 0.4848 = n2 × 0.5878
Dividing both sides by 0.5878 we get: n2 = (1 × 0.4848) / 0.5878
n2 = 0.82
Therefore, the index of refraction n of the substance is 0.82.
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An 85-g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 75 cm. What is the speed of the arrow as it leaves the bow? B) A 975−kg sports car accelerates from rest to 95 km/h in 6.4 s. What is the average power delivered by the engine? Problem 2: A ball of mass 0.440 kg moving east ( +x direction) with a speed of 3.80 m/s collides head-on with a 0.220−kg ball at rest. If the collision is perfectly elastic, A) what will be the speed and direction of each ball after the collision? B) What is the total kinetic energy after the collision? Problem 3: A 980-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.6 m before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80, calculates the speed of the sports car at impact. What was that speed?
1. The speed of the arrow as it leaves the bow is 109.7 m/s.
2. A) The speed of the ball 1 is 3.10 m/s towards east and B) the speed of ball 2 is 0.70 m/s towards east after the collision
3.The speed of the sports car before the collision is 3.3469 m/s.
Problem 1Given data;mass of the arrow, m = 85 g = 0.085 kgsForce applied by the bowstring, F = 105 NDisplacement, d = 75 cm = 0.75 mSpeed of the arrow, v can be calculated using work-energy theorem.W=K.E=(1/2)mv²initial K.E of the arrow is zero since it is at rest before it is shot from the bow.Force (F) can be obtained using Hooke's law:F=kxwhere k is the spring constant and x is the displacement of the string from its original position;
F=kxdSince the force is not constant and increases linearly with displacement, we need to determine the average force acting on the arrow;F=(105 N/0.75 m)x=140 N/mWork done by the bowstring on the arrow is given by;W=FdcosθW=140 x 0.75 x cos(0)W=105 JoulesK.E gained by the arrow is equal to the work done by the bowstringK.E=(1/2)mv²v=sqrt((2K.E)/m)v=sqrt((2 x 105)/0.085)v= 109.7 m/sTherefore, the speed of the arrow as it leaves the bow is 109.7 m/s.
Problem 2A ball of mass 0.440 kg moving east ( +x direction) with a speed of 3.80 m/s collides head-on with a 0.220−kg ball at rest. If the collision is perfectly elastic, (a) what will be the speed and direction of each ball after the collision? (b) What is the total kinetic energy after the collision?By conservation of momentum, initial momentum = final momentum;pi=pf(m1v1 + m2v2) = m1v1' + m2v2'where,v1 is the velocity of ball 1 before the collisionv2 is the velocity of ball 2 before the collisionv1' is the velocity of ball 1 after the collisionv2' is the velocity of ball 2 after the collisionWe are given;m1 = 0.440 kg, m2 = 0.220 kgv1 = 3.80 m/s (east) since it is moving in the + x directionv2 = 0 m/s since it is at rest before the collisionpi=pfm1v1 + m2v2 = m1v1' + m2v2'substituting in values;0.440 x 3.80 = 0.440v1' + 0.220v2'v1' = 3.80 - v2' ...................(1).
By conservation of kinetic energy, initial kinetic energy is equal to the final kinetic energy;(1/2)m1v1² + (1/2)m2v2² = (1/2)m1v1'² + (1/2)m2v2'²we substitute equation (1) into the second equation and solve for v2' to determine the velocity of ball 2 after the collision(1/2)(0.440)(3.80)² = (1/2)(0.440)(3.80 - v2')² + (1/2)(0.220)v2'²simplifying the equation above;0.83664 = 1.676(v2') - 0.22(v2')²2.199(v2')² - 1.676(v2') + 0.83664 = 0Using the quadratic formula;v2' = 0.70 m/s and v2' = 2.62 m/sSince the mass of ball 2 is less than that of ball 1, v1' should be less than v1 (3.80 m/s).
Therefore the solution with v2' = 0.70 m/s is valid. Thus;Ball 1 moves to the right with velocity;v1' = 3.80 - v2' = 3.80 - 0.70 = 3.10 m/s (east)Ball 2 moves to the right with velocity;v2' = 0.70 m/s (east)Total Kinetic energy after the collision;K.E = (1/2)m1v1'² + (1/2)m2v2'²= (1/2)(0.440)(3.10)² + (1/2)(0.220)(0.70)²= 0.887 JoulesTherefore, the speed of the ball 1 is 3.10 m/s towards east and the speed of ball 2 is 0.70 m/s towards east after the collision. Total kinetic energy after the collision is 0.887 J.
Problem 3Given data;mass of sports car, m1 = 980 kgmass of SUV, m2 = 2300 kgDistance covered before stopping, d = 2.6 mCoefficient of kinetic friction between tires and road, μk = 0.80Initial velocity of the sports car, u = ?Final velocity of the sports car, v = 0 m/s.
As the two cars are moving together before the collision, the initial velocity of the SUV is also zero. Therefore by conservation of momentum,pi = pf(m1u + m2(0)) = (m1 + m2)v0.980 u = 3280 vu = 3280/980u = 3.3469 m/sTherefore the speed of the sports car before the collision is 3.3469 m/s.
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