A 1.75 µF capacitor and a 6.00 µF capacitor are connected in series across a 3.00 V battery. How much charge (in µC) is stored on each capacitor?

The speed equals half of its maximum speed at positions ±1.50 cm from the equilibrium for an amplitude of 3.00 cm.

Speed equals half of its maximum speed when the particle is at a position where it is halfway through its amplitude.

This occurs at x = ±0.5X from the equilibrium position, considering X as the amplitude of the motion.

For an amplitude of 3.00 cm, the speed equals half of its maximum speed at position ±1.50 cm.

Answer:

v_th = 3.1 * 10^5 m/s

Explanation:

Given:

- mass of 238-Uranium m_u = 3.952 *10^-25 kg

- mass of alpha particle m_a = 6.64 * 10^-27 kg

- mass of thorium particle m_th = 3.885*10^-25 kg

- velocity of 238-Uranium v_u = 5.0 *10^5 m/s

- velocity of alpha particle v_a = 1.4 *10^7 m/s

Find:

- The recoil velocity of the thorium particle.

Solution:

- To solve this problem we will use conservation of momentum in both x and y direction.

- Momentum conservation in x-direction:

                                        P_i = P_f

                  m_u*v_u = m_a*v_a*cos(Q) + m_th*v_th,x

where v_th,x is the x component of thorium velocity:

                 P_i = 3.952 *10^-25 * 5.0 *10^5 = 1.976*10^-19

      P_f = 6.64 * 10^-27*1.4 *10^7*cos(25.4) + 3.885*10^-25*v_th,x

                 P_f = 8.3974*10^-20 + 3.885*10^-25*v_th,x

Hence,

             1.976*10^-19 = 9.296*10^-20 + 3.885*10^-25*v_th,x

                         v_th,x = 2.92473 * 10^5 m/s        

- Momentum conservation in y-direction:

                                        P_i = P_f

                         0 = m_a*v_a*sin(Q) + m_th*v_th,y

where v_th,x is the x component of thorium velocity:

                         v_th,y = m_a*v_a*sin(Q) / m_th    

        v_th,y = 6.64 * 10^-27*1.4 *10^7*sin(25.4) / 3.885*10^-25

Hence,

                            v_th,y = 1.02635 * 10^5 m/s        

- The magnitude of recoil velocity:

                        v_th = sqrt ( v_th,x ^2 + v_th,y ^2 )

            v_th = sqrt ( (2.92473 * 10^5)^2 + (1.02635 * 10^5)^2 )

                                  v_th = 3.1 * 10^5 m/s

Answer:

[tex]A'=2A[/tex]

Explanation:

According to the law of conservation of energy, the total energy of the system can be expresed as the sum of the potential energy and kinetic energy:

[tex]E=U+K=\frac{kA^2}{2}\\E=\frac{kx^2}{2}+\frac{mv^2}{2}=\frac{kA^2}{2}[/tex]

When the spring is in its equilibrium position, that is [tex]x=0[/tex], the object speed its maximum. So, we have:

[tex]\frac{k(0)^2}{2}+\frac{mv_{max}^2}{2}=\frac{kA^2}{2}\\A^2=\frac{mv_{max}^2}{k}\\A=\sqrt{\frac{mv_{max}^2}{k}}[/tex]

In order to double its maximum speed, that is [tex]v'{max}=2v_{max}[/tex]. We have:

[tex]A'=\sqrt{\frac{m(v'_{max})^2}{k}}\\A'=\sqrt{\frac{m(2v_{max})^2}{k}}\\A'=\sqrt{\frac{4mv_{max}^2}{k}}\\A'=2\sqrt{\frac{mv_{max}^2}{k}}\\A'=2A[/tex]

The factor we must increase the amplitude of the vibration to double its maximum speed is 2.

Apply the principle of conservation of energy to determine the amplitude;

[tex]U = K.E\\\\\frac{1}{2} KA^2 = \frac{1}{2}mv^2\\\\KA^2 = mv^2\\\\A^2 = \frac{m}{K} v^2\\\\\frac{A_1^2}{v_1^2 } = \frac{A_2^2}{v_2^2 } \\\\A_2 = \frac{v_2 A_1}{v_1} \\\\A_2 = \frac{2v_1 \times A_1}{v_1} \\\\A_2 =2A_1[/tex]

Thus, the factor we must increase the amplitude of the vibration to double its maximum speed is 2.

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Final answer:

A 26-N force must act down and parallel to the incline for the bin to move down at constant velocity since this force would balance out the 26-N frictional force acting up the incline.

Explanation:

The subject of this question is Physics, specifically the concepts related to mechanics and forces on inclines with friction. The student is in High School level, most likely studying the fundamentals of Newtonian mechanics.

Given that the bin is already being pushed up the incline with a 26-N force and moves at a constant speed, this means the net force on the bin is zero, therefore the force of friction must also be 26 N but acting downwards along the incline. When the bin moves down at a constant velocity, the force of friction still acts up the incline (opposite to the bin's movement) and therefore, to maintain a constant velocity, the force applied must be equal in magnitude to the frictional force but directed down the incline. Hence, a 26-N force must be applied down and parallel to the incline for the bin to move down the incline at constant velocity.

Answer

given,

Volume of Depression, V = 75000 m³

borrow soil

Density of soil,γ = 1540 Kg/m³

water content,w = 8 % = 0.08

Specific gravity of solid,G = 2.66

Final in-place

dry density = 1790 kg/m³

water content = 13 % = 0.13

a) Volume of borrow soil require to fill the depression

   Mass of solid solid,m= dry density of inplace soil x Volume of depression

                              = 1790 x 75000

                           m = 1.3425 x 10⁸ Kg

dry density of the borrow pit

 [tex]\gamma_d = \dfrac{\gamma}{1 + w}[/tex]

 [tex]\gamma_d = \dfrac{1540}{1 + 0.08}[/tex]

 [tex]\gamma_d = 1425.93\ kg/m^3[/tex]

Volume of borrow soil required

    [tex]V = \dfrac{m}{\gamma_d}[/tex]

    [tex]V = \dfrac{1.3425\times 10^8}{1425.93}[/tex]

           V = 94149 m³

b) Water required

  [tex]W = 1790\times 75000\times 0.13 - 1425.93\times 94149\times 0.08[/tex]

 W = 6.71 x 10⁶ Kg

Water required to achieve 13% moisture is equal to W = 6.71 x 10⁶ Kg

Final answer:

To fill a 75,000 m³ depression, 93,590.46 m³ of borrow soil is required when accounting for the dry density change. To achieve a final water content of 13%, an additional 17,452,500 kg of water mass is needed.

Explanation:

To answer this question, we must first understand the relationship between the initial and final states of the borrow soil in terms of their volumes, densities, and moisture contents. The question is divided into two parts: calculating the volume of borrow soil needed and determining the water mass addition required to achieve the desired moisture content.

a) How many m³ of borrow soil are needed to fill the depression?

Since we want to fill a depression with a volume of 75,000 m³, we need to consider the change in density from the borrow state to the in-place dry density. The in-place dry density is given as 1790 kg/m³. The dry density of the borrow soil can be calculated by subtracting the mass of water from the total mass.

First, we convert the water content to a decimal by dividing by 100:
Water content = 8% / 100 = 0.08

Using the formula for dry density (
dry density = total density / (1 + water content)), we find:
Dry density of borrow soil = 1540 kg/m³ / (1 + 0.08) = 1425.93 kg/m³

The volume of the borrow soil needed can be calculated by the volume of the depression divided by the ratio of the final in-place dry density to the dry density of the borrow soil:
Volume of borrow soil needed = 75,000 m³ x (1790 kg/m³ / 1425.93 kg/m³) = 93,590.46 m³

b) Assuming no evaporation loss, what water mass is needed to achieve 13% moisture?

For the final water content of 13%, we'll use the in-place dry density to determine the total mass, then calculate the water mass needed:
Total in-place mass = volume x in-place dry density
Total in-place mass of fill = 75,000 m³ x 1790 kg/m³= 134,250,000 kg

Convert 13% to decimal:
Water content = 13% / 100 = 0.13

We want 13% of the total mass to be water, so the water mass required is:
Water mass = total mass x water content
Water mass needed = 134,250,000 kg x 0.13 = 17,452,500 kg

This is the additional water mass required to achieve the desired moisture content of 13%.

Answer:

Density change, Δρ = 2.4 × 10⁻⁴ kg/m³

Temperature Change, ΔT = 0.0258 K

Velocity Change, Δc = 0.0148 m/s

Explanation:

For sound waves moving through the air,

Pressure and Temperature varies thus

(P₀/P) = (T₀/T)^(k/(k-1))

Where P₀ = initial pressure of air = 101KPa = 101000 Pa

P = final pressure of air due to the change brought about by the moving sound wave = 101000+30 = 101030 Pa

T₀ = initial temperature of air = 30°C = 303.15 K

T = final temperature of air = ?

k = ratio of specific heats = Cp/Cv = 1.4

(101000/101030) = (303.15/T)^(1.4/(1.4-1))

0.9990703 =(303.15/T)^(3.5)

Solving This,

T = 303.1758 K

ΔT = T - T₀ = 303.1758 - 303.15 = 0.0258 K

Density can be calculate in two ways,

First method

Δρ = ρ - ρ₀

P₀ = ρ₀RT₀

ρ₀ = P₀/RT₀

R = gas constant for air = 287 J/kg.k

where all of these are values for air before the wave propagates

P₀ = 101000 Pa, R = 287 J/kg.K, T₀ = 303.15K

ρ₀ = 101000/(287 × 303.15) = 1.1608655 kg/m³

ρ = P/RT

P = 101030 Pa, T = 303.1758K

ρ = 101030/(287×303.1758) = 1.1611115 kg/m³

Δρ = ρ - ρ₀ = 1.1611115 - 1.1608655 = 0.00024 kg/m³ = 2.4 × 10⁻⁴ kg/m³

Second method

(ρ₀/ρ) = (T₀/T)^(1/(k-1))

Where ρ₀ is initially calculated from ρ₀ = P₀/RT₀, then ρ is then computed and the diff taken.

Velocity Change

c₀ = √(kRT₀) = √(1.4 × 287 × 303.15) = 349.00669 m/s

c = √(kRT) = √(1.4 × 287 × 303.1758) = 349.0215415 m/s

Δc = c₀ - c = 349.0215415 - 349.00669 = 0.0148 m/s

QED!

In this exercise we have to use the pressure knowledge to calculate the velocity, temperature and density so we have:

The variation of temperature and pressure is given by the formula of:

[tex](P_0/P) = (T_0/T)^{(k/(k-1))}[/tex]

From the formula given above we can identify:

Solving the formula  for temperature we find:

[tex](101000/101030) = (303.15/T)^{(1.4/(1.4-1))}\\0.9990703 =(303.15/T)^{(3.5)}\\T = 303.1758 K\\\Delta T = T - T_0 = 303.1758 - 303.15 = 0.0258 K[/tex]

They are using the formula for density is:

[tex]\rho_0 = 101000/(287 * 303.15) = 1.1608655 kg/m^3\\\rho = P/RT\\\rho = 101030/(287*303.1758) = 1.1611115 kg/m^3\\\Delta \rho = 2.4 * 10^{-4} kg/m^3[/tex]

Calculating the speed we find that:

[tex]c_0 = \sqrt{kRT_0} = 349.00669 m/s\\c = \sqrt{kRT} = 349.0215415 m/s\\\Delta c = 0.0148 m/s[/tex]

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Answer:

R = 4.18 * 10^8ohms

Explanation:

R=resistance in ohms=?

ρ=resistivity of material in ohms meters = 1.68*10^-8 oh ohm meters

L= length of the object (m) = 4.4m

A = cross-sectional area of the object in square meters (m^2)= πr^2

r = (1.5mm/1000)/2= 0.00075m

A=π*(0.00075)^2 = 1.76714586764426*10^-6

Approximately, A = 1.767m^2

R = ρL/A= (1.68*10^8Ω⋅m) * (4.4m)/(1.767m^2)

R = 418336162.988115 ohms

Approximately, R = 4.18 * 10^8ohms.

The resistivity of copper will be "4.18 × 10⁸ Ω (ohms)".

According to the question,

Resistivity of ohms, ρ = 1.68 × 10⁻⁸ Ω.m

Object's length, L = 4.4 m

Radius, r = [tex]\frac{\frac{1.5}{1000} }{2}[/tex]

               = 0.00075 m

We know,

The cross-sectional area,

A = πr²

By substituting the values,

  = π × (0.00075)²

  = 1.767 × 10⁻⁶

  = 1.767 m²

hence,

The resistivity will be:

→ R = [tex]\frac{\rho L}{A}[/tex]

      = [tex]\frac{1.68\times 10^{-8}\times 4.4}{1.767}[/tex]

      = 4183316162.9 Ω or,

      = 4.18 × 10⁸ Ω  

Thus the above answer is correct.

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Answer:

3.8 × 10 ⁻¹⁴ m

Explanation:

The alpha particle will be deflected when its kinetic energy is equal to the potential energy

Charge of the alpha particle q₁= 2 × 1.6 × 10⁻¹⁹ C = 3.2 × 10⁻¹⁹ C

Charge of the gold nucleus q₂= 79 × 1.6 × 10⁻¹⁹ = 1.264 × 10⁻¹⁷C

Kinetic energy of  the alpha particle = 5.97 × 10⁶ × 1.602 × 10⁻¹⁹ J ( 1 eV) =  9.564 × 10⁻¹³

k electrostatic force constant = 9 × 10⁹ N.m²/c²

Kinetic energy = potential energy =   k q₁q₂ / r where r is the closest distance the alpha particle got to the gold nucleus

r = (  9 × 10⁹ N.m²/c² × 3.2 × 10⁻¹⁹ C × 1.264 × 10⁻¹⁷C) / 9.564 × 10⁻¹³ = 3.8 × 10 ⁻¹⁴ m

Answer:

13.01 m/s²

Explanation:

The period of a simple pendulum is given as

T = 2π√(L/g) .......................... Equation 1

Where T = Period of the simple pendulum, L = Length of the pendulum, g = acceleration due to gravity of the planet.

Given; T = 132/103 = 1.28 s, L = 54 cm = 0.54 m, π = 3.14

Substitute into equation 1

1.28 = (2×3.14)√(0.54/g)

1.28 = 6.28√(0.54/g)

Making g the subject of the equation,

√(0.54/g) = 1.28/6.28

√(0.54/g) = 0.2038

0.54/g = (0.2038)²

0.54/g = 0.0415

g = 0.54/0.0415

g = 13.01 m/s²

Hence the value of the acceleration due to gravity on the planet = 13.01 m/s²

Answer:

Explanation:

We are given that Sarah throws a tennis ball as far as she can.

At the moment the ball reaches its maximum height.

We have to find the true statement if air resistance is neglect.

When air resistance is negligible then the force act on the ball is force due to gravity.

The ball throw vertically then the acceleration act on the ball is acceleration due to gravity.

The value of g=-9.8 m/square sec

It acts on the ball in downward direction .

Therefore, the ball's vertical acceleration is downwards.

The horizontal acceleration is zero because the ball reaches at maximum height then there is no force which act in horizontal direction on the ball.

Therefore, horizontal acceleration of the ball is zero.

Hence, option a and e are true.

The statement B is correct which states that "The ball's vertical acceleration is downwards".

The statement E is correct which states that "The ball's horizontal acceleration is zero".

The given condition is that Sarah throws a tennis ball as far as she can. Also given that the air resistance is negligible.

Let consider that the ball is thrown vertically. In this case, the force on the ball is the force due to gravitational acceleration on the ball.

Hence the vertical acceleration on the ball is equivalent to the gravitational acceleration of 9.8 m/s2. The vertical force acting on the ball will be in a downward direction because this is the force due to gravity.

Hence the statement B is correct which states that "The ball's vertical acceleration is downwards".

When the ball reaches its maximum height, the horizontal acceleration will be zero on the ball, as there is no force acting on the ball in the horizontal direction.

Hence the statement E is correct which states that "The ball's horizontal acceleration is zero".

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We have that the  the rock's initial launch speed  is mathematically given as

From the question we are told

Generally the equation for the Motion   is mathematically given as

[tex]H=ut+0.5gt^2\\\\Therefore\\\\4.9=0*t+0.5*9.8*t^2\\\\t=1s\\\\Therefore\\\\sx=uxt\\\\20=ux*1\\\\[/tex]

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To determine the initial launch speed of the rock, we can use the equations of projectile motion. Since the rock is thrown horizontally, and it travels 20 m horizontally in 1 second, the initial launch speed can be calculated as 20 m/s.

To determine the initial launch speed of the rock, we can use the equations of projectile motion. Since the rock is thrown horizontally, its initial vertical velocity is zero. The only force acting on the rock in the vertical direction is gravity, which causes it to accelerate downward at a rate of 9.8 m/s². Using the formula:

-yf = yi + vit - 1/2gt²

where yi represents the initial height and yf represents the final height, we can solve for the initial downward velocity. Since the rock reaches the ground (yf = 0) and starts at a height of 4.9 m, the equation becomes:

0 = 4.9 + 0 - 1/2(9.8)t²

Simplifying the equation gives:

t² = 4.9/4.9

t = 1 second.

Since the horizontal distance traveled by the rock is 20 m, and it takes 1 second to reach the ground, the horizontal speed (initial launch speed) of the rock can be calculated using the formula:

v = d/t = 20 m/1 s = 20 m/s.

Therefore, the rock's initial launch speed is 20 m/s.

Explanation:

The temperature of the solar nebula was decreasing as it moved away from its center. Therefore, only heavy elements could condense in the inner solar system and terrestrial planets could not form with light elements, such as gases. In the outer solar system, the Jovian planets formed mostly with gases, since temperatures were too low to allow rocky compositions.

Answer:

The correct option is that both the ends will remain neutral.

Explanation:

As the rod is grounded all of the prolonged charge will be converted to ground so the overall charge on the rod in absence of the charged ball will be equal to zero.Such that the both ends will not bear any charge.

The correct option is: There is negative charge spread evenly on both ends.

 When a conducting rod is brought into contact with a charged ball, charge transfer occurs due to the process of induction and conduction. Initially, when the charged ball (let's assume it is positively charged) is brought close to the rod, it will induce a negative charge on the end of the rod nearest to the ball (end A) and a positive charge on the far end (end B) due to the separation of charges within the conductor. This is because the positive charge on the ball will repel the positive charges within the rod, pushing them away, and attract the negative charges, pulling them towards the ball.

 Once the rod is actually touched to the ball, the negative charges on the rod (which were induced by the ball's positive charge) will be neutralized by the positive charges from the ball, leaving the rod with a net negative charge after the ball is moved away. This negative charge will be evenly distributed along the rod initially.

 However, since the rod is not grounded, the negative charge will not have a path to escape and will remain on the rod. Due to the repulsion between like charges (negative-negative), these charges will repel each other and spread out as far as possible from each other along the length of the rod. This means that the negative charge will be spread evenly on both ends of the rod, as this arrangement minimizes the repulsive forces between the charges and results in the lowest potential energy configuration.

 Therefore, the final arrangement of any charge on the rod, when the charged ball is far away, will be an even distribution of negative charge on both ends A and B of the rod. This is consistent with the principle of electrostatic equilibrium, where charges within a conductor will redistribute themselves to minimize repulsion and achieve the lowest energy state.

Final answer:

The seed takes approximately 0.47 seconds to reach its maximum height when launched at 90° with the highest possible initial speed of 4.6 m/s.

Explanation:

When a seed is launched with an initial speed of 4.6 m/s and at an angle of 90°, the vertical component of the initial velocity is 4.6 m/s. The time it takes for a projectile to reach its maximum height can be calculated using the equation for vertical motion:

t = (Vf - Vi) / g

Where:

t is the time

Vf is the final velocity (which is 0 m/s at the maximum height)

Vi is the initial velocity

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

Using the given information, we can calculate the time it takes for the seed to reach its maximum height:

t = (0 - 4.6) / -9.8 = 0.469 s

Therefore, the seed takes approximately 0.47 seconds to reach its maximum height.

Explanation:

As the force is given as 5.30 kN or [tex]5.30 \times 1000 N[/tex]. Hence, mass will be calculated as follows.

           [tex]F_{w}[/tex] = mg

         [tex]5.30 \times 10^{3} = m \times 9.8 m/s^{2}[/tex]

                   m = 540.816 kg

(a)  At the top, centripetal force [tex]F_{c}[/tex] is acting upwards and the weight of the riders and car, [tex]F_{w}[/tex] will be acting downwards.

Therefore, force on the car by the boom will be calculated as follows.

                   [tex]F_{B} = F_{w} - F_{c}[/tex]

or,          [tex]F_{B} = mg - \frac{mv^{2}}{r}[/tex]

                         = [tex]5.30 \times 1000 N - \frac{540.816 kg \times (3.60)^{2}}{12}[/tex]

                         = 4715.919 N

Hence, the force [tex]F_{B}[/tex] on the car from the boom is 4715.919 N. This means that the car will be hanging on the boom and the boom will exert an upward force.

(b)   Now at the top, centripetal force [tex]F_{c}[/tex] will be acting upwards and the weight of cars and car riders will be acting in the downwards direction.

Hence, we will calculate the force on car by the boom as follows.

       [tex]F_{B} = F_{w} - F_{c}[/tex]

or,         [tex]F_{B} = mg - \frac{mv^{2}}{r}[/tex]

                        = [tex]5.30 \times 1000 N - \frac{540.816 kg \times (14.0)^{2}}{12}[/tex]

                        = -3533.33 N

Therefore, car will be pushing on the boom and the boom will exert a downward force.  

Answer:

[tex]s=274.2857\ m[/tex] is the distance from the target before which the pilot must release the canister.

Explanation:

Given:

Time taken by the canister to hit the ground:

using equation of motion

[tex]h=u_y.t+\frac{1}{2} g.t^2[/tex]

where:

[tex]u_y=[/tex] initial vertical velocity of the canister = 0 (since the the object is dropped from a horizontally moving plane)

[tex]t=[/tex] time taken to hit the ground

[tex]90=0+0.5\times 9.8\times t^2[/tex]

[tex]t=4.2857\ s[/tex]

Now the horizontal distance travelled by the canister after dropping:

[tex]s=v_x\times t[/tex]

[tex]s=64\times 4.2857[/tex]

[tex]s=274.2857\ m[/tex] is the distance from the target before which the pilot must release the canister.

Answer:

The amplitude of the resultant wave will be 0.

Explanation:

Suppose the first wave has an amplitude of A. Its angle is given as wt.

The second way will also have the same amplitude as that of first.

After the reflection, a phase shift of π is added So the wave is given as

[tex]W_1=W_2=Acos(\omega t)\\W_1^{'}=Acos(\omega t+ \pi)[/tex]

Adding the two waves give

                               [tex]W_1'+W_2=Acos(\omega t+ \pi)+Acos(\omega t)\\W_1'+W_2=-Acos(\omega t)+Acos(\omega t)\\W_1'+W_2=0[/tex]

So the amplitude of the resultant wave will be 0.

The average acceleration from t=0 to t=5.00 s is 0.5 m/s^2. instantaneous acceleration at t=0 is 0 m/s^2 and at t=5.00 s is 1.0 m/s^2. The velocity-time graph is parabolic, and the acceleration-time graph shows linear growth.

The velocity of a car as a function of time is given by vx(t) = \\(\alpha + \beta t^2\\), where \\(\alpha = 3.00 m/s\\) and \beta = 0.100 m/s^3. To find the average acceleration for the time interval from t = 0 to t = 5.00 s, we need the change in velocity over the time period. Using the given function, we can calculate the initial velocity at t = 0, which is v(0) = 3.00 m/s, and the final velocity at t = 5.00 s, which is v(5) = 3.00 + 0.100(5)^2 = 3.00 + 2.5 = 5.5 m/s. The average acceleration is then (5.5 m/s - 3.0 m/s) / (5.0 s - 0 s) = 0.5 m/s^2.

For instantaneous acceleration at any given time, we differentiate the velocity function with respect to time to obtain a(t) = dvx/dt = 2\beta t. Therefore, the instantaneous acceleration at t = 0 is a(0) = 2(0.100)(0) = 0 m/s^2, and at t = 5.00 s is a(5) = 2(0.100)(5) = 1.0 m/s^2.

When drawing the vx-t and ax-t graphs, for the velocity-time graph, the curve will start at 3.00 m/s and rise parabolically due to the \(t^2\) term in the function. For the acceleration-time graph, the line will start at 0 m/s^2 and increase linearly with time, passing through 1.0 m/s^2 at 5 seconds.

Answer:

238 Degree

Explanation:

Data given

F1=(3.3,-0.5) and F2=(-3.8,-0.3).

To determine the angle F1+F2 makes with the positive x-axis, we need to determine the magnitude of the force F1+F2.

Since force is a vector quantity, we add the vectors component by component

[tex]F_{1}+F_{2}=<3.3,-0.5> +<-3.8,-0.3>\\F_{1}+F_{2}=<3.3+(-3.8), -0.5+(-0.3)>\\ F_{1}+F_{2}=<-0.5,-0.8>\\F_{1}+F_{2}=(-0.5,-0.8)[/tex]

To determine the angle, we use

[tex]F=(x,y)\\\alpha=arctan(\frac{y}{x} )\\Hence for \\F_{1}+F_{2}=(-0.5,-0.8)\\\alpha=arctan(\frac{-0.8}{-0.5} )\\\alpha=58^{0}[/tex]

Since the component of the force F1+F2 is a negative y and negative x which are located in the  3rd quadrant,  the angle can be calculated as

∝=58+180=238 degree

Hence The angle is measured counterclockwise from the positive x-axis and must be in the range from 0 to 360 degrees is 238 Degree

The angle measured counterclockwise from the positive x-axis is θ = 57.99°

Here we know that:

First, we need to add the forces, we will get:

F1 + F2 =  (3.3, -0.5) + (-3.8, -0.3) = (3.3 - 3.8, -0.5 - 0.3))

F1 + F2 = (-0.5, -0.8)

Now, the angle measured counterclockwise from the positive x-axis of a vector

(a, b) is given by:

θ = Atan(b/a).

Where Atan(x) is the inverse tangent function.

So in this case the angle will be:

θ = Atan(-0.8/-0.5) = 57.99°

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Answer:

greatest speed=0.99c

least speed=0.283c

Explanation:

To solve this problem, we have to go to frame of center of mass.

Total available energy fo π + and π - mesons will be difference in their rest energy:

[tex]E_{0,K_{0} }-2E_{0,\pi } =497Mev-2*139.5Mev\\[/tex]

                       =218 Mev

now we have to assume that both meson have same kinetic energy so each will have K=109 Mev from following equation for kinetic energy we have,

K=(γ-1)[tex]E_{0,\pi }[/tex]

[tex]K=E_{0,\pi}(\frac{1}{\sqrt{1-\beta ^{2} } } -1)\\\frac{1}\sqrt{1-\beta ^{2} }}=\frac{K}{E_{0,\pi}}+1\\ {1-\beta ^{2}=\frac{1}{(\frac{K}{E_{0,\pi}}+1)^2}}\\\beta ^{2}=1-\frac{1}{(\frac{K}{E_{0,\pi}}+1)^2}}\\\beta = +-\sqrt{\frac{1}{(\frac{K}{E_{0,\pi}}+1)^2}}\\\\\\beta =+-\sqrt{1-\frac{1}{(\frac{109Mev}{139.5Mev+1)^2}}[/tex]

[tex]u'=+-0.283c[/tex]

note +-=±

To find speed least and greatest speed of meson we would use relativistic velocity addition equations:

[tex]u=\frac{u'+v}{1+\frac{v}{c^{2} } } u'\\u_{max} =\frac{u'_{+} +v_{} }{1+\frac{v}{c^{2} } } u'_{+} \\u_{max} =\frac{0.828c +0.9c }{1+\frac{0.9c}{c^{2} } } 0.828\\ u_{max} =0.99c\\u_{min} =\frac{u'_{-} +v_{} }{1+\frac{v}{c^{2} } } u'_{-}\\u_{min} =\frac{-0.828c +0.9c }{1+\frac{0.9c}{c^{2} } } -0.828c\\u_{min} =0.283c[/tex]

Final answer:

The greatest and least speeds of the π mesons resulting from the decay of a high-speed K0 meson are calculated using the relativistic laws of velocity addition. The maximum speed is (85/86)c, and the minimum speed is (5/14)c.

Explanation:

The question revolves around a high-speed K0 meson decaying into two π mesons (π+ and π-). To determine the greatest and least speeds that the π mesons can have, we employ the relativistic addition of velocities. The formula used in this context is Vx=(Vx'+V)/(1+(Vx'V/c²)), where V is the speed of the K0 meson, Vx' the speed of π mesons in the K0 meson's rest frame, and c is the speed of light.

The maximum speed is achieved when the π meson is emitted in the same direction as the K0 meson's movement, while the minimum speed occurs when the π meson is emitted in the opposite direction. Considering the K0 meson's speed (0.9c) and the π mesons' speed in the rest frame (0.8c), the equations result in the maximum speed being (85/86)c and the minimum speed being (5/14)c respectively.

Answer:

assume nitrogen is an ideal gas with cv=5R/2

assume argon is an ideal gas with cv=3R/2

n1=4moles

n2=2.5 moles

t1=75°C   in kelvin t1=75+273

t1=348K

T2=130°C  in kelvin t2=130+273

t2=403K

u=пCVΔT

U(N₂)+U(Argon)=0

putting values:

=>4x(5R/2)x(Tfinal-348)=2.5x(3R/2)x(T final-403)

by simplifying:

Tfinal=363K

To determine the change in entropy when the gases mix adiabatically and completely, we'll follow these steps:

1. Calculate the initial entropy of each gas before mixing.

2. Calculate the final entropy of the combined gases after mixing.

3. Calculate the change in entropy.

Given information:

For nitrogen gas:

- Moles of nitrogen gas, [tex]n_N2[/tex] = 4 mol

- Temperature of nitrogen gas, [tex]T_N2[/tex] = 75°C = 75 + 273.15 K

- Pressure of nitrogen gas, [tex]P_N2[/tex] = 30 bar = 30 * 100 kPa

For argon gas:

- Moles of argon gas, [tex]n_Ar[/tex] = 2.5 mol

- Temperature of argon gas, [tex]T_Ar[/tex] = 130°C = 130 + 273.15 K

- Pressure of argon gas, [tex]P_VR[/tex] = 20 bar = 20 * 100 kPa

Given specific heat capacities :

- [tex]CV_N2[/tex] = (5/2)R for nitrogen gas

- [tex]CV_Ar[/tex] = (3/2)R for argon gas

The change in entropy, ΔS, can be calculated using the equation:

[tex]\[ΔS = n_{\text{N2}} \cdot C_{V_{\text{N2}}} \cdot \ln\left(\frac{T_f}{T_{\text{N2}}}\right) + n_{\text{Ar}} \cdot C_{V_{\text{Ar}}} \cdot \ln\left(\frac{T_f}{T_{\text{Ar}}}\right)\][/tex]

where [tex]\(T_f\)[/tex]  is the final temperature after mixing.

1. Calculate the initial entropy of each gas:

[tex]\[S_{\text{N2}} = n_{\text{N2}} \cdot C_{V_{\text{N2}}} \cdot \ln\left(\frac{T_{\text{N2}}}{T_0}\right)\][/tex]

[tex]\[S_{\text{Ar}} = n_{\text{Ar}} \cdot C_{V_{\text{Ar}}} \cdot \ln\left(\frac{T_{\text{Ar}}}{T_0}\right)\][/tex]

where [tex]\(T_0\)[/tex] is the reference temperature (usually taken as 298.15 K).

2. Calculate the final temperature after mixing using the adiabatic process equation:

[tex]\[T_f = \left(\frac{n_{\text{N2}} \cdot C_{V_{\text{N2}}} \cdot T_{\text{N2}} + n_{\text{Ar}} \cdot C_{V_{\text{Ar}}} \cdot T_{\text{Ar}}}{n_{\text{N2}} \cdot C_{V_{\text{N2}}} + n_{\text{Ar}} \cdot C_{V_{\text{Ar}}}}\right)\][/tex]

3. Calculate the change in entropy using the formula above.

Let's plug in the values and calculate the change in entropy. First, we'll calculate the initial entropies of each gas:

[tex]\[S_{\text{N2}} = 4 \times \left(\frac{5}{2}\right)R \times \ln\left(\frac{75 + 273.15}{298.15}\right)\][/tex]

[tex]\[S_{\text{Ar}} = 2.5 \times \left(\frac{3}{2}\right)R \times \ln\left(\frac{130 + 273.15}{298.15}\right)\][/tex]

Now, calculate [tex]\(T_f\)[/tex] using the adiabatic process equation:

[tex]\[T_f = \left(\frac{4 \times \left(\frac{5}{2}\right)R \times 75 + 2.5 \times \left(\frac{3}{2}\right)R \times 130}{4 \times \left(\frac{5}{2}\right)R + 2.5 \times \left(\frac{3}{2}\right)R}\right)\][/tex]

Finally, calculate the change in entropy:

[tex]\[ΔS = 4 \times \left(\frac{5}{2}\right)R \times \ln\left(\frac{T_f}{75 + 273.15}\right) + 2.5 \times \left(\frac{3}{2}\right)R \times \ln\left(\frac{T_f}{130 + 273.15}\right)\][/tex]

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Answer:

The constant force exerted on the bullet is 1590.87 N.

Explanation:

It is given that,

Mass of the bullet, m = 8.6 g

Initial speed of the bullet, u = 0          

Final speed of the bullet, v = 430.1 m/s

We need to find the force exerted on the bullet while it is traveling down the 0.5 m length of the barrel of the rifle. Let a is the acceleration of the bullet. So,

[tex]v^2-u^2=2ad[/tex]

[tex]v^2=2ad[/tex]

[tex]a=\dfrac{v^2}{2d}[/tex]

[tex]a=\dfrac{(430.1)^2}{2\times 0.5}[/tex]

[tex]a=184986.01\ m/s^2[/tex]

Let F is the force exerted. It is given by :

[tex]F=ma[/tex]

[tex]F=8.6\times 10^{-3}\times 184986.01[/tex]

F = 1590.87 N

So, the constant force exerted on the bullet is 1590.87 N. Hence, this is the required solution.                                                  

Answer:

The speed the glow-worm would have accelerated after 10 years if released into free space and assumed to live is 3.09 X 10³⁴ m/s

Explanation:

Energy associated with photon of lights, is given as

E = hc/λ

Where;

h is Planck's constant =  6.626 x 10⁻³⁴ m2 kg/s

C is the speed of light = ?

λ is the light's wavelength = 650nm = 650 X10⁻⁹ m

Also Energy = Power X time

Time given = 10 years

Time in seconds = 10 yrs X 365 days X 24 hrs X 60mins X 60 secs

Time in seconds = 315360000 seconds

P*t = hc/λ

c = (P*t*λ)/h

c = (0.1 X 315360000 X 650 X 10⁻⁹)/(6.626 x 10⁻³⁴)

c = 3.09 X 10³⁴ m/s

The speed the glow-worm would have accelerated after 10 y if released into free space and assumed to live is 3.09 X 10³⁴ m/s

The electric field strength between parallel conducting plates can be calculated using the formula E = V/d, where E is the electric field strength, V is the potential difference between the plates, and d is the spacing between the plates. For a spacing of 1.00 mm, the electric field strength is 7.08 × 10^-8 N/C. For a spacing of 2.00 mm, the electric field strength is 3.54 × 10^-8 N/C.

The electric field strength between parallel conducting plates can be calculated using the formula:

E = V/d

Where E is the electric field strength, V is the potential difference between the plates, and d is the spacing between the plates. In this case, the potential difference is given as 0.708 nC, or 0.708 × 10-9 C.

To find the electric field strength, we need to convert the charge to coulombs and divide it by the spacing between the plates. For a spacing of 1.00 mm (or 0.01 cm), the electric field strength is:

E = (0.708 × 10-9 C) / (0.01 cm) = 7.08 × 10-8 N/C

For a spacing of 2.00 mm (or 0.02 cm), the electric field strength is:

E = (0.708 × 10-9 C) / (0.02 cm) = 3.54 × 10-8 N/C

The electric field strength inside the capacitor is 2000 N/C. The potential difference is 2 V for a 1.00 mm separation and 4 V for a 2.00 mm separation between the plates.

To find the electric field strength (E) and potential difference (V) across a parallel-plate capacitor, we use the following steps:

Therefore, the electric field strength inside the capacitor is 2000 N/C, the potential difference is 2 V for 1.00 mm separation, and 4 V for 2.00 mm separation.

Answer:

0.12959085 J

Explanation:

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

q = Charge = 1.55 μC

d = Distance between charge = 0.5 m

Electric potential energy is given by

[tex]U=k\dfrac{q^2}{d}[/tex]

In this system with three charges which are equidistant from each other

[tex]U=k\dfrac{q^2}{d}+k\dfrac{q^2}{d}+k\dfrac{q^2}{d}[/tex]

[tex]\\\Rightarrow U=k\dfrac{3q^2}{d}\\\Rightarrow U=8.99\times 10^9\times \dfrac{3\times (1.55\times 10^{-6})^2}{0.5}\\\Rightarrow U=0.12959085\ J[/tex]

The potential energy of the system is 0.12959085 J

Light takes 320 minutes to reach Earth from Pluto

Explanation:

We can solve this problem by applying the rule of three. We can do as follows:

- We call [tex]x[/tex] the distance between the Sun and the Earth

- Then the distance between the Earth and Pluto is 40 times this distance, so [tex]40x[/tex]

- Light takes about 8 minutes to cover the distance x between Sun and Earth

Therefore, calling T the time that light takes to cover distance between Earth and Pluto (40x), we can write:

[tex]\frac{x}{8}=\frac{40x}{T}[/tex]

And solving for T,

[tex]T=\frac{8\cdot 40 x}{x}=320 min[/tex]

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Answer:

V = 31.62 m V

Explanation:

Given,

Voltage across thick membrane = ?

Thickness of the membrane, d = 5.5 n m

Electric field = 5.75 M V/m

we know

[tex]E = \dfrac{V}{d}[/tex]

   V = E d

   V = 5.75 x 10⁶ x 5.5 x 10⁻⁹

   V = 31.62 x 10⁻³ V

   V = 31.62 m V

Hence, The Voltage across the membrane is equal to 31.62 m V.

The voltage across the membrane is dependent on the electric field and thickness of the membrane. The voltage across the membrane is 0.0316 V.

The voltage is defined as the difference in electric potential between two points.

Given that the electric field E is 5.75 mv/m and the thickness of the membrane d is 5.5 nm.

The voltage across the membrane is calculated as given below.

[tex]E = \dfrac {V}{d}[/tex]

[tex]V = Ed[/tex]

[tex]V = 5.75 \times 10^6 \times 5.5 \times 10^{-9}[/tex]

[tex]V = 0.0316 \;\rm V[/tex]

Hence we can conclude that the voltage across the membrane is 0.0316 V.

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Answer:

No. The crown is made of lead

Explanation:

From Archimedes principle,

D/D' = W/U

Where D = density of the crown, D' = Density of water, W = weight of the crown in air, U = Upthrust of the crown in water.

make D the subject of the equation,

D = D'(W/U)................... Equation 1

W = mg

Where g = 9.8 m/s², m = 14.7 kg.

W = 14.7(9.8) = 144.06 N.

U = 9.8(14.7-13.4) = 12.74 N.

Constant: D' = 1000 kg/m³

Substitute into equation 1

D = 1000(144.06/12.74)

D = 11307.69 kg/m³

Density of the crown = 11307.69 kg/m³

And Density of Gold = 19300 kg/m³

From the above, The crown is not made of Gold

To determine if the crown is made of gold, we need to use Archimedes' principle and calculate its density. If the density is greater than the density of water, the crown is made of gold.

When a crown of mass 14.7kg is completely submerged in water, an accurate scale reading reads only 13.4kg. To determine if the crown is made of gold, we need to use Archimedes' principle. According to this principle, when an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the object.

If the crown is made of gold, its density should be equal to or greater than the density of water. We can calculate the density of the crown by dividing its mass by the volume of water it displaces. If the density of the crown is greater than the density of water (1kg/L), then the crown is made of gold.

In this case, the mass of the crown is 14.7kg, and the mass of the water displaced is 13.4kg (since the scale reading is lower). By using the equation density = mass/volume, we can calculate the volume of water displaced by the crown and determine its density.

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As show the figure, peak of each blackbody curves shifts towards higher frequency. Then energy emitted by the black body is directly proportional to the frequency incident radiation.

Hence peak is shifted to higher energies.

Therefore the correct option is B: Toward higher energies.

The blackbody curve of a star moving towards Earth would see its peak shifted toward higher energies due to the Doppler Effect, creating a 'blue shift' in the light we observe.

The blackbody curve of a star moving toward Earth undergoes a phenomenon known as the Doppler Effect. When a star moves towards the observer, this effect causes the light from that star to be blue-shifted, which means the light moves towards higher energies, or shorter wavelengths. Therefore, the correct answer is (b) - the peak of the blackbody curve shifts towards higher energies.

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Answer:

29.79 N/m

Explanation:

A spring connected in series behaves like a capacitor connected in series.

Note: Spring and capacitor are alike because they both store energy, While the former store mechanical energy, the later store electrical energy.

From the above, the effective spring connected in series is given by the formula below

1/Kt = 1/K1 + 1/K2 ........................ Equation 1

Where Kt = effective spring constant, K1 = spring constant of spring 1, K2 = spring constant of spring 2

Making K2 the subject of the equation,

K2 = KtK1/(K1-Kt)..................... Equation 2

Given: K1 = 61 N/m, Kt = 20 N/m.

Substitute into equation 2

K2 = (61×20)/(61-20)

K2 = 1220/41

K2 = 29.76 N/m.

Hence the spring constant in the second spring = 29.79 N/m

The spring constant for Spring #2 is 369.17 N/m.

To find the spring constant of Spring #2, we need to use the formula for the effective spring constant of springs in series. The formula is:

1/keff = 1/k1 + 1/k2

Given that k1 = 61.0 N/m and the effective spring constant (keff) is 20.0 N/m, we can plug these values into the formula and solve for k2:

1/20 = 1/61 + 1/k2

Now we can solve for k2:

1/k2 = 1/20 - 1/61

k2 = 1/(1/20 - 1/61)

k2 = 369.17 N/m

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