Equipotential lines are perpendicular to electric field lines. When a charge is moved on an equipotential line, no work is done and the potential energy remains constant.
Equipotential lines and electric field lines are related to the concept of electric potential, which is the electric potential energy per unit charge at a point in space due to a source charge.
Electric potential is measured in volts (V) and is a scalar quantity, meaning it has only magnitude and no direction.
Equipotential lines are lines in space that connect points having the same electric potential. In other words, all points on an equipotential line have the same electric potential value.
Electric field lines, on the other hand, represent the direction and strength of the electric field at every point in space. They point in the direction of the force that would be exerted on a positive test charge placed at that point in the field.
The relationship between equipotential lines and electric field lines is that they are always perpendicular to each other. This means that at any point on an equipotential line, the electric field vector is perpendicular to the line.
This can be mathematically proven using the gradient operator, which relates the electric field to the electric potential.
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What difference does being wet or dry have on the severity of shock if a person comes into contact with 120 v?.
When a person comes into contact with electricity, the severity of the shock can be affected by whether they are wet or dry.
If a person is wet, the water on their skin can conduct electricity and allow it to pass through their body more easily, increasing the severity of the shock.
On the other hand, if a person is dry, the resistance to the flow of electricity is higher, reducing the severity of the shock.
In the case of a 120 V electrical shock, the severity of the shock can vary depending on the conditions.
It is important to note that any electric shock can be dangerous and potentially life-threatening, regardless of whether a person is wet or dry.
If someone comes into contact with electricity, it is crucial to seek medical attention immediately.
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Units of Planck's constant are {{c1::J s}}
The units of Planck's constant are Joule seconds (J*s).
Planck's constant is a fundamental physical constant that plays a crucial role in quantum mechanics. It relates the energy of a photon to its frequency through the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency. The unit of energy is Joules (J), and the unit of frequency is Hertz (Hz), so the unit of Planck's constant is J*s.
The significance of Planck's constant lies in its ability to bridge the gap between classical physics and quantum mechanics. It helps explain phenomena such as wave-particle duality, where particles can behave as waves and vice versa. Additionally, it is used in calculations related to atomic and subatomic particles, including the energy levels of electrons in atoms and the behavior of photons in lasers.
Overall, the units of Planck's constant demonstrate its importance as a fundamental constant in the field of quantum mechanics and its role in bridging the gap between classical physics and the mysterious realm of the subatomic world.
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what gauge pressure in the water pipes is necessary if a fire hose is to spray water to a height of 16 m?
1.568 atm gauge pressure in the water pipes is necessary if a fire hose is to spray water to a height of 16 m
Gauge pressure = ρgh
= 1000 × 9.8 × 16
= 156800 Pa
= 1.568 atm
The pressure measured in relation to the surrounding atmospheric pressure is known as gauge pressure. A diaphragm sensor can be used to detect gauge pressure, with one side of the diaphragm exposed to the pressure medium that has to be measured and the other exposed to the surrounding atmospheric pressure.
It should be noted that while measuring gauge pressure, unless the measuring site is exposed to ambient air pressure, the measured gauge pressure will alter with fluctuations in barometric pressure caused by changes in weather patterns.
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An object 0.1 m tall is placed 0.4 m from a convex mirror with a focal length of 0.3 m. What is the height of the image?
Answer: no exact answer
=
Explanation:
Michael has a constant elasticity of substitution (CES) utility function, U(q 1
,q 2
)=(q 1
rho
+q 2
rho
) rho
1
, where rho
=0 and rho≤11 14
Given that Michael's rho<1, what are his optimal values of q 1
and q 2
in terms of his income and the prices of the two goods? Answer 1. Substitute the income constraint into Michael's utility function to eliminate one control variable. Michael's constrained utility maximization problem is max q 1
,q 2
U(q 1
,q 2
)=(q 1
rho
+q 2
rho
) rho
1
s.t. Y=p 1
q 1
+p 2
q 2
We can rewrite Michael's budget constraint as q 2
=(Y−p 1
q 1
)/p 2
. Substituting this expression into his utility function, we can express Michael's utility maximization problem as: max q 1
U(q 1
, p 2
Y−p 1
q 1
)=(q 1
rho
+[ p 2
Y−p 1
q 1
] rho
) 1/rho
. By making this substitution, we have converted a constrained maximization problem with two control variables into an unconstrained problem with one control variable, q 1
2. Use the standard, unconstrained maximization approach to determine the optimal value for q 1
. To obtain the first-order condition, we use the chain rule and set the derivative of the utility function with respect to q 1
equal to zero: rho
1
(q 1
rho
+[ p 2
Y−p 1
q 1
] rho
) rho
1−rho
(rhoq 1
rho−1
+rho[ p 2
Y−p 1
q 1
] rho−1
[−− p 2
p 1
])=0 Using algebra, we can solve this equation for Michael's optimal q 1
as a function of his income and the prices: 15 (3.18) q 1
= p 1
1−σ
+p 2
1−σ
Yp 1
−σ
where σ=1/[1−rho]. By repeating this analysis, substituting for q 1
instead of for q 2
, we derive a similar expression for his optimal q 2
: (3.19) q 2
= p 1
1−σ
+p 2
1−σ
Yp 2
−σ
Thus, the utility-maximizing q 1
and q 2
are functions of his income and the prices.
The optimal values of \(q_1\) and \(q_2\) are determined by these equations, which are functions of Michael's income and the prices of the goods.
The given problem describes Michael's utility maximization problem with a constant elasticity of substitution (CES) utility function. The objective is to find the optimal values of \(q_1\) and \(q_2\) in terms of Michael's income (Y) and the prices of the two goods (\(p_1\) and \(p_2\)).
1. Substitute the income constraint into Michael's utility function:
\(U(q_1, q_2) = (q_1^\rho + q_2^\rho)^(1/\rho)\)
s.t. \(Y = p_1q_1 + p_2q_2\)
We can rewrite Michael's budget constraint as \(q_2 = (Y - p_1q_1)/p_2\). Substituting this expression into his utility function, we have:
\(U(q_1, p_2, Y) = (q_1^\rho + [p_2(Y - p_1q_1)/p_2]^\rho)^{(1/\rho)\)
By making this substitution, we have converted the constrained maximization problem with two control variables (\(q_1\) and \(q_2\)) into an unconstrained problem with one control variable \((q_1)\).
2. Use the standard unconstrained maximization approach to determine the optimal value for \(q_1\). To obtain the first-order condition, we differentiate the utility function with respect to \(q_1\) and set it equal to zero:
\(\delta U / \delta q_1 = \rho(q_1^{(\rho-1)} + \rho[p_2(Y - p_1q_1)/p_2]^{(\rho-1)}(-p_1/p_2)) = 0\)
Simplifying and solving for \(q_1\):
\(\rho q_1^{(\rho-1)} - \rho(p_1/p_2)[p_2(Y - p_1q_1)/p_2]^{(\rho-1)} = 0\)
\(\rho q_1^{(\rho-1)} - \rho(p_1/p_2)[Y - p_1q_1]^{(\rho-1)} = 0\)
\(\rho q_1^{(\rho-1)} = \rho(p_1/p_2)[Y - p_1q_1]^{(\rho-1)}\)
\(q_1^{(\rho-1)} = (p_1/p_2)[Y - p_1q_1]^{(\rho-1)\)
\(q_1^{(\rho-1)} = (p_1/p_2)^{(1-\rho)}[Y - p_1q_1]^{(\rho-1)}\)
\(q_1^{(\rho-1)} = (p_1/p_2)^{(1-\rho)}(Y - p_1q_1)^{(\rho-1)}\)
\(q_1^{(\rho-1)} = (p_1/p_2)^{(1-\rho)}(Y^{(\rho-1)} - (\rho-1)p_1q_1(Y - p_1q_1)^{(\rho-2)})\)
This equation represents Michael's optimal \(q_1\) as a function of his income (Y) and the prices (\(p_1\) and \(p_2\)).
3. Similarly, we can derive a similar expression for his optimal \(q_2\):
\(q_2^{(\rho-1)} = (p_2/p_1)^(1-\rho)(Y^{(\rho-1)} - (\rho-1)p_2q_2(Y - p_1q_2)^{(\rho-2)})\)
This equation represents Michael's optimal \(q_2\) as a function of his income (Y) and the prices (\(p_1\) and \(p_2\)).
Therefore, these equations, which depend on Michael's income and the prices of the commodities, determine the ideal values of \(q_1\) and \(q_2\).
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Waves start as a disturbance (an input of energy) in a medium (water, air, the earth etc). The energy of the disturbance is moved from the starting point to other places in the medium by what means
A surge is a nonstop and repeating disturbance of a medium and a palpitation is a single disturbance.
What's surge?
A surge is a disturbance that propagates through a medium. There are three words in that description that may need discharging disturbance, propagate, and medium.
A disturbance, in the sense used in this description, is a change from the current state of a measurable volume at some position. For illustration
a change in a kinematic variable like position, haste, or acceleration;
a change in an ferocious property like pressure, viscosity, or temperature;
a change in field strength like electric field strength, glamorous field strength, or gravitational field strength.
To propagate, in the sense used in this description, is to transmit the influence of commodity in a particular direction. Antonyms for propagate include spread, transmit, communicate, and broadcast. The noun form of the word is propagation.
A medium is the substance through which a surge can propagate. Water is the medium of ocean swells. Air is the medium through which we hear sound swells. The electric and glamorous fields are the medium of light. People are the medium of a colosseum surge. The Earth is the medium of seismic swells( earthquake swells). Cell membranes are the medium of whim-whams impulses. Transmission lines are the medium of interspersing current electric power. Medium is the form of the noun. Media is the plural form( although mediums is by some people).
Let's list a many crucial exemplifications of surge marvels and also connect them to this description. The first illustration that comes to mind when utmost people hear the word surge are the kinds of swells that one sees on the face of a body of water deep water swells in the ocean or ripples in a billabong
. The most important kinds of swells for humans are the swells we use to smell the world around us sound and light.
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Why is thermal energy classified as kinetic energy?
Answer:
Thermal energy is an example of kinetic energy, as it is due to the motion of particles, with motion being the key. Thermal energy results in an object or a system having a temperature that can be measured. Thermal energy can be transferred from one object or system to another in the form of heat.
Explanation:
Answer:
Thermal energy is an example of kinetic energy, as it is due to the motion of particles, with motion being the key. Thermal energy results in an object or a system having a temperature that can be measured. Thermal energy can be transferred from one object or system to another in the form of heat.
Explanation:
How many more avocados Does city of Phoenix?
More avocados Does city of Phoenix is Therefore, in Phoenix they consume 60% more avocados than in Denver.
Given that in Denver they consume 75 units of avocados, while in Phoenix they consume 120 units of said fruit, to determine the percentage difference between the consumption of both cities it is necessary to perform the following calculation:
75 = 100
120 = X
((120 x 100) / 75) = X
(12,000 / 75) = X
160 = X
160 - 100 = 60
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A spaceship hovering over the surface of Venus drops an object from a height of 24 m. How much longer does it take to reach the surface than if dropped from the same height on Earth? Neglect air resistance in both cases. [The acceleration due to gravity on Venus is 90.7% of that on Earth,
gVenus = (0.907)g.]
The time taken is 2.3s for a spaceship hovering over the surface of Venus to drop an object from a height of 24m, and 2.21s for the same spaceship hovering over the surface of Earth to drop an object from the same height.
What is the time taken?To solve this problem, we will use the motion equation to calculate the time of flight of an object on the surface of Venus and the Earth. The height is related by the following equation of motion:
h = v₀t + gt²/2
Because the object's initial velocity before dropping is zero, we can simplify the equation to:
h = gt²/2
We know the height h of the spaceship hovering, and Venus's gravity is g = 9.07m/s². Substituting the following values into the equation:
24m = (9.07 m/s²t²)/2
To calculate the time it takes an object dropped by a spaceship hovering from a height of 24m to reach the surface of Venus, we must remove t from the equation above, yielding:
t = \(\sqrt{2(24m)/9.07m/s^{2} }\)
= \(\sqrt{48m/9.07m/s^{2} }\)
= 2.3s
Similarly, to calculate the time it takes an object dropped from a height of 24m to reach the Earth's surface, and the gravity of the Earth is g = 9.81m/s² .
t = \(\sqrt{2(24m)/9.81m/s^{2} }\)
= \(\sqrt{48/9.81m/s^{2} }\)
= 2.21s
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I NEED THIS ANSWERED ASAP WITH THE RIGHT ANSWER
A bicycle with a 15 kg mass is easily stopped when a force of 60 N is applied. How much force would be needed to stop the bicycle if the time allowed to stop the bike was doubled?
Answer:
Going downhill at a steady speed, the forces along the direction of motion are:
net force = m * a = 0
W * sin(7) - Fdrag = 0
so Fdrag = m * g * sin(7) where m=62.kg and g=9.8m/sec^2
Explanation:
How many years would it take to reach the planet saturn travelling at {=5*5} thousand miles per hour?
To calculate the time it would take to reach Saturn, we need to know the distance between Earth and Saturn and the speed of the journey.
The average distance from Earth to Saturn varies due to the elliptical nature of the planet's orbit. On average, Saturn is about 1.4 billion kilometers (869 million miles) away from Earth. Let's use this as a rough estimate.Given that the speed is 5 * 5 thousand miles per hour (25,000 miles per hour), we can calculate the time it would take to travel the distance.Therefore, it would take approximately 3.97 years (or about 4 years) to reach Saturn at a speed of 25,000 miles per hour. Keep in mind that this is an approximation, as the distance and speed values are average values and may vary depending on the actual trajectory and speed of the spacecraft.
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Give an example of periods (periodic table)
Answer:
helium, hydrogen, nitrogen, oxygen,iron,uranium,
Explanation:
there are 118 elements but here are some
do you expect the friction force to have a larger effect on the two accelerations if the angle is larger or smaller? why?
As the angle of incline is increased then acceleration will also increase.
As the angle of the incline is increased, normal force is decreases, which decreases the frictional force also. The incline can be raised until the object begins to slide. The force acting parallel to the plane causes the object to accelerate down the incline and the force of friction opposes the object's motion so it acts upward.
When the angle is increased, the component of weight along the plane also (mg sin θ) increases. To balance this, the frictional force has to increase.
Angular acceleration represents time rate of change of angular velocity. The units for this acceleration is rads/s2 or degrees/s2.
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How would the mass and weight of an object on the moon compare to the mass and weight of the same object on earth?.
The mass would stay the same, but the weight would be only 16.5%.
Explanation:
Mass is the amount of "matter" in an object, which stays the same everywhere. Weight is the force acting on an object by gravity, which changes depending on nearby bodies. In this case, weight on the moon is 16.5% of that on Earth.
If two objects collide and stick together, what must increase?
A. the acceleration due to gravity
B. the friction between the two objects
C. the force of gravity between the objects
D. the two objects' momentum
Help pls.
How much heat is required to convert 500g of liquid water at 28°C into steam at 150 °C? Take the specific heat capacity of water to be 4183 J/Kg °C and the latent heat of vaporization to be 2.26 × 10^6 J/Kg.
Answer:
Q = 1.404 × 10^(5) KJ
Explanation:
We are given:
Mass;m = 500 g = 0.5kg
Temperature 1;T1 = 28 °C
Temperature:T2 = 150 °C
Specific heat capacity;c_p = 4183 J/Kg °C
Latent heat of vaporization;L = 2.26 × 10^(6) J/Kg.
The heat energy needed is given by;
Q = sensible heat energy + Latent heat
Formula for sensible heat is;
Sensible heat energy = mc(t2 - t1)
Formula for Latent heat is ;
Latent heat = mL
Thus:
Q = mc(t2 - t1) + mL
Q = m[c(t2 - t1) + L]
Q = 0.5((4183(159 - 28) + (2.26 × 10^(6)))
Q = 1.404 × 10^(8) J = 1.404 × 10^(5) KJ
Joel the firefighter, standing 45 m from a burning building, directs a stream of water from a ground level fire hose at an angle of 25 degrees above the horizontal. The Vi of the stream as it leaves the hose is 55 m/s. What is the water’s Vix?
: where are you likely to find a cooling plant located in the duckwork? A. single family home B. A large apartment C. large factory D. A medium sized offices
Answer:
Option C
Explanation:
In a large sized factory, it is essential to create cooling system based on duct work because it can then be able to regulate cooling of any section of the factory from one place. Also, ductwork cooling is preferred in large spaces such as big offices building, towers, factories etc.
Hence, option C is correct
47. explain why the abundances of heavy elements in stars correlate with their positions in the galaxy.
The presence of heavy elements in stars correlates with their positions because these elements show which star formed first, and a star's brightness and spectral type show where it is in the galaxy.
Heavy elements are present in population I stars. It is concentrated in the discs of spiral galaxies and typically contains the sun. It is also hot, youthful, and brilliant. The majority of them are found in spiral arms. In globular clusters and the outer galactic halo, where the concentration of heavy elements is relatively low, population II stars are born. The important information to compare the locations of the stars is therefore their abundances of heavy metals. Energy must be added in order for elements heavier than iron and nickel to develop.
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The most economical sources for electricity in remote locations would be
Solar and Wind Electricity Production.
Petroleum and Nuclear-produced Electricity.
Tidal and Biomass-fuel Electricity.
Hydrogen Fuel Cells and Biomass.
The most economical sources for electricity in remote locations would be solar and wind electricity production.
What is Solar Electricity Production?Solar power is the process of converting sunlight into electricity using photovoltaic (PV) cells or panels. The photovoltaic effect, discovered by French physicist Edmond Becquerel in 1839, converts sunlight into electricity.
Solar panels generate direct current (DC) electricity when exposed to sunlight. The current is transferred to an inverter, which converts the DC electricity to alternating current (AC) electricity, which is used in homes and buildings.
What is Wind Electricity Production?Wind energy is a type of renewable energy generated by the movement of the wind. Wind turbines are used to convert wind energy into electricity. Wind energy is the most commonly used renewable energy source, accounting for around 5% of all electricity generated worldwide. Wind turbines can be installed on land or offshore, depending on the location.
In conclusion, solar and wind electricity production are the most economical sources for electricity in remote locations.
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What does the idea of multifactorial causation in drug effects suggest?
Taking several different drugs at the same time produces unpredictable effects.
Each person will have the same physiological response to a drug, but there can be different psychological responses.
Taking several different drugs at once increases their effectiveness. T
he effects of a drug depend on the dosage and the user’s personal makeup and experience.
According to the theory of multifactorial causation in drug effects, a drug's effects are influenced by dosage, the user's individual characteristics, and past experiences.
Dosage forms, sometimes also referred to as unit doses, are pharmaceutical drug products in the form in which they are marketed for use, with a particular combination of active ingredients and inactive ingredients (excipients), in a particular configuration (such as a capsule shell, for example), and apportioned into a specific dose. If two items are both amoxicillin, for instance, one comes in 500 mg capsules and the other in 250 mg chewable tablets. When each drug product is separately packaged, the term "unit dose" can also occasionally refer to non-reusable packaging), while the FDA distinguishes between the two terms as "packaging" or "dispensing" of unit doses. Depending on the situation, the term "multiple-unit dosage" might apply to various drug products packaged together or to a single drug product containing multiple pharmaceuticals.
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Which of the following systems
is a good example of potential
energy being transformed into
kinetic energy and back again?
The movement of a pendulum is a good example of potential energy being transformed into kinetic energy and back again.
Kinetic Energy: The energy of a body due to its motion is called kinetic energy.
Potential Energy: Potential energy is the energy that is stored in an object due to its position relative to some zero position. Electrostatic energy, Gravitational energy, etc. are some examples of potential energy.
When the pendulum is at its extremes, all the energy of the pendulum is converted into potential energy. And when it is at the center it has only kinetic energy. At other positions, the total energy is the sum of potential and kinetic. But the total energy of the pendulum is always conserved.
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The correct question is-
Give an example of a system where potential energy is transformed into kinetic energy and back again. Explain.
A baseball goes from zero to 32 m/s in 0.164 s.
What is its average acceleration?
Answer in units of m/s².
Answer: Доброе утро, ответ 5+5="4
Explanation:
Your bone structure, metabolism and body fat distribution all contribute to your ______ A.Body type B. Height C. Energy levels D. Exercise habits
Answer:
Maybe [A]
Explanation:
True or False: The variables in the equation 4x-(5y)2=64x-(5y)2=6 are 4, 5, and 6
Answer:
true
Explanation:
Answer:
true
Explanation:
an electron moves with a speed of what are the magnitude and direction of the magnetic force on the electron
To determine the magnitude and direction of the magnetic force on an electron moving with a certain speed, we need additional information, specifically the strength and direction of the magnetic field in which the electron is moving. The magnetic force experienced by a charged particle depends on the velocity of the particle, the magnetic field strength, and the angle between the velocity vector and the magnetic field vector.
If we have the value and direction of the magnetic field, we can use the formula for the magnetic force on a charged particle:
F = q * v * B * sin(theta)
Where:
F is the magnitude of the magnetic force
q is the charge of the electron
v is the velocity of the electron
B is the magnetic field strength
theta is the angle between the velocity vector and the magnetic field vector
If you provide the necessary information, such as the magnetic field strength and its direction, we can calculate the magnitude and direction of the magnetic force on the electron.
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Complete the missing information for each of the following.
Calculate the energy, height, and velocity at each of the following
points for the 1.5 kg weighted basketball.
(at the top) PE=29.4J KE= h=
(at the halfway point) PE= KE= v= h=
(on the floor) PE= KE= v=
Answer:
3.141592653 im king of NYC
Explanation:
a section of hollow pipe and a solid cylinder have the same radius, mass, and length. they both rotate about their long central axes with the same angular speed. which object has the higher rotational kinetic energy?
Answer:
Teniendo dos tuberías, una hueca y otra sólida, con la misma masa, radio y longitud, ambas poseen la misma rapidez angular, entonces el cuerpo hueco tendrá mayor energía cinética rotacional que la tubería sólida, esto es debido a que el tubo hueco tiene mayor inercia rotacional.
La ecuación de energía rotacional viene dada como:
Er = (1/2)·I·ω
De esta manera tenemos que el tubo hueco tiene mayor inercia rotacional que el tubo sólido, por ende, mayor energía.
Explanation:
What beat frequencies are possible with tuning forks of frequencies 255, 258, and 260 hz ?
Possible beat frequencies with tuning forks of frequencies 255, 258, and 260 Hz are 2, 3 and 5 Hz respectively.
The beat frequency refers to the rate at which the volume is heard to be oscillating from high to low volume. For example, if two complete cycles of high and low volumes are heard every second, the beat frequency is 2 Hz. The beat frequency is always equal to the difference in frequency of the two notes that interfere to produce the beats. So if two sound waves with frequencies of 256 Hz and 254 Hz are played simultaneously, a beat frequency of 2 Hz will be detected. A common physics demonstration involves producing beats using two tuning forks with very similar frequencies. If a tine on one of two identical tuning forks is wrapped with a rubber band, then that tuning forks frequency will be lowered. If both tuning forks are vibrated together, then they produce sounds with slightly different frequencies. These sounds will interfere to produce detectable beats. The human ear is capable of detecting beats with frequencies of 7 Hz and below.
A piano tuner frequently utilizes the phenomenon of beats to tune a piano string. She will pluck the string and tap a tuning fork at the same time. If the two sound sources - the piano string and the tuning fork - produce detectable beats then their frequencies are not identical. She will then adjust the tension of the piano string and repeat the process until the beats can no longer be heard. As the piano string becomes more in tune with the tuning fork, the beat frequency will be reduced and approach 0 Hz. When beats are no longer heard, the piano string is tuned to the tuning fork; that is, they play the same frequency. The process allows a piano tuner to match the strings' frequency to the frequency of a standardized set of tuning forks.
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1. A ball is thrown straight up.if the launch velocity is 15 m/s, at what velocity will the ball return to the thrower's hand?
2. A boat moves across a river going 18 m/s. At the same time there is a current flowing at a right angle to the boat at 6 m/s. What is the resulting velocity of the boat?
Answer:
1) The velocity of the ball return to the thrower's hand is -15 meters per second.
2) The resulting velocity of the boat is \(\vec v_{B} = 6\,\hat{i}+18\,\hat{j}\,\left[\frac{m}{s} \right]\).
Explanation:
1) Let suppose that ball experiments a free fall, that is an uniform accelerated motion, in which effects from gravity and Earth's rotation can be neglected. The velocity of the ball is represented by the following equations of motion:
Position
\(v_{o}\cdot t -\frac{1}{2}\cdot g\cdot t^{2} = 0\)
\(t\cdot \left(v_{o}-\frac{1}{2}\cdot g\cdot t \right) = 0\) (1)
Velocity
\(v = v_{o}-g\cdot t\) (2)
Where:
\(t\) - Time, measured in seconds.
\(g\) - Gravitational acceleration, measured in meters per square second.
\(v_{o}\) - Initial velocity of the ball, measured in meters per second.
\(v\) - Final velocity of the ball, measured in meters per second.
From (1), we get the time when the ball returns to the thrower's hand:
\(v_{o}-\frac{1}{2}\cdot g\cdot t = 0\)
\(t = \frac{2\cdot v_{o}}{g}\)
And then we apply this result in (2):
\(v = v_{o}-g\cdot \left(\frac{2\cdot v_{o}}{g} \right)\)
\(v = -v_{o}\) (3)
Then, the velocity of the ball return to the thrower's hand is -15 meters per second.
2) The resulting velocity of the boat (\(\vec v_{B}\)) is represented by the vectorial sum of the velocity of the boat relative to the river (\(\vec v_{B/R}\)) and the velocity of the river (\(\vec v_{R}\)), both measured in meters per second, that is:
\(\vec v_{B} = \vec v_{R}+\vec {v}_{B/R}\) (4)
If we know that \(\vec v_{R} = 6\,\hat{i}\,\left[\frac{m}{s} \right]\) and \(\vec v_{B/R} = 18\,\hat{j}\,\left[\frac{m}{s} \right]\), then the resulting velocity of the boat is:
\(\vec v_{B} = 6\,\hat{i}+18\,\hat{j}\,\left[\frac{m}{s} \right]\)
The resulting velocity of the boat is \(\vec v_{B} = 6\,\hat{i}+18\,\hat{j}\,\left[\frac{m}{s} \right]\).