An object’s velocity is measured to be vx(t) = α - βt2, where α = 4.00 m/s and β = 2.00 m/s3. At t = 0 the object is at x = 0. (a) Calculate the object’s position and acceleration as functions of time. (b) What is the object’s maximum positive displacement from the origin?

Answer:

Explanation:

Given

initial Pressure [tex]P_1=90\ psia[/tex]

elevation [tex]z_1=16\ ft[/tex]

Final Pressure [tex]P_2=30\ psia[/tex]

elevation [tex]z_2=450\ ft[/tex]

Pressure after Pumping(pump inlet pressure ) is given by

[tex]P_{after\ pump}=P_{top}+\Delta P[/tex]

[tex]\Delta P=\rho gh[/tex]

where [tex]\rho [/tex]=density of water[tex](62.2\ lbm\ft^3)[/tex]

g=acceleration due to gravity[tex](32.2\ ft/s^2)[/tex]

h=elevation

[tex]\Delta P=62.2\times 32.2\times (16+450)\times \frac{1\ lbf}{32.174\ lbm\ ft}[/tex]

[tex]\Delta P=28,985\ lbf/ft^2[/tex]

[tex]\Delta P=201.3\ lbf/in.^2[/tex]

[tex]P_{after\ pump}=30+201.3=231.3\ lbf/in.^2[/tex]

Pressure required to be applied

[tex]\Delta P_{pump}=P_{after\ pump}-P_{bottom}[/tex]

[tex]\Delta P_{pump}=231.3-90=141.3\ psi[/tex]                  

Answer:

Explanation:

Given

Initial Volume [tex]v_1=1\ m^3[/tex]

final Volume [tex]v_2=3\ m^3[/tex]

Heat added at constant Pressure [tex]Q=200\ kJ[/tex]

Decrease in Energy of System [tex]\Delta U=-340\ kJ[/tex]

According to First law of thermodynamics

[tex]Q=\Delta U+W[/tex]

[tex]W=Q-\Delta U[/tex]

[tex]W=200-(-340)[/tex]

[tex]W=540\ kJ[/tex]

Work done in a constant Pressure Process is given by

[tex]W=P\Delta V[/tex]

where P is the constant Pressure

[tex]540=P\times (3-1)[/tex]

[tex]P=270\ kPa[/tex]                          

Answer:

Both hit at the same time

Explanation:

If air resistant is ignored, then gravitational acceleration g is the only thing that affect the bullets vertical motion, no matter what their horizontal motion is. Since both bullets are starting from rest, vertically speaking their speeds are 0 initially, they are both subjected to the same acceleration g, then they travel at the same rate and would reach the ground at the same time.

Answer:

B. 0.0254m

Explanation:

A is the amplitude of the oscillation, i.e. the

maximum displacement of the object from

equilibrium, either in the positive or negative x-direction. Simple harmonic motion is repetitive.

The period T is the time it takes the object tocomplete one oscillation and return to the startingposition.

d = 2A = 2×0.0127

Answer:

[tex]Q=4704000J\\Q=4.7MJ[/tex]

Explanation:

Given data

Heat capacity c=4.2 J/gC

Water weigh m=1000g

Temperature Risen T=20°C to 90°C

To find

Heat Q

Solution

The formula to find Heat is:

Q=mcΔT

Where

m is mass

c is heat capacity

ΔT is change in temperature

So

[tex]Q=4.2(16*1000)*(90-70)\\Q=4704000J\\Q=4.7MJ[/tex]

The automobile cooling system would absorb approximately 4,699,840 Joules of heat when its temperature rises from 20°C to 90°C, according to the formula for heat absorption Q = mcΔT.

To answer this question, we apply the formula for heat absorption: Q = mcΔT.

 Here, 'Q' is the total heat absorbed, 'm' is the mass of the substance (water in this case), 'c' is the specific heat capacity of the substance, and 'ΔT' is the change in temperature. The mass can be defined as the volume of water multiplied by the density of water (1 g/cm^3). So for 16L, m = 16000g. The specific heat capacity of water is approximately 4.186 J/g °C, and the change in temperature, ΔT, is 70°C (90°C - 20°C).

Plugging in these values, we get Q = 16000g× 4.186 J/g °C × 70°C = 4,699,840 Joules. Therefore, it's estimated that the automobile cooling system would absorb around 4,699,840 Joules of heat.

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Answer:

Th steps is as shown in the attachment

Explanation:

from the diagram, its indicates that twelve identical charges are distributed evenly on the circumference of the circle. assuming one of gthe charge is shifted to the centre of he circle alomng the x axis, as such the charge is unbalanced and there is need ot balanced all the identical charges for the net force to be equal to zero.

The mathematical interpretation is as shown in the attachment.

Answer:

The initial speed of the ball is 30 m/s.

Explanation:

It can be assumed that the ball is thrown at an angle of 45 degrees to the ground. The ball lands 90 m away. We need to find the initial speed of the ball. We know that the horizontal distance covered by the projectile is called its range. It is given by :

[tex]R=\dfrac{u^2\ sin2\theta}{g}[/tex]

u is the initial speed of the ball.

[tex]v=\sqrt{\dfrac{Rg}{sin2\theta}}[/tex]

[tex]v=\sqrt{\dfrac{90\times 9.8}{sin2(45)}}[/tex]

v = 29.69 m/s

or

v = 30 m/s

So, the initial speed of the ball is 30 m/s. Hence, this is the required solution.                                      

The exact initial speed requires the angle [tex]\( \theta \)[/tex]; assuming [tex]\( \theta = 45^\circ \)[/tex], the initial speed is 29.71 m/s.

To solve this problem, we need to use the kinematic equations for projectile motion. The horizontal distance (range) \( R \) that the ball travels can be found using the following equation:

[tex]\[ R = \frac{{v_0^2 \sin(2\theta)}}{g} \][/tex]

Given that the ball lands 90 m away and assuming that the angle of projection [tex]\( \theta \)[/tex] is known (but not provided in the question), we can rearrange the equation to solve for [tex]\( v_0 \)[/tex]:

[tex]\[ v_0^2 = \frac{R \cdot g}{\sin(2\theta)} \][/tex]

[tex]\[ v_0 = \sqrt{\frac{R \cdot g}{\sin(2\theta)}} \][/tex]

However, since the angle [tex]\( \theta \)[/tex] is not given, we cannot calculate the exact initial speed [tex]\( v_0 \)[/tex] without additional information. The problem as stated is incomplete because it requires the value of [tex]\( \theta \)[/tex] to find [tex]\( v_0 \)[/tex].

If we assume that the angle [tex]\( \theta \)[/tex] is such that [tex]\( \sin(2\theta) \)[/tex] is maximized (which occurs at [tex]\( \theta = 45^\circ \)[/tex], then [tex]\( \sin(2 \cdot 45^\circ) = \sin(90^\circ) = 1 \)[/tex], and the equation simplifies to:

[tex]\[ v_0 = \sqrt{R \cdot g} \][/tex]

[tex]\[ v_0 = \sqrt{90 \, \text{m} \cdot 9.81 \, \text{m/s}^2} \][/tex]

[tex]\[ v_0 = \sqrt{882.9 \, \text{m}^2/\text{s}^2} \][/tex]

[tex]\[ v_0 =29.71 \, \text{m/s} \][/tex]

 Therefore, under the assumption that [tex]\( \theta = 45^\circ \)[/tex], the initial speed of the ball would be [tex]\( 29.71 \, \text{m/s} \)[/tex].

In conclusion, without the value of [tex]\( \theta \)[/tex], we cannot determine the exact initial speed of the ball. However, if we assume the most favorable condition for maximum range, where [tex]\( \theta = 45^\circ \)[/tex], the initial speed [tex]\( v_0 \)[/tex] would be [tex]\( 29.71 \, \text{m/s} \)[/tex].

Answer:

They both move towards South direction.

Explanation:

NB: The car traveling due east must be backing west and the car traveling due south must be backing north.

From their momentum in calculation below:

Momentum of the car traveling east is MU= 1500x90x1000/3600= 37,500kgm/s.

Momentum of car traveling due South =MU = 3000x60x1000/3600= 50,000. So, from here we see the the momentum of the car moving south has the high momemtum so they stick together and move towards the direction of south.

Answer:

223 degree

Explanation:

We are given that

Magnitude of resultant vector= 8 units

Resultant vector makes an angle with positive -x in counter clockwise direction

[tex]\theta=43^{\circ}[/tex]

We have to find the magnitude and angle of the equilibrium vector.

We know that equilibrium vector is equal in magnitude  and in opposite direction  to the given vector.

Therefore, magnitude of equilibrium vector=8 units

x-component of a  vector=[tex]v_x=vcos\theta[/tex]

Where v=Magnitude of vector

Using the formula

x-component of resultant  vector=[tex]v_x=8cos43=5.85[/tex]

y-component of resultant vector=[tex]v_y=vsin\theta=8sin43=5.46[/tex]

x-component of equilibrium vector=[tex]v_x=-5.85[/tex]

y-component of equilibrium vector=[tex]-v_y=-5.46[/tex]

Because equilibrium vector lies in III quadrant

[tex]\theta=tan^{-1}(\frac{v_x}{v_y})=tan^{-1}(\frac{-5.46}{-5.85})=43^{\circ}[/tex]

The angle [tex]\theta'[/tex]lies in III quadrant

In III quadrant ,angle =[tex]\theta'+180^{\circ}[/tex]

Angle of equilibrium vector measured from positive x in counter clock wise direction=180+43=223 degree

3 trams must be added

Explanation:

In this problem, there are 12 trams along the ring road, spaced at regular intervals.

Calling L the length of the ring road, this means that the space between two consecutive trams is

[tex]d=\frac{L}{12}[/tex] (1)

In this problem, we want to add n trams such that the interval between the trams will decrease by 1/5; therefore the distance will become

[tex]d'=(1-\frac{1}{5})d=\frac{4}{5}d[/tex]

And the number of trams will become

[tex]12+n[/tex]

So eq.(1) will become

[tex]\frac{4}{5}d=\frac{L}{n+12}[/tex] (2)

And substituting eq.(1) into eq.(2), we find:

[tex]\frac{4}{5}(\frac{L}{12})=\frac{L}{n+12}\\\rightarrow n+12=15\\\rightarrow n = 3[/tex]

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Answer: 3

Explanation:

We need to do (1/(5-1))x12

When substances with different heat capacities are brought into thermal contact, heat transfers until they reach thermal equilibrium. In this case, the final temperature will be closer to the initial temperature of substance A.

When two substances with different heat capacities are brought into thermal contact, heat will transfer from the substance with a higher initial temperature to the substance with a lower initial temperature until they reach thermal equilibrium. In this case, since substance A has a much greater heat capacity than substance B, it can absorb and transfer a larger amount of heat. Therefore, the final temperature of the two substances will likely be closer to the initial temperature of substance A and lower than the initial temperature of substance B.

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Final answer:

To maintain the total charge below 953 µC for capacitors connected in parallel to a 3.00 V battery, a maximum of 45 capacitors, each with a capacitance of 7.00 µF, can be used. This is calculated by dividing the total charge by the charge per capacitor (953 µC / 21 µC per capacitor).

Explanation:

To find out the maximum number of 7.00 µF capacitors that can be connected in parallel while keeping the total charge under 953 µC, we use the formula for charge on a capacitor, Q=CV, where C is the capacitance and V is the voltage. Since the capacitors would be connected in parallel, the voltage across each capacitor would be the same as the voltage of the battery, which is 3.00 V.

The total charge Q for one capacitor is given by:

Q = C × V = 7.00 µF × 3.00 V = 21.00 µC per capacitor.

To find out how many such capacitors we could connect in parallel without exceeding the total charge of 953 µC, we divide the total permissible charge by the charge per capacitor:

Number of capacitors = Total charge / Charge per capacitor = 953 µC / 21.00 µC per capacitor ≈ 45.38.

Since we can't have a fraction of a capacitor, the maximum number you can use is 45 capacitors.

The maximum number of 7.00 µF capacitors that can be connected in parallel with a 3.00 V battery while keeping the total charge stored under 953 μC is 45 capacitors.

To determine the maximum number of 7.00 µF capacitors that can be connected in parallel with a 3.00 V battery while keeping the total charge stored within the capacitor array under 953 μC, follow these steps:

Since the number of capacitors must be an integer, the maximum number of 7.00 µF capacitors that can be connected in parallel is 45 capacitors.

Answer:

6908316.619 N/C

25087609.3949 N/C

6652357.02259 N/C

690831.6619 N/C

Explanation:

x = Distance from the ring

R = Radius of ring = 10 cm

q = Charge = 78 μC

k = Coulomb constant = [tex]8.99\times 10^{9}\ Nm^2/C^2[/tex]

Electric field at a point x is from a ring given by

[tex]E=\dfrac{kqx}{(x^2+r^2)^{1.5}}[/tex]

For 1 cm

[tex]E=\dfrac{kqx}{(x^2+r^2)^{1.5}}\\\Rightarrow E=\dfrac{8.99\times 10^9\times 78\times 10^{-6}\times 0.01}{(0.01^2+0.1^2)^{1.5}}\\\Rightarrow E=6908316.619\ N/C[/tex]

The electric field is 6908316.619 N/C

For 5 cm

[tex]E=\dfrac{kqx}{(x^2+r^2)^{1.5}}\\\Rightarrow E=\dfrac{8.99\times 10^9\times 78\times 10^{-6}\times 0.05}{(0.05^2+0.1^2)^{1.5}}\\\Rightarrow E=25087609.3949\ N/C[/tex]

The electric field is 25087609.3949 N/C

For 30 cm

[tex]E=\dfrac{kqx}{(x^2+r^2)^{1.5}}\\\Rightarrow E=\dfrac{8.99\times 10^9\times 78\times 10^{-6}\times 0.3}{(0.3^2+0.1^2)^{1.5}}\\\Rightarrow E=6652357.02259\ N/C[/tex]

The electric field is 6652357.02259 N/C

For 100 cm

[tex]E=\dfrac{kqx}{(x^2+r^2)^{1.5}}\\\Rightarrow E=\dfrac{8.99\times 10^9\times 78\times 10^{-6}\times 1}{(1^2+0.1^2)^{1.5}}\\\Rightarrow E=690831.6619\ N/C[/tex]

The electric field is 690831.6619 N/C

Answer:

The Double Slits Experiment and Michelson Interferometer

The questions will be answered for the double slits experiment.

(b) Mathematically, the double slits experiment equation can be given as: d sin 0 = m(Wavelength). d is the separation distance, 0 is the diffraction angle, m = 1,2,3,...

(c) assumptions

The width of the slits is lesser than the microwave's wavelength. This is to set up a standing wave between the microwave source and detector.

m is an positive integer. To obtain a constructive interference of the E-M wave(microwave)

(d) The uncertainties are:

(i)The zero error in the reading of the multiplier will disrupt the value of the wavelength by small percentage. This can be adjusted to obtain more accurate result.

(ii) The angle as obtained by the gionometer can not be measured to highest level of accuracy, as there are some approximations. High sensitive equipment should be used to obtain accurate result

Explanation:

Double Slits Experiment.

The double slits is a pair of 2cm wide slits with a 6cm separation, cut in a metal foil, with the double slits at the centre of turnable with the microwave source and microwave detector through the slits. The shunt on the microammeter is adjusted to

so that a large scale deflection is obtained. The interference pattern may then be explored by moving the receiver arm, whilst keeping the transmitter arm fixed. A graph of intensity (current) versus θ(Measured through the gionometer) should be plotted, and an estimate made of the microwave wavelength by measuring the angular separation of adjacent maxima in the interference pattern and the separation of the two slits.

Michelson Interferometer

Setting up the interferometer, the mirrors are metal sheets and the beamsplitter is a hardboard sheet. The metal sheets should be carefully positioned to be perpendicular to the microwave beam, and the beamsplitter positioned at the centre of the turntable and at 45° to the beam. Any slight change in the path length of either of the beams leaving the beamsplitter changes the interference pattern at the receiver. One of the metal sheets is mounted in a frame which slides along rails. If this sheet is slowly moved, maxima and minima of current will be recorded by the receiver and a plot of “intensity” against distance can be made. The distance between successive maxima (or minima) is one half of the microwave wavelength which can thus be measured.

Answer:

Q = 177J

Explanation:

Specific heat capacity of lead=0.13J/gc

Q=MCΔT

ΔT=T2-T1,where T1=22degrees Celsius and T2=37degree Celsius.

ΔT=37 - 22 = 15

Q = Change in energy

M = mass of substance

C= Specific heat capacity

Q = (91g) * (0.13J/gc) * (15c)= 177.45J

Approximately, Q = 177J

In space, spatial coordinates can be roughly divided into measures of Right ascension and declination. The declination is measured in degrees while the ascent is measured in hours, minutes, seconds. When you have objects in space such as those of the characteristics presented we will have to they are not necessarily close together in the sky because we can find two stars on the same right ascension but on different declination lines (Which means they can be very far apart from each other)

Answer:

In New York is at ocean level while Denver is at an altitude of 1 mile from ocean level. In this way, the breaking point of water is lower in Denver than in New York, that is, the water will boil at lower temperatures in Denver than in New York. If the breaking point of the water decreases, at this point it will put aside more effort to cook an egg. Now the time required to cook the egg is higher in Denver than in New York.

An egg cooks slower in Denver because water boils at a lower temperature due to the city's higher altitude and lower atmospheric pressure. This lower boiling point decreases the rate of the cooking chemical reactions.

An egg cooks more slowly in boiling water in Denver than in New York City due to differences in atmospheric pressure and the effect this has on boiling point temperatures. Denver, known as the Mile-High City, is approximately one mile above sea level. This altitude results in a lower atmospheric pressure than New York City which is virtually at sea level. Because of this, water boils at a slightly lower temperature in Denver (approximately 202 degrees Fahrenheit) than it does in New York City (approximately 212 degrees Fahrenheit).

Since cooking is essentially a series of chemical reactions, and these reactions occur faster at higher temperatures, an egg will cook more slowly in Denver’s boiling water than in New York's. This is because the boiling water in Denver is at a lower temperature due to the effect of pressure on boiling point.

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The magnitude of the gravitational force exerted by the Sun on Mars is approximately 1.49x10^22 N, while the magnitude of the gravitational force exerted by Mars on the Sun is approximately 5.92x10^15 N.

To calculate the magnitude of the gravitational force exerted by the Sun on Mars, we can use Newton's law of gravitation. The formula is given by F = G * (m1 * m2) / r^2, where F is the magnitude of the force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them.

Plugging in the values for the Sun's mass (2x10^30 kg), Mars' mass (6.4x10^23 kg), and the distance between them (2.3x10^11 m), we get

F = (6.673x10^-11 N·m²/kg²) * ((2x10^30 kg) * (6.4x10^23 kg)) / (2.3x10^11 m)^2

Simplifying the equation and performing the calculations, the magnitude of the gravitational force exerted by the Sun on Mars is approximately 1.49x10^22 N.

Similarly, to calculate the magnitude of the gravitational force exerted by Mars on the Sun, we can use the same formula with the masses and distance reversed. Plugging in the values, we get

F = (6.673x10^-11 N·m²/kg²) * ((6.4x10^23 kg) * (2x10^30 kg)) / (2.3x10^11 m)^2

Simplifying and calculating the equation, the magnitude of the gravitational force exerted by Mars on the Sun is approximately 5.92x10^15 N.

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Answer:

[tex]v = 21.77\ m/s[/tex]

Explanation:

Given,

Air resistance, = 0.45 m/s² for each 1 m/s

Air resistance for velocity v = 0.45 v.

Terminal velocity = ?

acceleration due to gravity= g = 9.8 m/s²

 now,

a = 0.45 v

we know,

[tex]a =\dfrac{dv}{dt}[/tex]

[tex]\dfrac{dv}{dt}=0.45 v[/tex]

[tex]\dfrac{dv}{v} = 0.45 dt[/tex]

integrating both side

[tex]\int \dfrac{dv}{v} = 0.45\int dt[/tex]

[tex] ln(v) = 0.45 t[/tex]

[tex]t = e^{0.45 t}[/tex]

When a body is moving at terminal velocity, Force due to gravity is equal to  force due to air resistance.

 m a = m g

    0.45 v = 9.8

     [tex]v = \dfrac{9.8}{0.45}[/tex]

    [tex]v = 21.77\ m/s[/tex]

Hence, the terminal velocity of the skydiver is equal to 21.77 m/s.

Answer:

[tex] f_s = 640 Hz[/tex]

Explanation:

For this case we know that the speed of the sound is given by:

[tex] V_s = 340 m/s[/tex]

And we have the following info provided:

[tex] v_c = 25 m/s [/tex] represent the car leading

[tex] v_s= 25 m/s[/tex] represent the car behind with the source

[tex] f_o = 640 Hz[/tex] is the frequency for the observer

And we can find the frequency of the source [tex] f_s[/tex] with the following formula:

[tex] f_s = \frac{v-v_o}{v-v_s} f_o [/tex]

And replacing we got:

[tex] f_s = \frac{340-25}{340-25} *640 Hz = 640 Hz[/tex]

So then the frequency for the source would be the same since the both objects are travelling at the same speed.

[tex] f_s = 640 Hz[/tex]

Answer:

a.Velocity=[tex]\vec{v}=(6t)\hat{j}+8\hat{k}[/tex]

b.[tex]\vec{a}=6\hat{j}[/tex]

Explanation:

We are given that

A particle moves so that its position (in m) as  function of time is

[tex]\vec{r}=2\hat{i}+(3t^2)\hat{j}+(8t)\hat{k}[/tex]

a.We have to find its velocity

We know that

Velocity,[tex]v=\frac{dr}{dt}[/tex]

Using the formula

Velocity,v=[tex]\frac{d(2i+3t^2j+8tk)}{dt}[/tex]

Velocity=[tex]\vec{v}=(6t)\hat{j}+8\hat{k}[/tex]

b.Acceleration[tex]=\vec{a}=\frac{d\vec{v}}{dt}[/tex]

[tex]\vec{a}=\frac{d((6t)\hat{j}+8\hat{k})}{dt}[/tex]

[tex]\vec{a}=6\hat{j}[/tex]

The velocity and acceleration of a particle can be calculated from its position function using differentiation in respect to time. The first derivative gives the velocity, while the second derivative gives the acceleration. Unit vector notation is used to display the results.

The question involves the calculations of a particle's velocity and acceleration over time, represented in unit vector notation. This scenario falls within the realm of physics, specifically kinematics. Given a particle's motion described by a position function, we can calculate its velocity and acceleration as functions of time.

V(t), the velocity of the particle, is the first derivative of the position function. Hence, to find the velocity, we simply differentiate the position function with respect to time. Now, to calculate A(t), the acceleration of the particle, we take the first derivative of the velocity function, or equivalently, the second derivative of the position function. This gives us the rate of change of velocity, which represents acceleration.

For example, suppose the particle's position function r(t) is given by r(t) = 3t^2 + 2 squares. Differentiating the position function once gives us v(t) = 6t + 2, the velocity function. Differentiating v(t) gives us a(t) = 6, the acceleration function. Both velocity and acceleration are given in unit vector notation.

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Answer:

[tex]q = 2.17\times 10^{-15}~{\rm C}[/tex]

Explanation:

The relation between the acceleration and the electrical force between the spheres can be found by Newton's Second Law.

[tex]\vec{F} = m\vec{a}\\\vec{F} = (1.4\times 10^{-3})(110) = 0.154~N[/tex]

This is equal to the Coulomb's Force.

[tex]F = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{2q}{(1.6 \times 10^{-2})^2} = 0.154\\q = 2.17\times 10^{-15}~C[/tex]

Answer:

E = 39.68 V

Explanation:

EMF ( Electromotive force) : This is defined as the magnitude of the potential difference of both the external circuit and the inside of the cell. The S.I unit is Volt.

The Expression for E.M.F is given as,

E = I(R+r) ................... Equation 1

Where E = EMF, I = current, R = External Resistance, r = internal resistance.

Also,

P = I²R

I = √(P/R) ..................... Equation 2

Where P = power, R = External resistance.

Given: P = 92 J/s, R = 14 Ω.

Substitute into equation 2

I = √(92/14)

I = √(6.57)

I = 2.56 A.

Also Given: r = 1.5 Ω.

Substitute into equation 1

E = 2.56(1.5+14)

E = 2.56(15.5)

E = 39.68 V.

A battery has emf E and internal resistance r = 2.00 Ω. A 11.5 Ω resistor is connected to the battery, and the resistor consumes electrical power at a rate of 96.0 J/s.

Part APart complete

What is the emf of the battery?

Express your answer with the appropriate units.

E =

39.0 V

Answer:

V= 12.56 m/s

Explanation:

Given that

R= 1 m

The angular speed ,ω = 2 rev/s

We know that 1 rev = 2 π rad

Therefore

ω = 4 π rev/s

The linear speed V is given as

V= ω x R

V = 4 π x 1 m/s

V= 4 π m/s

Therefore the speed of the bicycle will be 4 π m/s

We know that , π = 3.14

V= 12.56 m/s

The linear speed will be 12.56 m/s

Answer:

a. 6.69m/s

b. y=4.48m

c. t=1.43secs

Explanation:

Data given, acceleration,a=35m/s^2

distance covered,d=64cm=0.64m,

a. to determine the speed, we use the equation of motion

initial velocity,u=0m/s

if we substitute values we arrive at

[tex]v^{2}=u^{2}+2as\\v^{2}=0+2*35*0.64\\v^{2}=44.8m/s\\v=\sqrt{44.8}\\ v=6.69m/s\\[/tex]

b. After taking the shot,the acceleration value is due to gravity i.e a=9.81m/s^2

and the distance becomes (y-2.2) above the ground. When it reaches the maximum height, the final velocity becomes zero and the initial velocity becomes 6.69m/s.

Hence we can write the equation above again

[tex]v^{2}=u^{2}-2a(y-2.2)\\[/tex]

if we substitute values we have

[tex]v^{2}=u^{2}-2a(y-2.2)\\0=6.69^{2}-2*9.81(y-2.2)\\y-2.2=\frac{44.76}{19.62} \\y=2.28+2.2\\y=4.48m[/tex]

c. the time it takes to arrive at 1.83m is obtain by using the equation below

[tex]1.83-2.2=6.69t-\frac{1}{2} *9.81t^{2}\\4.9t^{2}-6.69t-0.37\\using \\t= \frac{-b±\sqrt{b^{2}-4ac} }{2a}\\ where \\a=4.9, b=-6.69, c=-0.37[/tex]

if we insert the values, we solve for t , hence t=1.43secs

(a) The speed of the shot when Sam releases it 6.75 m/s.

(b) The height risen by the shot above the ground is 4.52 m.

(c) The time taken for the shot to return to 1.8 m above the ground is 1.44 s.

The given parameters;

The speed of the shot when Sam releases it is calculated as;

[tex]v^2 = u^2 + 2as\\\\v^2 = 0 + 2a(\Delta h)\\\\v = \sqrt{2a(\Delta h)} \\\\v = \sqrt{2\times 35(0.65)} \\\\v = 6.75 \ m/s[/tex]

The height risen by the shot is calculated as follows;

[tex]v^2 = u^2 + 2gh\\\\at \ maximum \ height , v = 0\\\\0 = (6.75)^2 + 2(-9.8)h\\\\19.6h = 45.56 \\\\h = \frac{45.56}{19.6} \\\\h =2.32 \ m[/tex]

The total height above the ground = 2.20 m + 2.32 m = 4.52 m.

The time taken for the shot to return to 1.8 m above the ground is calculated as follows;

the time taken to reach the maximum height is calculated as;

[tex]h = vt - \frac{1}{2} gt^2\\\\2.32 = 6.75t - (0.5\times 9.8)t^2\\\\2.32 = 6.75t -4.9t^2\\\\4.9t^2 -6.75t + 2.32=0\\\\a = 4.9, \ b = -6.75, \ c = 2.32\\\\t = \frac{-b \ \ + /- \ \ \sqrt{b^2-4ac} }{2a} \\\\t = \frac{-(-6.75) \ +/- \ \ \sqrt{(-6.75)^2 - 4(4.9\times 2.32)} }{2(4.9)} \\\\t = 0.72 \ s\ \ or \ \ 0.66 \ s[/tex]

[tex]t \approx 0.7 \ s[/tex]

height traveled downwards from the maximum height reached = 4.52 m - 1.8 m = 2.72 m

[tex]h = vt + \frac{1}{2} gt^2\\\\h = 0 + \frac{1}{2} gt^2\\\\t = \sqrt{\frac{2h}{g} } \\\\t = \sqrt{\frac{2(2.72)}{9.8} } \\\\t = 0.74 \ s[/tex]

The total time spent in air;

[tex]t = 0.7 \ s \ + \ 0.74 \ s\\\\t = 1.44 \ s[/tex]

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Answer:

Explanation:

A ball is thrown vertically upward with a certain Kinetic Energy in the absence of air resistance and while returning it experiences air resistance.

Air resistance causes the ball to lose its kinetic energy as it provides resistance which will convert some of its kinetic energy to heat energy.

So in a way total energy is conserved but not kinetic energy as some portion of it is lost in the form of heat.              

Final answer:

A ball thrown upward with air resistance will return with less kinetic energy due to the negative work done by air resistance, which transforms some of its kinetic energy into heat. This does not violate the conservation of energy as the energy is still conserved but in different forms. Mechanical energy decreases but is compensated by the increase in thermal energy in the air.

Explanation:

When a ball is thrown vertically upward in the absence of air resistance, it will return to its original level with the same kinetic energy (KE) because energy is conserved in a system where no external forces do work. However, when air resistance is a factor, it will return with less kinetic energy. This is because air resistance does negative work on the ball, converting some of its Kinetic Energy into thermal energy as it dissipates heat into the air.

The conservation of energy principle is not contradicted by this scenario. The total energy (kinetic plus potential plus any energy converted into heat due to air resistance) is still conserved. However, the mechanical energy of the ball (the sum of its kinetic and potential energy) decreases due to the work done by air resistance. This reduction in mechanical energy is exactly balanced by the increase in thermal energy of the air.

If an object like a feather is thrown upward, it will experience significant air resistance, and it will definitely return with less kinetic energy than it had when it was thrown up. Again, this doesn't violate the conservation of energy; instead, the energy is simply transformed into different forms, primarily heat, due to interactions with the air.

Answer:

Number of electrons, [tex]n=9.98\times 10^{19}[/tex]

Explanation:

A strong lightning bolt transfers an electric charge of about 16 C to Earth, q = 16 C

We need to find the number of electrons that transferred. Let there are n electrons transferred. It is given by using quantization of electric charge as :

q = ne

[tex]n=\dfrac{q}{e}[/tex]

e is elemental charge

[tex]n=\dfrac{16}{1.602\times 10^{-19}}[/tex]

[tex]n=9.98\times 10^{19}[/tex]

So, there are [tex]9.98\times 10^{19}[/tex] electrons that gets transferred. Hence, this is the required solution.

A strong lightning bolt that transfers an electric charge of about 16 C to Earth, transfers 1.0 × 10²⁰ electrons (1.7 × 10⁻⁴ moles of electrons).

A strong lightning bolt transfers an electric charge of about 16 C to Earth (or vice versa).

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field.

We want to calculate the number of electrons that have a charge of 16 C. We have to consider the following relationships:

16 C × 1 electron/1.602 × 10⁻¹⁹ C = 1.0 × 10²⁰ electron

1.0 × 10²⁰ electron × 1 mol/6.022 × 10²³ electron = 1.7 × 10⁻⁴ mol

A strong lightning bolt that transfers an electric charge of about 16 C to Earth, transfers 1.0 × 10²⁰ electrons (1.7 × 10⁻⁴ moles of electrons).

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Answer:

(a) the electric field of the Earth will be directed towards the negatively charged water droplet.

(b) E = (9.8*m)/q

Explanation:

Part (a) the direction of the Earth’s electric field

Electric field is always directed towards negatively charged objects.

The water droplet has a negative charge ‘-q’, therefore the electric field of the Earth will be directed towards the negatively charged water droplet.

Part (b) an expression for the Earth’s electric field in terms of the mass and charge of the droplet

The magnitude of Electric field is given as;

E = F/q

where;

f is force and q is charge

Also from Newton's law, F = mg

where;

m is mass and g is acceleration due to gravity = 9.8 m/s²

E =  F/q = mg/q

E = (9.8*m)/q, Electric field in terms of mass and charge of the droplet.

Answer:

d. the same within the uncertainty of each measurement method

Explanation:

The density of an object and in general any physical property, has the same value regardless of the method used to measure it, either directly or indirectly. Since two completely different valid methods are used, the results must be the same, taking into account the level of precision of each of the methods.

Answer:

884 balloons

Explanation:

Assume ideal gas, since temperature is constant, then the product of pressure and volume is constant.

So if pressures reduces from 100 to 1.2, the new volume would be

[tex]V_2 = \frac{P_1V_1}{P_2} = \frac{100*0.15}{1.2} = 12.5 m^2[/tex]

The spherical volume of each of the balloon of 30cm diameter (15 cm or 0.15 m in radius) is

[tex]V_b = \frac{4}{3}\pir^3 = \frac{4}{3}\pi 0.15^3 = 0.014 m^3[/tex]

The number of balloons that 12.5 m3 can fill in is

[tex]V_2/V_b = 12.5 / 0.014 = 884[/tex]

Final answer:

Based on the volume of the tank, temperature, and pressure, we can calculate the number of moles of helium. By dividing this number by the number of moles of helium in one balloon, we can determine how many balloons can be blown up.

Explanation:

To find out how many balloons can be blown up with the given amount of helium, we need to calculate the total volume of helium in the tank and divide it by the volume of each balloon.

Given:

The volume of the tank is 0.150 m³

The temperature of the helium in the tank is 27.0°C

The pressure of the helium in the tank is 100 atm

The volume of each balloon is 30.0 cm in diameter, which is equivalent to a radius of 15.0 cm or 0.15 m

The temperature of the filled balloon is 27.0°C

The absolute pressure of the filled balloon is 1.20 atm

First, we need to convert the volume of the tank to liters:

0.150 m³ * 1000 L/m³ = 150 L

Next, we need to calculate the number of moles of helium in the tank using the ideal gas law:

P * V = n * R * T

n = (P * V) / (R * T)

where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Using the given values:

Pressure (P) = 100 atm

Volume (V) = 150 L

Ideal gas constant (R) = 0.0821 L·atm/K·mol

Temperature (T) = 27.0°C + 273.15 = 300.15 K

Calculating the number of moles:

n = (100 atm * 150 L) / (0.0821 L·atm/K·mol * 300.15 K) = 6.19 moles

Now we can calculate the number of balloons that can be blown up:

Number of balloons = (Number of moles of helium) / (Number of moles of helium in one balloon)

The number of moles of helium in one balloon can be found using the ideal gas law:

P * V = n * R * T

n = (P * V) / (R * T)

Using the given values:

Pressure (P) = 1.20 atm

Volume (V) = (4/3) * π * (0.15 m)³ = 0.141 m³

Ideal gas constant (R) = 0.0821 L·atm/K·mol

Temperature (T) = 27.0°C + 273.15 = 300.15 K

Calculating the number of moles:

n = (1.20 atm * 0.141 m3) / (0.0821 L·atm/K·mol * 300.15 K) = 0.006 moles

Finally, calculating the number of balloons:

Number of balloons = 6.19 moles / 0.006 moles = 1031

Therefore, 1031 balloons can be blown up with the given amount of helium.