A proton accelerates from rest in a uniform electric field of 680 N/C. At one later moment, its speed is 1.30 Mm/s (nonrelativistic because v is much less than the speed of light). (a) Find the acceleration of the proton.

Answer:

I. The hollow sphere has the greater angular momentum.

Explanation:

Given that the mass and radius of hollow sphere and solid sphere are same. Let the mass and radius of two spheres be m and r respectively. The two spheres are rotating having same angular velocity ω .

Moment of inertia of solid sphere, I₁ = [tex]\frac{2}{5}\times{m}r^{2}[/tex]

Moment of inertia of hollow sphere, I₂ = [tex]\frac{2}{3}\times{m}r^{2}[/tex]

Since, I₁ and I₂ are not equal. Therefore, the statement iv is wrong.

The relation between angular momentum, moment of inertia and angular velocity is :

L = Iω

Let L₁ and L₂ be the angular momentum of solid sphere and hollow sphere respectively.

L₁ = I₁ω     and   L₂ = I₂ω

As ω is same for both spheres but I₂ is greater than I₁, hence L₂ is greater than L₁.

Therefore, statement I is correct that the hollow sphere has the greater angular momentum.

The angular momentum of the hollow sphere is greater than that of solid sphere.

The moment of inertia of solid sphere is given as follows;

[tex]I_{ss} = \frac{2}{5} mr^2 = 0.4mr^2[/tex]

The moment of inertia of hollow sphere is given as follows;

[tex]I_{hs} = \frac{2}{3} mr^2 = 0.67 mr^2[/tex]

The angular momentum of each sphere is calculated as follows;

[tex]L = I\omega \\\\L_{ss} = 0.4mr^2 \omega \\\\L_{hs} = 0.67 mr^2 \omega[/tex]

Thus, we can conclude that the angular momentum of the hollow sphere is greater than that of solid sphere.

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the magnitude of the normal force is greater than the magnitude of the weight of the person

Explanation:

As the elevator moves down, there are two forces acting on the person:

- The weight of the person, [tex]W=mg[/tex], where m is the mass of the person and g the acceleration due to gravity, acting downward

- The normal reaction exerted by the floor of the lift on the person, N, acting upward

This means that Newton's second law can be written as

[tex]\sum F = N-W= ma[/tex] (1)

where we chose upward as positive direction, and a is the acceleration of the elevator.

Here we know that the elevator is moving downward and it is slowing down: this means that the velocity is negative (upward), and the acceleration is in the opposite direction (upward), so the acceleration is positive.

Eq.(1) can be rewritten as

[tex]N=W+ma[/tex]

and as we said that [tex]a>0[/tex], this means that

[tex]N>W[/tex]

So the magnitude of the normal force is greater than the magnitude of the weight of the person.

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The magnitude of the normal force on a person in a downward-slowing elevator is greater than the magnitude of the weight force.

When a person is standing in an elevator that is moving downward and slowing down, the magnitude of the normal force on the person is greater than the magnitude of the weight force on the person.

According to Newton's third law, the normal force and the weight force are equal in magnitude and opposite in direction. In this case, since the elevator is moving downward and slowing down, the acceleration of the person is in the upward direction. Therefore, the normal force exerted by the floor of the elevator on the person must be greater than the weight force in order to cause the upward acceleration.

It is important to note that if the elevator is in free-fall and accelerating downward at the acceleration due to gravity, the normal force becomes zero and the person appears to be weightless.

Answer

given,

height of the jump = 0.44 m

acceleration due to gravity, g = 9.8 m/s²

velocity at the height point = 0 m/s

initial speed = ?

Using equation of motion for speed calculation

v² = u² + 2 g h

0 = u² - 2 x 9.8 x 0.44

u = √8.624

u = 2.94 m/s

time taken to reach the highest point

v = u + g t

0 = 2.94 - 9.8 x t

t = 0.3 s

total time of flight will be equal to double of the time taken to reach the maximum height.

Total time = 2 x 0.3 = 0.6 s

A flea jumps straight up to a height of 0.440 m with an initial speed of 2.94 m/s and is in the air for 0.600 s.

A flea jumps straight up to a height (h) of 0.440 m. At the maximum height, the final speed (v) is zero. Given that the gravity (g) is 9.81 m/s², we can calculate the initial speed (u) using the following kinematic expression (We will assume y+ as the positive direction)

[tex]v^{2} = u^{2} + 2 \times g \times h \\\\(0m/s)^{2} = u^{2} + 2 \times (-9.81m/s^{2} ) \times 0.440m\\\\u = 2.94 m/s[/tex]

We can calculate the time (t) required to reach the maximum height using the following kinematic expression.

[tex]v = u + g \times t\\\\0m/s = 2.94m/s - 9.81m/s^{2} \times t\\\\t = 0.300 s[/tex]

The time in the air will be double the time to reach the maximum height.

[tex]t_{total} = 2 \times 0.300 s = 0.600 s[/tex]

A flea jumps straight up to a height of 0.440 m with an initial speed of 2.94 m/s and is in the air for 0.600 s.

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Answer:

[tex] d = \frac{v^2_i}{2a}= \frac{(20m/s)^2}{2* 3.43 m/s^2}=58.309m [/tex]

Explanation:

For this case  we can use the second law of Newton given by:

[tex] \sum F = ma[/tex]

The friction force on this case is defined as :

[tex] F_f = \mu_k N = \mu_k mg [/tex]

Where N represent the normal force, [tex]\mu_k [/tex] the kinetic friction coeffient and a the acceleration.

For this case we can assume that the only force is the friction force and we have:

[tex] F_f = ma[/tex]

Replacing the friction force we got:

[tex] \mu_k mg = ma[/tex]

We can cancel the mass and we have:

[tex] a = \mu_k g = 0.35 *9.8 \frac{m}{s^2}= 3.43 \frac{m}{s^2}[/tex]

And now we can use the following kinematic formula in order to find the distance travelled:

[tex] v^2_f = v^2_i - 2ad[/tex]

Assuming the final velocity is 0 we can find the distance like this:

[tex] d = \frac{v^2_i}{2a}= \frac{(20m/s)^2}{2* 3.43 m/s^2}=58.309m [/tex]

The hockey puck will travel approximately 58.5 meters across the floor before it stops.

To solve this problem, we will use the concepts of kinetic energy and the work-energy theorem.

Initially, the hockey puck has a kinetic energy of KE1 = 0.5*m*v^2 = 0.5*0.3 kg*(20 m/s)^2 =60 J

As it travels across the floor, the friction does work on it until it stops. The work done by the friction force is W = μ*m*g*d, where: μ is the coefficient of kinetic friction (0.35), m is the mass (0.3 kg), g is the gravitational constant (9.8 m/s^2), and d is the distance traveled.

The work done by the friction is equal to the initial kinetic energy (work-energy theorem), so we can solve for d: 60 J = 0.35 * 0.3 kg * 9.8 m/s^2 * d. Simplifying this equation gives d = 60 J / (0.35 * 0.3 kg * 9.8 m/s^2) = 58.5 m

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The film thickness needs to be increased by C. λfilm/2 to achieve destructive interference.

When light reflects off a thin film, it can undergo constructive or destructive interference depending on the thickness of the film and the wavelength of the light. For constructive interference, the path difference between the reflected rays should be an integer multiple of the wavelength, while for destructive interference, the path difference should be an odd multiple of half the wavelength.

Interference in thin films is a phenomenon where light waves reflect off both the top and bottom surfaces of a thin film, leading to constructive or destructive interference patterns, creating colors. Therefore, to achieve destructive interference, the film thickness needs to be increased by λfilm/2. Thus, the correct answer is option C.

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To switch from constructive to destructive interference, the film thickness must be increased by λ[tex]_f_i_l_m[/tex]/2.

In thin film interference, the thickness of the film and the wavelength of light are crucial in determining whether the interference is constructive or destructive. Constructive interference happens when the path length difference for the two reflected rays is an integral multiple of the wavelength inside the film (i.e., 2t = nλ[tex]_f_i_l_m[/tex], where n is an integer). Conversely, destructive interference occurs when this path length difference is a half-integral multiple of the wavelength (i.e., 2t = (2n + 1)λ[tex]_f_i_l_m[/tex] / 2).

Given that the film currently gives constructive interference for a wavelength of λ[tex]_f_i_l_m[/tex], the path length difference is an integral multiple of λ[tex]_f_i_l_m[/tex]. To switch to destructive interference, the thickness must be increased so the total path length difference becomes a half-integral multiple of λ[tex]_f_i_l_m[/tex]. This increase should be λ[tex]_f_i_l_m[/tex]/2.

The film thickness would need to be increased by λ[tex]_f_i_l_m[/tex]/2 to give destructive interference.

Answer:

The direction of the magnetic field is in the  z-direction

Explanation:

Applying right hand rule, which states that when the thumb, index finger and the middle finger are held mutually at right angle to each other, with the index finger pointing in the direction of the moving charge (y- direction), the thumb pointing in the direction of the magnetic force (x-direction) pushing on the moving charge and then the middle finger pointing in the direction of the magnetic field (z-direction).

z direction.

Using John Ambrose Fleming's right hand rule which asks to position the middle finger, the thumb and the index finger as follows;

                                                 | thumb

                                                 |

                                                 |

index                                       |

                                                /

                                            /

                                        /

                                    /

                               middle

With this arrangement shown above, if you point your index finger in the direction in which the charge is moving, and then the middle finger in the direction of the magnetic field, then your thumb will point in the direction of the magnetic force.

For example;

If a charge is moving in the x direction, and the magnetic force is moving in the y direction, then the above figure is rotated to make sure that the index finger is pointing in the direction of the moving charge so as to get the direction of the magnetic field as follows;

                                                               y|       thumb (magnetic force)

                                                                 |

(direction of charge)                                |

index                     x                               |

                                                                /

                                                            /

                                                        /

                                                    / z

                               middle (direction of field)

Therefore, the magnetic field is in the z direction.

Now to the question, if the charge(electron) is moving in the y direction at right angles to a magnetic field and it experiences a magnetic field in the x direction, the diagram above can be rotated to depict this situation as follows;

                                                               y|       index (direction of charge)

                                                                 |

                                                                 |

                                                                  |                                 x

                                                                /                  thumb (direction of force)      

                                                            /

                                                        /

                                                    / z

                               middle (direction of field)

Therefore the magnetic field will move in the z direction.

Note:

i. For every rotation of right-hand, the index finger must always point in the direction of the charge.

ii. Since the question does not specify which direction the electron is moving in (whether positive or negative), the direction of the magnetic field (whether positive or negative) might not be determined either. But in either case, the field will move in the z direction.

Answer:

Average speed of the car will be 27.582 km/hr

Explanation:

We have given that in first 5 minutes distance traveled by car is 2 miles

After 14 minutes it has travel 4.5 miles

And finally after 21 minutes distance traveled by car is 6 miles

So total time of traveling t = 21 minutes

As we know that 1 hour = 60 minutes

So 21 minutes [tex]=\frac{21}{60}=0.35hour[/tex]

Total distance traveled = 6 miles

As 1 miles = 1.609 km

So 6 miles [tex]=6\times 1.609=9.654km[/tex]

Average speed is equal to the ratio of total distance and total time

So average speed [tex]v=\frac{9.654}{0.35}=27.582km/hr[/tex]

Answer:

average speed = 0.146 m/s

Explanation:

given data

distance = 30600 km

time = January 26, 1977 to September 18, 1983

solution

we get here time that is January 26, 1977 to September 18, 1983

so it is = 6 year and 7 months and 22 days

and  that is = 2422 days = 2.09 × [tex]10^{8}[/tex] seconds

and distance is = 30600 km = 30600 × 10³ m

so here average speed will be as

average speed = [tex]\frac{distance}{time}[/tex]        ...........1

average speed = [tex]\frac{30600*10^3}{2.09*10^8}[/tex]

average speed = 0.146 m/s

Final answer:

Upon removing 12000 electrons from a neutral balloon, it will acquire a positive charge of 1.92 microcoulombs (1.92 µC). This is calculated by multiplying the number of electrons by the elemental charge and converting to the appropriate unit.

Explanation:

When 12000 electrons are removed from a neutral balloon, it obtains a positive charge because electrons carry a negative charge. To determine the charge the balloon now carries, we multiply the number of electrons removed by the elementary charge of an electron.

The charge (Q) on the balloon can be calculated using the formula:

Q = n × e

Where:

Let's plug in the values into the formula:

Q = 12000 × 1.6 × 10⁻¹⁹ C

Q = 1.92 × 10⁻¹⁵ C

Converting coulombs (C) to microcoulombs (µC) by multiplying by 10¶ gives:

Q = 1.92 × 10⁻¹⁵ C × 10¶ µC/C

Q = 1.92 µC

The balloon will carry a charge of 1.92 µC after 12000 electrons have been removed.

Answer:

v = 30.15 m/s

a = 60.07 m/s2

Explanation:

Velocity is derivative of position with respect to time

[tex]v_x = x' = 3*0.25t^2 = 0.75t^2[/tex]

[tex]v_z = z' = 5*0.75t^4/2 = 1.875t^4[/tex]

Acceleration is derivative of velocity with respect to time

[tex]a_x = v_x' = 2*0.75 t = 1.5t[/tex]

[tex]a_z = v_z' = 4*1.875t^3 = 7.5t^3[/tex]

At t = 2 seconds

[tex]v_x = 0.75*2^2 = 3m/s[/tex]

[tex]v_z = 1.875*2^4 = 30m/s[/tex]

[tex]v = \sqrt{v_x^2 + v_z^2} = \sqrt{3^2 + 30^2} = \sqrt{909} = 30.15 m/s[/tex]

[tex]a_x = 1.5*2 = 3 m/s^2[/tex]

[tex]a_z = 7.5*2^3 = 60 m/s^2[/tex]

[tex]a = \sqrt{a_x^2 + a_z^2} = \sqrt{3^2 + 60^2} = \sqrt{3609} = 60.07 m/s[/tex]

Answer:

[tex]Q = T1 - T2 (KA)/L[/tex]

Explanation:

Where Q is the co-efficient of heat transfer.

T-1 is initial temperature of exchanger

T-2 is final temperature of sink where has to to be dissipated

k is the co-efficient of thermal conductivity

A is the total area of exchanger surface...

and L is the total length of exchanger...

Answer:

Explanation:

Given

Apparent  frequency [tex]f'=1370\ Hz[/tex]

Velocity of sound [tex]v=343\ m/s[/tex]

speed of observer [tex]v_o=35\ m/s[/tex]

Using Doppler effect Apparent frequency when source is approaching is given by

[tex]f'=f(\frac{v-v_0}{v-v_s})[/tex]

[tex]1370=f(\frac{343-35}{343-v_s})---1[/tex]

Apparent frequency when source moves away from observer

[tex]f''=f(\frac{v+v_0}{v+v_s})[/tex]

[tex]1330=f(\frac{343+35}{343+v_s})---2[/tex]

Divide 1 and 2 we get

[tex]\frac{f'}{f''}=\frac{\frac{343-35}{343-v_s}}{\frac{343+35}{343+v_s}}[/tex]

[tex]\frac{1370}{1330}=\frac{343-35}{343-v_s}\times \frac{343+v_s}{343+35}[/tex]

[tex]v_s=40\ m/s[/tex]

Thus speed of sound of police car is 40 m/s

The problem involves the Doppler effect and can be solved by applying the relevant formulas for when the source of sound is approaching and when it is receding. By setting up a system of two equations with the speed of the police car as the unknown, one can solve for it algebraically.

This is a classic problem related to the Doppler effect, which describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave. It requires us to know some principles about sound waves, including that their speed in the medium is constant, and their frequency and wavelength are inversely proportional.

To solve the problem, we need to apply the formula of the Doppler effect both when the police car is approaching and when it is moving away. The formula for the perceived frequency (f) when the source is moving towards the observer in a medium, like air, is given by f = f0*(v + vo) / (v - vs), where f0 is the emitted frequency, v is the speed of the medium (sound in this case), vo is the observer's speed, and vs is the source's speed.

When the source (police car) is moving away, the formula differs slightly and is given by f = f0 * (v - vo) / (v + vs). Here, using the frequencies when the car was approaching (1370 Hz) and moving away (1330 Hz), along with your speed (35.0 m/s) and the speed of sound (343 m/s), we have two equations with vs (the speed of the police car) as the unknown. This system of equations can be solved algebraically, yielding the speed of the police car.

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Answer:

17 hours

Explanation:

k = Thermal conductivity of concrete = 1.1 W/m°C

A = Area = [tex]8.9\ m^2[/tex]

l = Thickness = 0.18 m

[tex]\Delta T[/tex] = Change in temperature = 20.8-10.1

Power is given by

[tex]P=\dfrac{kA\Delta T}{L}\\\Rightarrow P=\dfrac{1.1\times 8.9\times (20.8-10.1)}{0.18}\\\Rightarrow P=581.961\ W[/tex]

Time required to produce 1 kWh

[tex]t=\dfrac{3600\times 10^3}{581.961}\\\Rightarrow t=6185.98153485\ s[/tex]

For one dollar

[tex]t=\dfrac{6185.98153485}{0.1}\\\Rightarrow t=61859.8153485\ s\\\Rightarrow t=\dfrac{61859.8153485}{60\times 60}\\\Rightarrow t=17.1832820412\ hours[/tex]

The time taken is 17 hours

Answer:

No, the deviation in the path of asteroid is not by 22°

Explanation:

Given:

velocity of asteroid, [tex]v_a=21\ km.s^{-1}[/tex]

acceleration of the rocket, [tex]a=0.035\ km.s^{-2}[/tex]

time of acceleration, [tex]t=40\ s[/tex]

Now, the final velocity of the asteroid:

using the equation of motion,

[tex]v=u+a.t[/tex]

where:

[tex]v=[/tex] final velocity

[tex]u=[/tex] initial velocity in the direction

[tex]v=0+0.035\times 40[/tex]

[tex]v=1.4\ km.s^{-1}[/tex]

Now direction of the resultant velocity:

[tex]\tan\beta=\frac{v_a}{v}[/tex]

[tex]\tan\beta=\frac{21}{1.4}[/tex]

[tex]\beta=86.186^{\circ}[/tex]

So, the deviation in the asteroid:

[tex]\theta=90-\beta[/tex]

[tex]\theta=90-86.186[/tex]

[tex]\theta=3.814^{\circ}[/tex]

Answer:

0.015%

Explanation:

Data provided in the question:

Length by which the measuring tape can be off, δL = 0.42 cm

Total measured length for which Error of  δL is observed, L = 28 m  

Now,  

we know,  

1 m = 100 cm  

Thus,

28 m = 28 × 100 = 2800 cm

Percent uncertainty = [δL ÷ L] × 100%

= [0.42  ÷ 2800] × 100%

= 0.015%

Answer:

[tex]w=\sqrt{\frac{k}{m} }[/tex]

b. [tex]5.6rad/s[/tex]

Explanation:

a. from the spring-mass system which is explicitly describe by hooks law

from

F=-kx

which is in comparison to newtons general law of motion

F=ma

where the displacement x is expressed as

[tex]x=Asin(wt)\\[/tex]

and the acceleration is the second derivative of the displacement

[tex]a=-Aw^{2}sin(wt)\\[/tex]

hence final expression after substituting for the acceleration and the displacement  is expressed as

[tex]w=\sqrt{\frac{k}{m} }[/tex]

b. for k=105N/m and m=3.4kg

we have the angular frequency to be

[tex]w=\sqrt{\frac{105}{3.4}}\\\\w=5.6rad/s[/tex]

Final answer:

The angular frequency (ω) can be found using the formula ω = sqrt(k/m). By plugging in the given values of mass (3.4 kg) and spring constant (105 N/m), the angular frequency is calculated to be approximately 5.56 rad/s.

Explanation:

The subject question pertains to simple harmonic motion (SHM) and specifically relates to the oscillatory motion of a mass attached to a horizontal spring. The question involves determining the angular frequency and calculating it in radians per second.

Part (a)

To write an equation for the angular frequency (ω) of the oscillation, we use the equation:

ω = sqrt(k/m)

where ω is the angular frequency, k is the spring constant, and m is the mass.

Part (b)

Plugging in the given values:

m = 3.4 kg

k = 105 N/m

We find ω using the formula:

ω = sqrt(105 N/m / 3.4 kg) = sqrt(30.8824 s-2) ≈ 5.56 rad/s

Therefore, the angular frequency of the oscillation is approximately 5.56 rad/seconds.

Answer:

1/3

Explanation:

Take the displacement and divide it by the speed of light. Meters will cancel leaving you with seconds. :)

110m / 343m/s = 0.32069971 seconds or 1/3 seconds

The time elapses between when the sound reaches the nearest dancer and when it reaches the farthest dancer  1/3 second ( approx.). So, option (d) is correct.

The rate at which a body's displacement changes in relation to time is known as its velocity. Velocity is a vector quantity with both magnitude and direction. SI unit of velocity is meter/second.

Given The first dancer in the line is 10 m from the speaker playing the music; the last dancer in the line is 120 m from the speaker.

So, distance between the  first dancer in the line and the last dancer in the line be = 120 m -10 m = 110 m

Velocity of sound in air be = 343 m/s.

Hence, The time elapses between when the sound reaches the nearest dancer and when it reaches the farthest dancer = distance/velocity = 110/343 second = 1/3 second ( approx.).

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Answer: Meteor showers occur when the earth in its orbit around the Sun passes through debris left over from the destruction of comets.

Explanation:A meteor is a particle broken off an asteroid or comet orbiting the Sun, it burns up as it enters the Earth's atmosphere, creating the effect called shooting star. Cosmic debris of meteor is known as meteoroids. These meteoroids, entering Earth's atmosphere, at extremely high speeds on parallel trajectories is an event known as meteor shower.

Answer:

Explanation:

given

spring constant k = 880 N/m

mass m = 150 kg

Normal force will be equal to the component of weight of mass m which is perpendicular to the inclined surface

= mgcosθ

So normal force

FN = mgcosθ j , as it acts in out of plane direction .

b )

Fricrion force acting in upward direction = μs mgcosθ

component of weight acting in downward direction = mgsinθ

restoring force by spring on block in downward direction

= kd

= 880d

F( total ) = (μs mgcosθ -  mgsinθ -  880d )i

c )  

for balance

mgsinθ =  kd

sinθ = kd / mg

= 880 x .1 / 150 x 9.8

= 88 / 1470

.0598

θ = 3.4 degree

d )

d = mgsinθ / k

150 x 9.8 sin45 / 880

= 1.18 m

The problem involves the equations of force related to a block on an inclined plane with a spring. The solution involves using concepts of statics, the restoring force of a spring, the force of gravity on an inclined plane and trigonometric functions to derive the formulas and find the answers.

Part (a) The normal force, FN, is the product of the gravitational motion and the cosine of the angle. Therefore, FN=mgcosθ.

Part (b) Just before the block begins to slide up the plane, the sum of the forces in the x-direction is equal to the difference between the restoring force of the spring and the force due to gravity along the plane. Therefore, ΣFx = k*d - mgsinθ.

Part (c) For a frictionless plane, the angle of the plane is found by observing that the force due to gravity must be equal to the restoring force of the spring, which gives θ = arcsin(k*d/(mg)). To get the angle in degrees for d = 0.1 m, you would plug in the values: θ = arcsin(880*0.1/(150*9.81)) in radians, and convert to degrees.

Part (d) If θ = 45 degrees and the surface is frictionless, the distance the spring will be compressed, d, can be found from the equation mg*sinθ = k*d, which gives d = m*g*sinθ/k. Substituting in these values, d = 150*9.81*sin(45)/880.

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Answer:

option C

Explanation:

Given,

Change on the capacitor = Q

Separation is doubled

Energy stored in the capacitor,E = ?

we know,

[tex]E = \dfrac{Q^2}{2C}[/tex]

and [tex]C = \dfrac{\epsilon_0A}{d}[/tex]

now,

[tex]E = \dfrac{Q^2}{2\epsilon A}\ d[/tex].......(1)

where d is the separation between the two plates.

now, when the separation is doubled

[tex]E' = \dfrac{Q^2}{2\epsilon A}\ (2d)[/tex]

[tex]E' = 2(\dfrac{Q^2}{2\epsilon_0 A}\ d)[/tex]

From equation (1)

  E' = 2 E

Hence, the energy stored in the capacitor is doubled if the separation is increased.

The correct answer is option C.

Answer:

Final velocity will be equal to 0.321 m/sec

Explanation:

We have given mass of clay model of lion [tex]m_1=0.225kg[/tex]

Its speed is 0.85 m/sec, so [tex]v_1=0.85m/sec[/tex]

Mass of another clay model [tex]m_2=0.37kg[/tex]

It is given that second clay is motionless

So its velocity [tex]v_2=0m/sec[/tex]

Now according to conservation of momentum

Momentum before collision will be equal to momentum after collision

So [tex]m_1v_!+m_2v_2=(m_1+m_2)v[/tex], here v is velocity after collision

So [tex]0.225\times 0.85+0.37\times 0+(0.225+0.37)v[/tex]

[tex]0.595v=0.191[/tex]

v = 0.321 m/sec

So final velocity will be equal to 0.321 m/sec  

Answer:

See explanation below.

Explanation:

For this case we have the following function:

[tex] x(t)= ct^4 +dt^2 +f[/tex]

Where [tex] c= 6 m/s^4 , d = 8m/s^2 , f=-6m [/tex]

If we replace those values we got:

[tex] x(t) = 6t^4 + 8t^2 -6[/tex]

If we want to find the position after t = 2.358 seconds for example we ust need to replace in the position function t = 2.358 and we got:

[tex] x(t=2.358) = 6(2.358)^4 + 8(2.358)^2 -6 \approx 224 m[/tex]

If we want to find the velocity we need to take the derivate of the position function and we got:

[tex] \frac{dx}{dt}=v(t) = 4ct^3 + 2dt[/tex]

[tex] v(t)= 24 t^3 + 16 t[/tex]

If we want to find the instantaneous velocity we just need to replace on v(t) a value for t

And the accelaration would be given by the second derivate of the position:

[tex] \frac{dv}{dt}= 72 t^2[/tex]

If we want to find the instantaneous acceleration we just need to replace on v(t) a value for t

Final answer:

Explaining the position function of a particle in meters at a given time using a specific function with provided values for constants.

Explanation:

Position Function: The position of a particle in meters at time 't' is described by the function x(t) = ct² + dt² + f, with given values for c, d, and f.

Explanation: To find the position of the particle using this function, substitute the values of c, d, f, and the specific time 't'. This will give you the position of the particle at that time.

Example: If c = 6m/s², d = 8m/s², f = -6m, and t = 2s, then you can plug in these values to find the position of the particle at t = 2s.

Using the principle of buoyancy and the ideal gas law equation, we can calculate the temperature needed inside a hot air balloon to provide the required lift. The maximum altitude of the balloon is limited by air temperature, pressure, and other environmental factors.

The principle of buoyancy used by hot air balloons to create lift is based on Archimedes' Principle, which states that the buoyant or lifting force exerted on a body immersed in a fluid equals the weight of the fluid the body displaces. In order for a hot air balloon to gain lift, the air inside the balloon must be heated to a temperature that makes it less dense than the surrounding cooler air. This difference in air density is what actually provides the uplift.

To calculate the required temperature inside the balloon, we can use the ideal gas law equation P₁V₁ / P₂V₂. Given the values in the question and the atmospheric pressure, we can derive the temperature inside the balloon required to provide the needed lift.

The maximum altitude attainable by a hot air balloon is limited by the ambient air temperature and pressure, which decrease as altitude increases. If the balloon's lift decreases (because it can no longer heat the air inside it sufficiently to provide lift), the balloon will stop ascending. Wind currents and weather conditions can also affect the maximum altitude.

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a) The average true power is 318.3 W

b) The reactive power is 132.6 W

c) The apparent power is 344.8 W

d) The power factor is 0.92

Explanation:

a)

For a circuit made of a resistor and a capacitor, the average (true) power is given by the resistive part of the circuit only.

Therefore, the average true power is given by:

[tex]P=I^2R[/tex]

where

I is the current

R is the resistance

In this problem, we have

V = 67 V (rms voltage)

[tex]R=12 \Omega[/tex] is the resistance of the load

[tex]X=5\Omega[/tex] is the reactance of the circuit

First we have to find the impedance of the circuit:

[tex]Z=\sqrt{R^2+X^2}=\sqrt{12^2+5^2}=13 \Omega[/tex]

Then we can find the current in the circuit by using Ohm's law:

[tex]I=\frac{V}{Z}=\frac{67}{13}=5.15 A[/tex]

Therefore, the average true power is

[tex]P=I^2R=(5.15)^2(12)=318.3 W[/tex]

b)

The reactive power of a circuit consisting of a resistor and a capacitor is the power given by the capacitive part of the circuit.

Therefore, it is given by

[tex]Q=I^2X[/tex]

where

I is the current

X is the reactance of the circuit

In this circuit, we have

[tex]I=5.15 A[/tex] (current)

[tex]X=5 \Omega[/tex] (reactance)

Therefore, the reactive power is

[tex]Q=(5.15)^2(5)=132.6W[/tex]

c)

In a circuit with a resistor and a capacitor, the apparent power is given by both the resistive and capacitive part of the circuit.

Therefore, it is given by

[tex]S=I^2Z[/tex]

where

I is the current

Z is the impedance of the circuit

Here we have

I = 5.15 A (current)

[tex]Z=13 \Omega[/tex] (impedance)

Therefore, the apparent power is

[tex]S=I^2 Z=(5.15)^2(13)=344.8 W[/tex]

d)

For a circuit with a resistor and a capacitor, the power factor is the ratio between the true power and the apparent power. Mathematically:

[tex]PF=\frac{P}{S}[/tex]

where

P is the true power

S is the apparent power

For this circuit, we have

P = 318.3 W (true power)

S = 344.8 W (apparent power)

So, the power factor is

[tex]PF=\frac{318.3}{344.8}=0.92[/tex]

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Answer:

a) E = 3.99 × 10⁵J/mol = 3.99 × 10² KJ/mol = 399KJ = 400KJ/mol

b) E = 3.99 × 10⁴ J/mol = 3.99 × 10¹ KJ/mol = 39.9KJ/mol = 40 KJ/mol

c) E = (3.99 × 10^-10) J/mol = 3.99 × 10⁻⁷ KJ/mol

Explanation:

E = hf where E = energy, H = Planck's constant = 6.62607004 × 10⁻³⁴ J.s and f = 1/time period

a) Period = 1 fs = 1 × 10⁻¹⁵ s

f = 1/(10⁻¹⁵ ) = 10¹⁵ Hz

E = hf = 6.62607004 × 10⁻³⁴ × 10¹⁵ = 6.63 × ⁻10¹⁹ J = 6.63 × 10⁻¹⁶ KJ

In energy per mol, we multiply the energy with the avogadro's constant, 6.022 × 10²³ atoms/mol

E = 3.99 × 10⁵J/mol = 3.99 × 10² KJ/mol = 399KJ = 400KJ/mol

b) Period = 10 fs = 10 × 10⁻¹⁵ = 10⁻¹⁴ s

f = 1/period = 1/10⁻¹⁴ = 10¹⁴ Hz

E = hf = 6.62607004 × 10⁻³⁴ × 10¹⁴ = 6.63 × 10^-20 J = 6.63 × 10⁻¹⁷ KJ

In energy per mol, we multiply the energy with the avogadro's constant, 6.022 × 10²³ atoms/mol

E = 3.99 × 10⁴ J/mol = 3.99 × 10¹ KJ/mol = 39.9KJ/mol = 40 KJ/mol

c) Period = 1s

f = 1/period = 1.0 Hz

E = 6.62607004 × 10⁻³⁴ × 1 = 6.63 × 10⁻³⁴ J = 6.63 × 10⁻³¹ KJ.

In energy per mol, we multiply the energy with the avogadro's constant, 6.022 × 10²³ atoms/mol

E = (3.99 × 10^-10) J/mol = 3.99 × 10⁻⁷ KJ/mol

Answer:

(i) E =  6.626 X 10⁻¹⁹ J    = 400 KJ/mol

(ii) E = 6.626 X 10⁻²⁰ J   = 40 KJ/mol

(iii) E = 6.626 X 10⁻³⁴ J   = 4 X 10⁻¹³ KJ/mol

Explanation:

Energy associated with excitation of a quantum is given as;

E = hf

where;

E is the energy of excitation

h is Planck's constant = 6.626 X 10⁻³⁴Js⁻¹

f is the threshold frequency in s⁻¹

Thus, E = h/t

Part (i)

E = (6.626 X 10⁻³⁴)/(1 X 10⁻¹⁵)

E = 6.626 X 10⁻¹⁹ J  

In (KJ/mol) = 6.626 X 10⁻²² KJ X 6.022 X10²³ = 400 KJ/mol

Part (ii)

E = (6.626 X 10⁻³⁴)/(1 X 10⁻¹⁴)

E = 6.626 X 10⁻²⁰ J  

In (KJ/mol) = 6.626 X 10⁻²³ KJ X 6.022 X10²³ = 40 KJ/mol

Part (iii)

E = (6.626 X 10⁻³⁴)/(1)

E = 6.626 X 10⁻³⁴ J  

In (KJ/mol) = 6.626 X 10⁻³⁷ KJ X 6.022 X10²³ = 4 X 10⁻¹³ KJ/mol

Answer:

Explanation:

Reaction force of inclined surface = mg cosθ

= Friction force acting in upward direction = μ x mg cosθ

If F be force required in upward direction to keep the block at rest on the plane

F +  μ x mg cosθ = mg sinθ

F = mg sinθ -  μ x mg cosθ

F = mg( sinθ - μ cosθ)

= 2 x 9.8 ( sin34 - 0.2 cos34 )

= 19.6 ( .559 - .1658)

= 7.7 N

This is the minimum force required .

The additional force that needs to be applied to keep a 2 kg block from sliding down a 34 degree incline, given a coefficient of static friction of 0.2, is 7.89 N.

To find out how much additional force, F, needs to be applied to keep the 2 kg block from sliding down the 34° incline, we first need to calculate the gravitational force pulling the block down the incline. This force is given by F_gravity = m * g * sin(theta), where m = 2 kg, g = 9.8 m/s² (roughly the acceleration due to gravity on Earth), and theta = 34 degrees. Thus, F_gravity = 2 kg * 9.8 m/s² * sin(34°) = 11.10 N.

Next, we calculate the static frictional force that is preventing the block from sliding down the incline. This force is given by F_friction = m * g * cos(theta) * mu_s, where mu_s = 0.2 is the coefficient of static friction. Thus, F_friction = 2 kg * 9.8 m/s² * cos(34°) * 0.2 = 3.21 N.

The additional force, F, that must be applied to keep the block from sliding down the incline is the difference between the gravitational force and the frictional force. That's F = F_gravity - F_friction = 11.10 N - 3.21 N = 7.89 N.

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Answer:

Q1. corect is B, Q2. it is A, Q3.  E and Q4. A  

Explanation:

Q1 For this exercise we can use Newton's second law where acceleration is centripetal.

              F = m a

              a = v² / r.

              G m M / r² = m v² / r

               G M / r = v²

The velocity has a constant magnitude whereby we can divide the length of the circular orbit (2π r) between the period

               G M / r = (2π r / T)²

                r³ = G M T2 / 4π²

Let's calculate

              T = 103 day (24 h / 1 day) (3600 s / 1h) = 8,899 10⁶ s

              r³ = 6.67 10⁻¹¹  1.99 10³⁰ (8,899 10⁶) 2 / 4π²

              r = ∛ (266.25 10³⁰)

              r = 6.4 10¹⁰  m

The distance matches the value in part B

Q2 Astronauts have measured the acceleration of gravity, so we can use the second law with a body on the planet's surface

            F = m g

            G m M_p / R_p² = m g

             G M_p / R_p² = g

              M_p = g R_p² / G

They indicate that the radius of the planet is half the radius of the Earth

              R_p = ½ R_earth

              R_p = ½ 6.37 10⁶

              R_p = 3.185 10⁶ m

Let's calculate

             M_p = 8.2 (3,185 10⁶)² / 6.67 10⁻¹¹

             M_p = 1.25 10²⁴ kg

The correct answer is A

Q3 We use Newton's second law again, with part Q1, where M is the mass of the planet and m is the mass of the moon

                r³ = G M T² / 4π²

               T = 63 days (24h / 1day) (3600s / 1h) = 5.443 10⁶ s

               r³ = 6.67 10⁻¹¹ 1.25 10²⁴ (5.443 10⁶)² / 4π²

               r = ∛ (62.56807 10²⁴)

               r = 3.97 10⁸ m

The correct answer is E

Q4 To calculate this part let's use the conservation of mechanical energy,

Starting point The surface of the moon

          Em₀ = K + U = ½ m v2 - G m M / r

Final point. Infinity with zero speed

           [tex]Em_{f}[/tex] = 0

              Em₀ = Em_{f}

              ½ m v² - G m M / R = 0

              v² = 2 G M / r

              M = v2 r / 2G

              r = 2 G M / v²

Since we don't know the radius of the moon, we will also use the equation in part 2

              M = g r² / G

               r = √ GM / g

Let's replace

              2G M / v² = √ G M / g

              4 G M / v⁴ = 1 / g

              M = v⁴ / (g 4G)

              M = 3000⁴ / (2.7  4 6.67 10-11)

              M = 1.12 10²³ kg

corract is  A

Answer:

a) The positive plate is at a higher potential than the negative plate.

b) Electric field between the plates = 5.76 KV/m

Explanation:

a) The positive plate is at a higher potential than the negative plate because of convention. We could define the negative plate to be "high potential." However, that convention has to be consistent. The positive plate is at the high potential, and that potential causes positive charges to flow from high potential to low potential. If we had defined the convention in the other direction, then the negative terminal would be the high potential, and that potential would cause negative charges to flow from high potential to low potential.

b) Electric field, E = potential difference across plate/distance or separation between the plates

E = V/d

V = 490V, d = 8.5cm = 0.085m

E = 490/0.085 = 5,764.706 V/m = 5.76 KV/m

Hope this helps!!!

Answer:

(b) 9m/s south

Explanation:

Case 1: A motorcycle has a velocity of 15 m/s, due south

Case 2: A car with a velocity of 24 m/s, due north.

Let the velocity of the car due south = Vs↓

Let the velocity of the car due north = Vn↑

Magnitude of the velocity of the motorcycle as seen by the driver of the car = V

V = Vn - Vs

  = 24m/s - 15m/s = 9m/s↓

The magnitude and velocity of of the motorcycle as seen by the driver of the car = 9m/s south

The correct option is B

e. 39 m/s, south

Let the velocity of the motorcycle be [tex]V_{M}[/tex]

Let the velocity of the car be [tex]V_{C}[/tex]

Let the velocity of the motorcycle relative to the car be = [tex]V_{MC}[/tex]

According to relativity of velocities in one dimension;

[tex]V_{MC}[/tex] = [tex]V_{M}[/tex] - [tex]V_{C}[/tex]       --------------------------(i)

Now, take;

south to be negative (-ve)                  

north to be positive (+ve)

Therefore, we can say that;

[tex]V_{M}[/tex] = -15m/s        [since the velocity is due south]

[tex]V_{C}[/tex] = +24m/s       [since the velocity is due north]

Now, substitute the values of [tex]V_{M}[/tex] and [tex]V_{C}[/tex] into equation (i) as follows;

[tex]V_{MC}[/tex] = -15 - (+24)

[tex]V_{MC}[/tex] = -15 -24

[tex]V_{MC}[/tex] = -39 m/s

Remember that we have taken;

south to be negative (-ve)                  

north to be positive (+ve)

Since the result we got is negative, it means the speed is due south.

Therefore, the speed of the motorcycle as seen by the driver of the car is 3.9m/s, due south.

Explanation:

It is known that the molecular weight of ammonia ([tex]NH_{3}[/tex]) is as follows.

   Molecular weight ([tex]NH_{3}[/tex]) = [tex]14 + 3 \times 1[/tex] = 17

(a)   Therefore, we will calculate the mass as follows.

     [tex]0.5 kmol \times (\frac{1000 mol}{1 kmol}) \times (\frac{17 g}{1 mol})[/tex]

                       = 8500 g

Now, formula to calculate weight of the system in N is as follows.

            Weight = mass × g

             = [tex]8500 g \times (\frac{1 kg}{1000 g}) \times (9.8 m/s^{2})[/tex]

             = 83.3 N     (1 [tex]kg m/s^{2}[/tex] = 1 N)

Hence, the weight of the system is 83.3 N.

(b)   Relation between specific volume and number of moles is as follows.

            [tex]v (m^{3}/kmol) = \frac{V}{n}[/tex]

Therefore, calculate the specific volume as follows.

       [tex]V_m = \frac{6 m^{3}}{0.5 k mol}[/tex]

                   = 12 [tex]m^{3}/k mol[/tex]

Also,  

               [tex]v (m^{3}/kmol) = \frac{V}{m}[/tex]  

            v = [tex]\frac{6 m^{3}}{8.5 kg}[/tex]

               = 0.705882 [tex]m^{3}/kg[/tex]

Therefore, we can conclude that the value of specific volume is 12 [tex]m^{3}/k mol[/tex]  and 0.705882 [tex]m^{3}/kg[/tex].

Answer:

a) [tex]w=83.385\ N[/tex]

b) [tex]\bar V=12\ m^3.kmol^{-1}[/tex]

[tex]\b V=0.7059\ m^3.kg^{-1}[/tex]

Explanation:

Given:

no. of moles of ammonia in a closed system, [tex]n=0.5\ kmol=500\ mol[/tex]

volume of ammonia, [tex]V=6\ m^3[/tex]

We  know the molecular formula of ammonia: [tex]NH_3[/tex]

The molecular mass of ammonia:

[tex]M=14+3\times 1=18\ g.mol^{-1}[/tex]

Now the mass of given ammonia:

[tex]m=n.M[/tex]

[tex]m=500\times 17[/tex]

[tex]m=8500\ g=8.5\ kg[/tex]

a)

Now weight:

[tex]w=m.g[/tex]

[tex]w=8.5\times 9.81[/tex]

[tex]w=83.385\ N[/tex]

b)

Specific volume:

[tex]\bar V=\frac{6}{0.5}[/tex]

[tex]\bar V=12\ m^3.kmol^{-1}[/tex]

also

[tex]\b V=\frac{V}{m}[/tex]

[tex]\b V=\frac{6}{8.5}[/tex]

[tex]\b V=0.7059\ m^3.kg^{-1}[/tex]