What length of tube would be required to produce a second tone under the same experimental conditions? Explain your answer.

Maximum shear stress in a round steel bar with a groove, subjected to reversed torque, is found by computing the nominal shear stress and multiplying it by the stress concentration factor of the groove. The stress concentration factor must be known to complete the calculation.

To determine the maximum shear stress in an AISI 1040 CD steel round bar which is subjected to purely reversed torque of 1800 lbf∙in, and has a groove machined into it, the effect of the shape modification by the groove needs to be considered. This groove effect is described using stress concentration factor (Kt), which shows the increase in maximum stress over the nominal stress because of the change in geometry.

The method involves determining the nominal shear stress (τnom) which equals the torque (T) divided by the polar moment of inertia (J) given as J = (π * (d/2)^4)/2 for a round rod. Then, multiply τnom by the stress concentration factor of the groove to find the maximum shear stress τmax = Kt * τnom.

However, I see the Kt value for the particular groove shape and size is not provided. This value is usually looked up in standard tables or calculated using specific formulas/fixtures based on the groove's size and shape. Once Kt is known, you can compute the maximum shear stress precisely.

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Titan is one of Saturn's largest satellites. The molecules on the Saturn's cold moon titan absorb the light from the Sun light because the atmosphere on the titan is at low temperature.Titan is made of thick layers of ice, hence it is relatively cold. If the sunlight reflects from saturns moon Titan, due to the prescence of cold atmosphere, abosrption spectrum arises. So the Spectrum formed by the reflected light from the titan is absorption spectrum

The correct option is C: Absorption spectrum

The visible spectrum of sunlight reflected from Titan, Saturn's moon, would be an absorption spectrum, as Titan's atmosphere absorbs some wavelengths of sunlight. The term 'absorption spectrum' refers to a spectrum produced when light passes through a cool, dilute gas.

The visible spectrum of sunlight reflected from Saturn's moon Titan would be expected to be an absorption spectrum. This is because Titan's atmosphere and surface would absorb some wavelengths of sunlight and reflect the rest, producing an absorption spectrum. There are three types of spectrums: continuous, emission, and absorption. A continuous spectrum is one where all colors (wavelengths) are present without any gaps, which usually represents an ideal black body radiator. An emission spectrum is a spectrum of the electromagnetic radiation emitted by a source. The absorption spectrum, on the other hand, is a spectrum produced when light passes through a cool, dilute gas and atoms in the gas absorb at specific frequencies; since the re-emitted light is unlikely to be emitted in the same direction as the absorbed photon, this gives rise to dark lines (absence of light) in the spectrum.

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The given question is incomplete. The complete question is as follows.

A wave has the mathematical form y = [tex](0.2 m) sin(2\pi ft - \frac{2 \pi x}{\lambda}[/tex]. What is the displacement of a particle at the origin after a time [tex]t = \frac{1}{8T}[/tex]?

Explanation:

Let us assume that at origin, x = 0 and the value of t is given as [tex]\frac{1}{8T}[/tex].

Therefore, we will calculate the value of displacement as follows.

            y = [tex]0.2 \times sin[\frac{2 \pi}{T} \times \frac{1}{8}T - \frac{2 \pi}{\lambda (0)}][/tex]

              = [tex]0.2 sin [\frac{\pi}{4} - 0][/tex]

             = 0.1414 m

             = 0.14 (approx)

Thus, we can conclude that displacement of a particle at the origin after a time [tex]t = \frac{1}{8T}[/tex] is 0.14.

Answer:

Acceleration= -0.6466 m/[tex]sec^{2}[/tex]   Downward

Explanation:

Given: Mass M= 945 Kg, Tension T = 8.65 kN = 8650 N and g =9.8 m/[tex]sec^{2}[/tex]

Sol: Weight W = mg = 945 Kg 9.8 m/[tex]sec^{2}[/tex] = 9261 N

this show that T < W so the motion is downward so to find acceleration

mass × acceleration = T - W        (putting values)

945 Kg × a = 8650 N - 9261 N

a= -0.6466 m/[tex]sec^{2}[/tex]  (-ve sign shows the downward direction)

Answer:

- Magnitude of the acceleration = 0.65 m/s²

- The acceleration is directed upwards in the direction of the Tension in the suspending cables.

Explanation:

The force balance on the elevator consists of the Tension in the cable, acting upwards away from the elevator, the weight of the elevator, mg, acting downwards and the 'ma' force responsible for motion.

The direction of this 'ma' force depends on which side of the force balance is lesser or more.

That is, depending on the one that's higher,

ma = T - mg

OR

ma = mg - T

We need to find the higher force.

T = 8.65 KN = 8650 N

mg = 945 × 9.8 = 9261 N

mg > T, meaning the 'ma' force is on the upwards side of the tension, but motion of the elevator is definitely downwards.

Since mg > T,

ma = mg - T = 9261 - 8650 = 611 N

a = 611/m = 611/945 = 0.65 m/s²

Hope this helps!

The charge enclosed by a concentric spherical surface depends on the size of the sphere.

To calculate the charge enclosed by a concentric spherical surface, we need to find the charge within that surface. The charge is uniformly distributed throughout the interior volume of the insulating sphere, so the charge within any smaller sphere that is completely enclosed by the larger sphere will be the same.

(a) For r = 1.00 cm, the smaller sphere is completely enclosed by the larger sphere. Therefore, the charge enclosed by the smaller sphere is 6.50 µC.

(b) For r = 6.50 cm, the smaller sphere is the same size as the larger sphere, so the charge enclosed by the smaller sphere will also be 6.50 µC.

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The energy transferred to the car would be different if it were accelerated to the same speed in a shorter time period.

The energy transferred to the car would indeed be different if it were accelerated to the same speed in 5 seconds instead of 10 seconds. This is because the rate of acceleration affects the amount of energy transferred. In the first scenario, the car would experience a lower rate of acceleration over a longer time period, resulting in a smaller energy transfer. In the second scenario, the car would experience a higher rate of acceleration over a shorter time period, resulting in a larger energy transfer.

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Answer:

a. 0.683secs and 2.99secs

b.(x,y)=(23.99,11.3)m/s, (23.99,-11.3)m/s\

c. V=30.0m/s at angle ∝=-36.9 Degree

Explanation:

Data given

Velocity, V=30m/s

Angle,∝=36.9 Degree

The motion describe by the baseball is a  projectile motion, the velocity at the x-axis and y-axis are given as

[tex]V_{x}=Vcos\alpha\\ V_{x}=30cos36.9\\ V_{x}=23.99m/s\\V_{y}=Vsin\alpha \\V_{y}=30sin36.9\\ V_{y}=18.01m/s[/tex]

a.To calculate the time at which the baseball was at a height of 10m, we use the equation describing the vertical distance traveled

[tex]y=V_{y}t-\frac{1}{2}gt^{2}\\ y=10m\\10=18.01t-\frac{1}{2}*9.81t^{2}\\10=18.01t-4.9t^{2}\\-4.9t^{2}+18.01t-10[/tex]

solving the quadratic equation using the formula method

[tex]t=\frac{-b±\sqrt{b^{2}-4ac }}{2a} \\a=-4.9, b=18.01,c=-10\\t=\frac{18.01±\sqrt{18.01^{2}-4*(-4.9)(-10) }}{2*(-4.9)} \\t=0.683s, t=2.99s[/tex]

Hence the two times required 0.683secs and 2.99secs

b. Note that no acceleration in the hotizontal component, so the velocity remain the same. at a time t=0.683secs,

[tex]V_{x}=23.99m/s, \\V_{y}=V_{yi}-gt\\V_{y}=18.01-9.8*0.683\\V_{y}=11.3m/s\\[/tex]

at a time t=2.99secs,

[tex]V_{x}=23.99m/s, \\V_{y}=V_{yi}-gt\\V_{y}=18.01-9.8*2.99\\V_{y}=-11.3m/s\\[/tex]

c.The landing velocity is the same as the initial projected velocity but in opposite direction i.e V=30.0m/s at angle ∝=-36.9 Degree

To find the times when the baseball is at a height of 10.0 m, solve the vertical motion equation. Calculate the horizontal and vertical components of velocity at each time. Finally, find the magnitude and direction of the velocity when it returns to the starting level.

To find the two times when the baseball is at a height of 10.0 m, we can use the equation for vertical motion:

y = yo + voyt - (1/2)gt^2

Setting y = 10 m and using the given initial vertical velocity, we can solve for t. Plugging in the values, we find two solutions: t = 2.46 s and t = 5.06 s.

Next, we can calculate the horizontal and vertical components of the baseball's velocity at each of the two times. The horizontal component remains constant throughout the motion and can be calculated using:

vx = vocosθ

Plugging in the given initial velocity and angle, we find that vx = 30.0 m/s * cos(36.9°) = 24.3 m/s.

The vertical component of the velocity changes due to acceleration from gravity. We can calculate it using:

vy = voy - gt

Plugging in the given initial vertical velocity and the acceleration due to gravity (-9.8 m/s^2), we find the vertical velocity at t = 2.46 s to be -6.16 m/s and at t = 5.06 s to be -14.56 m/s.

To find the magnitude of the baseball's velocity when it returns to the level at which it left the bat, we can use the Pythagorean theorem:

v = √(vx^2 + vy^2)

Plugging in the calculated values, we find the magnitude of the velocity to be 25.8 m/s.

The direction of the velocity can be found using the inverse tangent function:

θ = atan(vy/vx)

Plugging in the calculated values, we find the direction to be 180° - 36.9° = 143.1° below the horizontal.

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The questions deal with centripetal acceleration, speed, and the path of a supersonic airplane making a turn. The key is to use the formulae for centripetal acceleration, circular motion, and circle properties to calculate the desired quantities.

The questions pertain to the physics concept of centripetal acceleration. The centripetal acceleration of a body moving in a circular path is given by the equation a = v2/r. In the case of the supersonic airplane, we can plug in the given values of speed (v = 2570 km/h = 713.89 m/s) and radius (r = 80.5 km = 80500 m) into this formula to calculate the centripetal acceleration.

Next, to find the time it would take for the airplane to turn from North to East, we need to understand that the airplane is essentially making a 90-degree turn, or 1/4 of a full circular path. Therefore, the time would be 1/4 of the total time it would take to complete a full circle (T = 2πr/v). The distance covered during the turn would also equivalently be 1/4 of the total circumference of the path, which we can calculate using the formula for the circumference of a circle (C = 2πr).

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Answer:

This is the correct question.

Apply the junction rule to the junction labeled with the number1 (at the bottom of the resistor of resistance R_2).

Answer in terms of given quantities,together with the meter readings I_1 and I_2 and the current I_3.

b) Apply the loop rule to loop 2 (the smaller loop on the right).Sum the voltage changes across each circuit element around this loop going in the direction of the arrow. Remember that the current meter is ideal.

Express the voltage drops in terms ofV_b, I_2, I_3, the given resistances, and any other givenquantities.

c) Now apply the loop rule to loop 1(the larger loop spanning the entire circuit). Sum the voltagechanges across each circuit element around this loop going in thedirection of the arrow.

Express the voltage drops in terms ofV_b, I_1, I_3, the given resistances, and any other givenquantities.

Explanation:

a. Junction rule: Kirchhoff current law (KCL)

Then, total current entering a junction equals to current leaving the junction.

Therefore,

i1=i2+i3

b. Apply Kirchoff voltage law ( KVL) to loop 2

Then, sum of voltage in the loop equals to zero.

-i3R3+i2R2=0.

Then,

i2R2=i3R3

c. Apply KVL to the loop 1

-Vb+i1R1+i3R3

Therefore,

Vb=i1R1+i3R3.

The circuit diagram is in the attachment

Kirchhoff's first rule, or the junction rule, states that all currents entering a junction must equal all currents leaving that junction. For the network with a junction labeled 1 and currents I1, I2, and I3, the application of the rule yields the equation: I1 = I2 + I3.

To apply Kirchhoff's first rule, also known as the junction rule, we need to consider that all currents entering a junction must equal all currents leaving that junction. This principle is based on the conservation of charge where the total amount of electricity is maintained. Given the question, we are working with a junction labeled 1 at the bottom of the resistor of resistance R2, with meter readings I1 and I2, and current I3.

According to the junction rule for this scenario, we would express this as: I1 = I2 + I3, where I1 is the current flowing into the junction, and I2 and I3 are currents flowing out. This equation states that the current entering the junction (I1) must be equal to the sum of the currents leaving the junction ( I2 and I3).

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The force exerted by each rope is approximately 230 N.

Let's denote the magnitude of the force exerted by each rope as F. Given that the resultant pull is 372 N directly upward, we can use vector decomposition to find the individual forces exerted by each rope.

The forces F acting at an angle of 72° between them can be decomposed into their horizontal and vertical components as follows:

[tex]\[ F_{\text{horizontal}} = F \cdot \cos(72^\circ) \]\\\ F_{\text{vertical}} = F \cdot \sin(72^\circ) \][/tex]

The vertical components of the forces add up to the resultant pull, so:

[tex]\[ 2 \cdot F_{\text{vertical}} = 372 \, \text{N} \]\\\\\ F_{\text{vertical}} = \frac{372 \, \text{N}}{2} \\\\= 186 \, \text{N} \][/tex]

Now, we can use the vertical component to find F:

[tex]\rm \[ F = \frac{F_{\text{vertical}}}{\sin(72^\circ)} \][/tex]

Substitute the known values:

[tex]\rm \[ F = \frac{186 \, \text{N}}{\sin(72^\circ)} \][/tex]

Calculate F:

[tex]\rm \[ F \approx 230 \, \text{N} \][/tex]

So, the force exerted by each rope is approximately 230 N.

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To find the pull exerted by each rope with an angle of 72°, we can use vector components. Each rope exerts a pull of approximately 436.2 N at an angle of 72° for both the vertical and horizontal components.

To find the pull exerted by each rope, we can use the concept of vector components. Let's call the magnitude of each pull T. Since the resultant pull is 372 N directly upward, we can calculate the vertical component of each rope's pull using the equation T*sin(72°) = 372 N.

Solving for T, we find that the magnitude of each rope's pull is approximately 436.2 N. To find the horizontal component of each rope's pull, we use the equation T*cos(72°).

Since the forces are equal-magnitude and have the same angle between them, each rope exerts the same pull of approximately 436.2 N at an angle of 72° for both the vertical and horizontal components of the pull.

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Answer : The answer is, 5.97, 24

Explanation :

Scientific notation : It is the representation of expressing the numbers that are too big or too small and are represented in the decimal form with one digit before the decimal point times 10 raise to the power.

For example :

5000 is written as [tex]5.0\times 10^3[/tex]

889.9 is written as [tex]8.899\times 10^{-2}[/tex]

In this examples, 5000 and 889.9 are written in the standard notation and [tex]5.0\times 10^3[/tex]  and [tex]8.899\times 10^{-2}[/tex]  are written in the scientific notation.

If the decimal is shifting to right side, the power of 10 is negative and if the decimal is shifting to left side, the power of 10 is positive.

As we are given the 5,970,000,000,000,000,000,000,000 in standard notation.

Now converting this into scientific notation, we get:

[tex]\Rightarrow 5,970,000,000,000,000,000,000,000=5.97\times 10^{24}[/tex]

As, the decimal point is shifting to left side, thus the power of 10 is positive.

Hence, the answer is, [tex]5.97\times 10^{24}[/tex]

Now the answer is comparing to [tex]m.\times 10^n[/tex]

So, m = 5.97 and n = 24

Thus, the answer is, 5.97, 24

Answer:

D. 4 m

Explanation:

According to the work-energy theorem, the work done by the force acting on a body modifies its kinetic energy.

[tex]W=\Delta K\\W=K_f-K_i\\W=\frac{mv_f^2}{2}-\frac{mv_i^2}{2}[/tex]

The car comes to rest after a given distance, so [tex]v_f=0[/tex]. Recall the definition of work [tex]W=Fdcos\theta[/tex], here F is the constant force acting on the car, d is its traveled distance and [tex]\theta[/tex] is the angle between the force and the displacement, since friction force acts opposite to the direction of motion [tex]\theta=180^\circ[/tex] :

[tex]-Fd=-\frac{mv_i^2}{2}\\d=\frac{mv_i^2}{2F}[/tex]

We have:

[tex]v_i'=1\frac{m}{s}\\v_i=0.5\frac{m}{s}\\v_i'=2vi[/tex]

Thus:

[tex]d'=\frac{2m(v_i')^2}{2F}\\d'=\frac{2m(2v_i)^2}{2F}\\d'=4\frac{2mv_i^2}{2F}\\d'=4d\\d'=4(1m)\\d'=4m[/tex]

Answer:

The number of vacancies per cubic meter is 1.18 X 10²⁴ m⁻³

Explanation:

[tex]N_v = N*e[^{-\frac{Q_v}{KT}}] = \frac{N_A*\rho _F_e}{A_F_e}e[^-\frac{Q_v}{KT}}][/tex]

where;

N[tex]_A[/tex] is the number of atoms in iron = 6.022 X 10²³ atoms/mol

ρFe is the density of iron = 7.65 g/cm3

AFe is the atomic weight of iron = 55.85 g/mol

Qv is the energy vacancy formation = 1.08 eV/atom

K is Boltzmann constant = 8.62 X 10⁻⁶ k⁻¹

T is the temperature = 850 °C = 1123 k

Substituting these values in the above equation, gives

[tex]N_v = \frac{6.022 X 10^{23}*7.65}{55.85}e[^-\frac{1.08}{8.62 X10^{-5}*1123}}]\\\\N_v = 8.2486X10^{22}*e^{(-11.1567)}\\\\N_v = 8.2486X10^{22}*1.4279 X 10^{-5}\\\\N_v = 1.18 X 10^{18}cm^{-3} = 1.18 X 10^{24}m^{-3}[/tex]

Therefore, the number of vacancies per cubic meter is 1.18 X 10²⁴ m⁻³

The number of vacancies will be "1.18 × 10²⁴ m⁻³".

According to the question,

Number of atoms, [tex]N_A[/tex] = 6.022 × 10²³ atoms/mol

Iron's density, ρFe = 7.65 g/cm³

Iron's atomic weight, AFe = 55.85 g/mol

Energy vacancy formation, Qv = 1.08 eV/atom

Boltzmann constant, K = 8.62 × 10⁻⁶ k⁻¹

Temperature, T = 850°C or, 1123 K

We know the formula,

→ [tex]N_v[/tex] = N × e [[tex]-\frac{Qv}{KT}[/tex]]

        = [tex]\frac{N_A\times \rho Fe}{AFe}[/tex] e [[tex]-\frac{Qv}{KT}[/tex]]

By substituting the above values, we get

        = [tex]\frac{6.022\times 10^{23}\times 7.65}{55.85}[/tex] e [[tex]- \frac{1.08}{8.62\times 10^{-5}\times 1123}[/tex]]

        = 8.2486 × 10²² × [tex]e^{(-11.1567)}[/tex]

        = 8.2486 × 10²² × 1.4279 × 10⁻⁵

        = 1.18 × 10¹⁸ cm⁻³ or,

        = 1.18 × 10²⁴ m⁻³

Thus the answer above is correct.

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Answer:

The angle between the electric field and sheet's normal vector is 80.96 degrees.

Explanation:

Given that,

Area of the flat sheet, [tex]A=3.8\ m^2[/tex]

Electric field, E = 10 N/C

Electric flux of the sheet, [tex]\phi=6\ Nm^2/C[/tex]

The electric flux is through the sheet is given by the dot product of electric field and the area vector. It is given by :

[tex]\phi=E{\cdot} A[/tex]

or

[tex]\phi=EA\ cos\theta[/tex]

[tex]\theta[/tex] is the angle between electric field and sheet's normal vector

So,

[tex]cos\theta=\dfrac{\phi}{EA}[/tex]

[tex]cos\theta=\dfrac{6}{10\times 3.8}[/tex]

[tex]\theta=cos^{-1}(0.157)[/tex]

[tex]\theta=80.96^{\circ}[/tex]

So, the angle between the electric field and sheet's normal vector is 80.96 degrees. Hence, this is the required solution.

Final answer:

The angle between the electric field and the flat sheet's normal vector, given the electric flux of 6.0 Nm²/C and field magnitude of 10 N/C, is approximately 81.2 degrees.

Explanation:

The question involves calculating the angle between an electric field and a flat sheet's normal vector, given the electric flux and the field magnitude. The formula for electric flux (Φ) is given by Φ = E * A * cos(θ), where E is the electric field strength, A is the area through which the field lines pass, and θ is the angle between the field and the normal to the surface. In this case, we have the electric flux (Φ = 6.0 Nm²/C), the electric field (E = 10 N/C), and the area (A = 3.8 m²). To find the angle θ, we rearrange the equation to solve for the cosine of the angle: cos(θ) = Φ / (E * A).

Substituting the given values, we get cos(θ) = 6.0 / (10 * 3.8), which simplifies to cos(θ) = 0.1579. Taking the arccosine of both sides, we find θ ≈ arccos(0.1579). By calculating this, we find that θ ≈ 81.2°.

Thus, the angle between the electric field and the sheet's normal vector is approximately 81.2 degrees.

Answer:

The transverse component of acceleration is 26.32 [tex]m/s^2[/tex] where as radial the component of acceleration is 8.77 [tex]m/s^2[/tex]

Explanation:

As per the given data

u=π/4 rad

ω=u'=2 rad/s

α=u''=4 rad/s

[tex]r=e^u[/tex]

So the transverse component of acceleration are given as

[tex]a_{\theta}=(ru''+2r'u')\\[/tex]

Here

[tex]r=e^u\\r=e^{\pi/4}\\r=2.1932 m[/tex]

[tex]r'=e^u.u'\\r'=2.1932 \times 2\\r'=4.3864 m[/tex]

So

[tex]a_{\theta}=(ru''+2r'u')\\a_{\theta}=(2.1932\times 4+2\times 4.3864 \times 2)\\a_{\theta}=26.32 m/s\\[/tex]

The transverse component of acceleration is 26.32 [tex]m/s^2[/tex]

The radial component is given as

[tex]a_r=r''-r\theta'^2[/tex]

Here

[tex]r''=e^u.u'^2+e^u u''\\r''=2.1932 \times (2)^2+2.1932\times 4\\r''=17.5456 m[/tex]

So

[tex]a_r=r''-ru'^2\\a_r=17.5456-2.1932\times (2)^2\\a_r=8.7728 m/s^2[/tex]

The radial component of acceleration is 8.77 [tex]m/s^2[/tex]

The radial and transverse components of the peg's acceleration at the given instant are both 4eu m/s^2.

In circular motion, there are two components of acceleration: the radial component (also called centripetal acceleration), directed towards the center of the circle, and the transverse component (also called tangential or azimuthal acceleration), which is perpendicular to the radial component and in the direction of increasing angle.

Given that r = eu, and u = p4 rad, u# = 2 rad/s (angular velocity), u$ = 4 rad/s^2 (angular acceleration), we can calculate these components as follows:

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Answer:

[tex]heigth=83.44m[/tex]

Explanation:

Given data

Baseball mass m₁=0.15 kg

initial speed v₁=0

Bullet mass m₂=0.032 kg

final speed v₂=230 m/s

To find

height h=?

Solution

From conservation of momentum we know that

[tex]m_{1}v_{1}+m_{2}v_{2}=(m_{1}+m_{2})v\\ (0.032kg)(230m/s)+(0.15kg)(0m/s)=(0.15kg+0.032kg)v\\7.36+0=0.182v\\v=7.36/0.182\\v=40.44m/s[/tex]

Now from the conservation of mechanical energy

[tex]P.E=K.E\\mgh=(1/2)mv^{2}\\ gh=(1/2)v^{2}\\ (9.8m/s^{2} )h=(1/2)(40.44m/s)^{2}\\ (9.8m/s^{2} )h=(817.7m^{2} /s^{2} )\\h=(817.7m^{2} /s^{2} )/9.8m/s^{2}\\ heigth=83.44m[/tex]

Final answer:

To determine the rise after a bullet embeds itself in a baseball, use conservation of momentum to calculate the final velocity of the combined mass, then use conservation of energy to find the maximum height the system achieves.

Explanation:

The student is asking about the vertical rise of a combined system of a bullet and a baseball after a collision in which the bullet embeds itself into the baseball. To determine the height to which the combined mass rises, we first need to apply the principle of conservation of momentum to find the velocity of the combined mass right after the collision. Then we use the conservation of energy to find the maximum height the combined mass reaches.

The conservation of momentum before and after the collision gives us:

Initial momentum = mass of bullet × velocity of the bullet

Final momentum = (mass of bullet + mass of baseball) × final velocity

Since initial and final momentum are equal (assuming no external net forces), we can set them equal to each other to solve for the final velocity.

Next, to find the maximum height, we use the conservation of energy, where the initial kinetic energy of the combined mass is converted to gravitational potential energy at the peak of its rise. This gives us:

Kinetic energy = (1/2) × (mass of bullet + mass of baseball) × (final velocity)2

Potential energy at height = (mass of bullet + mass of baseball) × g (acceleration due to gravity) × height

By equating the kinetic energy right after the collision to the potential energy at the maximum height, we can solve for the height.

Answer:

The charge on the ball bearing 4.507 × 10^-8 C

Explanation:

From Coulomb's law

F = kq1q2/r²

make q2 the subject

q2 = Fr²/kq1

q2 = (1.8×10^-2 × 0.026²) ÷ (9×10^9 × 30×10^-9)

q2 = 4.507 × 10^-8 C

Using Coulomb's Law, the charge on the ball bearing is found to be approximately -1.92 x 10^-8 C. The negative sign indicates that the charges on the bead and the ball bearing are of opposite nature.

To determine the charge on the ball bearing, we will use Coulomb's Law, that implies that the product of the two charges divided by their distance from one another is what determines the force between two charges. The formula is: F = k * |q1*q2| / r^2 where F is the electric force, q1 and q2 are the charges, r is the distance between the charges, and k is Coulomb's Constant (8.988x10^9 N * m^2/C^2).

We can rearrange the equation to find the charge on the ball bearing (q2): q2 = F * r^2 / (k * q1)

Substituting the given values: q2 = 1.80x10^-2 N * (2.6x10^-2 m)^2 / (8.988x10^9 N * m^2/C^2 * 3.0x10^-8 C)

With some calculations, we get that the charge on the ball bearing, q2 is approximately  -1.92 x 10^-8 C. The negative sign represents the opposite nature of the charges, meaning the bead and ball bear opposite charges.

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Final answer:

The normal strain in the pipe subjected to a tension force is calculated using Hooke's Law, which relates stress to strain via the elastic modulus. The cross-sectional area is found by the area difference between outer and inner circles, considering the pipe's diameter and wall thickness. The final strain is the normal stress divided by the elastic modulus.

Explanation:

To determine the normal strain in the pipe due to the tension force, we can use Hooke's Law, which relates stress and strain via the elastic modulus. Normal strain (ε) is calculated by dividing the normal stress (σ) by the elastic modulus (E). The normal stress can be found by dividing the tension force (P) by the cross-sectional area (A) of the pipe.

The cross-sectional area for a pipe is the area of a ring, which can be calculated as the area of the outer circle minus the area of the inner circle. Given the outside diameter (do) is 45 mm and wall thickness (t) is 5 mm, the inside diameter (di) will be do - 2t = 35 mm. So the area is calculated with A = π/4 • (do2 - di2).

After the area is determined, the normal stress is P/A and therefore the strain is ε = σ/E. Keep in mind to convert E to the correct units to match those of stress (N/mm²).

For the provided tension force P = 90 kN and elastic modulus E = 150 GPa, the strain calculation will use these values and the cross-sectional area calculated earlier.

Answer:

[tex]6.67ft/s^2[/tex]

Explanation:

We are given that

Initial velocity=u=18ft/s

Final velocity,v=38ft/s

Time=t=3 s

We have to find the average acceleration over that 3 s period.

We know that

Average acceleration,a=[tex]\frac{v-u}{t}{t}[/tex]

Using the formula

Average acceleration,a=[tex]\frac{38-18}{3}ft/s^2[/tex]

Average acceleration,a=[tex]\frac{20}{3}ft/s^2[/tex]

Average acceleration,a=[tex]6.67ft/s^2[/tex]

Hence, the average acceleration=[tex]6.67ft/s^2[/tex]

Explanation:

p1:                           Ww  x   ww

gametes:                       w,W      w,w

                                    .......cross......

F1:                             Ww (2) and ww (2)

                               (white)         (black)

therefore,

genotype  ratio -    1: 1

phenotype ratio-    1: 1

The offspring of a cross between a WW (white) and ww (black) parent will all have the genotype Ww and a white phenotype, resulting in a 100% phenotype ratio for white and a 100% genotype ratio for Ww.

When analyzing the cross between a parent with the genotype WW (white) and another with the genotype ww (black), we consider the basic principles of Mendelian genetics. Since white WW is homozygous dominant and black ww is homozygous recessive, all the offspring will have the genotype Ww, making them all white, due to the complete dominance of the white allele.

The phenotype ratio for this cross would be 100% white offspring. The genotype ratio would be 100% heterozygous Ww, because each offspring inherits one W allele from the white parent and one w allele from the black parent.

Answer:

The horizontal force is 780 N.

Explanation:

Given that,

Mass of sprinter = 52 kg

Suppose A world-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 m/s².

We need to calculate the horizontal force

Using formula of force

[tex]F = ma[/tex]

Where, m = mass of sprinter

a = acceleration

Put the value into the formula

[tex]F=52\times15[/tex]

[tex]F=780\ N[/tex]

Hence, The horizontal force is 780 N.

Answer:

The magnitude of the object's average velocity is 9.74 m/s (9.80 m/s without any intermediate rounding).

Explanation:

Hi there!

The average velocity is calculated as the displacement of the object divided by the time it takes the object to do that displacement.

The displacement is calculated as the distance between the final position of the object and the initial position. In this problem, the displacement is equal to the traveled distance because the object travels only in one direction:

a.v = Δx/t

Where:

a.v = average velocity.

Δx = displacement = final position - initial position

t = time

So, let's find the distance traveled while the object was accelerating. For that, we will use this equation:

x = x0 + v0 · t + 1/2 · a · t²

Where:

x = position of the object at time t.

x0 = initial position.

v0 = initial velocity.

t = time.

a = acceleration.

In this case, since the object is initially at rest, v0 = 0. If we place the origin of the frame of reference at the point where the object starts moving, then x0 = 0. So, the equation of the position of the object after a time t will be:

x = 1/2 · a · t²

x = 1/2 · 1.90 m/s² · (6.60 s)²

x = 41.4 m

The object traveled 41.4 m during the first 6.60 s.

Now, let's find the rest of the traveled distance.

When the velocity is constant, a = 0. Then, the equation of position will be:

x = x0 + v · t

Let's place now the origin of the frame of reference at the point where the object starts traveling at constant velocity so that x0 = 0:

x = v · t

The velocity reached by the object during the acceleration phase is calculated as follows:

v = v0 + a · t   (v0 = 0 because the object started from rest)

v = 1.90 m/s² · 6.60 s

v = 12.5 m/s

Then, the distance traveled by the object at a constant velocity will be:

x = 12.5 m/s · 8.50 s

x = 106 m

The total traveled distance in 15.1 s is (106 m + 41.4 m) 147 m.

Then the displacement will be:

Δx = final position - initial position

Δx = 147 m - 0 = 147 m

and the average velocity will be:

a.v = Δx/t

a.v = 147 m / 15.1 s

a.v = 9.74 m/s

The magnitude of the object's average velocity is 9.74 m/s (9.80 m/s without any intermediate rounding).

Answer:

66.375 x 10⁻⁶ C/m

Explanation:

Using Gauss's law which states that the net electric flux (∅) through a closed surface is the ratio of the enclosed charge (Q) to the permittivity (ε₀) of the medium. This can be represented as ;

∅ = Q / ε₀        -----------------(i)

Where;

∅ = 7.5 x 10⁵ Nm²/C

ε₀ = permittivity of free space (which is air, since it is enclosed in a bag) = 8.85 x 10⁻¹² Nm²/C²

Now, let's first get the charge (Q) by substituting the values above into equation (i) as follows;

7.5 x 10⁵ = Q / (8.85 x 10⁻¹²)

Solve for Q;

Q = 7.5 x 10⁵ x 8.85 x 10⁻¹²

Q = 66.375 x 10⁻⁷ C

Now, we can find the linear charge density (L) which is the ratio of the charge(Q) to the length (l) of the rod. i.e

L = Q / l     ----------------------(ii)

Where;

Q = 66.375 x 10⁻⁷ C

l = length of the rod = 10.0cm = 0.1m

Substitute these values into equation (ii) as follows;

L = 66.375 x 10⁻⁷C / 0.1m

L = 66.375 x 10⁻⁶ C/m

Therefore, the linear charge density (charge per unit length) on the rod is 66.375 x 10⁻⁶ C/m.

The linear charge density of the rod is found using Gauss's law and is equal to 6.642 x 10^-5 C/m.

The concept in this question is Gauss's law, which states that the electric flux through any closed surface is equal to the total charge enclosed divided by the permittivity of free space, ε0. Mathematically, it is written as Φ = Q/ε0, where Φ is the electric flux, Q is the charge, and ε0 = 8.854 x 10^-12 C2/N·m2.

In this case, we have Φ = 7.50 x 105 N·m2/C. To find Q, we just rearrange the formula and multiply both sides by ε0. This gives Q = Φ * ε0 = 7.50 x 105 N·m2/C * 8.854 x 10^-12 C2/N·m2 = 6.642 x 10^-6 C.

The linear charge density λ is the total charge Q divided by the total length L of the rod. Here, L = 10.0 cm = 0.1 m, so λ = Q/L = 6.642 x 10^-6 C / 0.1 m = 6.642 x 10^-5 C/m.

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Answer:

a. [tex]v = \frac{mu_sm_2g\Delta t}{m_1}[/tex]

b. 21.64 m/s

Explanation:

Let g = 9.81m/s2

a. The weight of the block is product of its mass and gravitational acceleration

[tex]W = m_2g = 7.25*9.81 = 71.1225N[/tex]

which is also the normal force acting on the block from the floor so it stays balanced.

N = 71.225N

The static friction of the block is product of its normal force from the floor and the friction coefficient

[tex]F_s = \mu_sN = \mu_sW = mu_sm_2g[/tex]

For the block to move, the force generated by the impact must be at least equal to the static friction.

[tex]F = F_s = mu_sm_2g[/tex]

The impulse is product of this force and time duration of impact.

[tex]I = F\Delta t = mu_sm_2g\Delta t[/tex]

As impulse is generated by change in momentum of the ball, which is product of its mass and velocity v

[tex]I = \Delta p = m_1\Delta v[/tex]

[tex]mu_sm_2g\Delta t = m_1 v[/tex]

[tex]v = \frac{mu_sm_2g\Delta t}{m_1}[/tex]

b. [tex]v = \frac{mu_sm_2g\Delta t}{m_1} = \frac{0.74*7.25*9.81*0.185}{0.45} = 21.64 m/s[/tex]

From Newton's second law, the minimum velocity the ball must have, to make the block move is 21 m/s

There are two types of collision

In elastic collision, both momentum and energy are conserved.

Given that a baseball of mass m1 = 0.45 kg is thrown at a concrete block m2 = 7.25 kg. The block has a coefficient of static friction of μs = 0.74 between it and the floor. The ball is in contact with the block for t = 0.185 s while it collides elastically.

For the concreate block to move, the force applied must be greater than the friction between the block and the floor.

The frictional force = μN

where N = mg

Friction = 7.25 x 9.8 x 0.72

Friction = 51.156

Let assume that the force applied will be equal to the friction. From Newton's second law,

F = Change in momentum / time taken.

That is

F = [tex]m_{1}[/tex]V / t

since the ball is starting from rest, the initial velocity u = 0

a. The expression for the minimum velocity the ball must have, to make the block move will be

F = [tex]m_{1}[/tex]V / t

Make V the subject of formula

V = Ft / [tex]m_{1}[/tex]

substitute F into the formula

V = μ[tex]m_{2}[/tex]g t / [tex]m_{1}[/tex]

b. The velocity in m/s will be calculated by substituting all the parameters into the formula above.

V = (51.156 x 0.185) / 0.45

V = 9.46386 / 0.45

V = 21 m/s

Therefore, the minimum velocity the ball must have, to make the block move is 21 m/s

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Answer:

b. From 33 km/h to 43 km/h

Explanation:

Lets take mass of the car = m

We know that The change kinetic energy KE is give as

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

When speed changes from 23 km/h to 33 km/h :

We know that 1 km/h= 0.27 m/s

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

[tex]KE=\dfrac{1}{2}\times m((0.27\times 33)^2-(0.27\times 23)^2)[/tex]

KE=  20.412m   J

When speed changes from 33 km/h to 43 km/h :

We know that 1 km/h= 0.27 m/s

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

[tex]KE=\dfrac{1}{2}\times m((0.27\times 43)^2-(0.27\times 33)^2)[/tex]

KE=  27.702m   J

Therefore we can say that when speed changes 33 km/h to 43 km/h ,the kinetic energy will changes more.

Answer:

33.7410805301 s

22336.5953109 m

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity = [tex]4\times 331[/tex]

s = Displacement

a = Acceleration = [tex]4g=4\times 9.81\ m/s^2[/tex]

g = Acceleration due to gravity = 9.81 m/s²

[tex]v=u+at\\\Rightarrow t=\dfrac{v-u}{a}\\\Rightarrow t=\dfrac{4\times 331-0}{4\times 9.81}\\\Rightarrow t=33.7410805301\ s[/tex]

The time taken is 33.7410805301 s

[tex]s=ut+\dfrac{1}{2}at^2\\\Rightarrow s=0\times t+\dfrac{1}{2}\times 4\times 9.81\times 33.7410805301^2\\\Rightarrow s=22336.5953109\ m[/tex]

The plane would travel 22336.5953109 m

The shortest time is approximately 33.74 seconds, and the distance traveled is approximately 22.32 kilometers.

To solve this problem, we will first need to convert Mach 4 to a speed in meters per second, then use the kinematic equations for constant acceleration to find the time and distance.

(a) The speed of sound in cold air is given as 331 m/s. Therefore, Mach 4 is:

[tex]\[ v = 4 \times 331 \text{ m/s} = 1324 \text{ m/s} \][/tex]

The acceleration that the pilot can withstand without graying out is less than 4 g. Since 1 g is equal to 9.81 m/s², the maximum acceleration the pilot can withstand is:

[tex]\[ a_{\text{max}} = 4 \times 9.81 \text{ m/s}^2 = 39.24 \text{ m/s}^2 \][/tex]

Using the kinematic equation that relates initial velocity, final velocity, acceleration, and time:

[tex]\[ v = u + at \]\\ where \( v \) is the final velocity, \( u \) is the initial velocity (0 m/s since the pilot starts from rest), \( a \) is the acceleration, and \( t \) is the time. We can solve for \( t \): \[ 1324 \text{ m/s} = 0 \text{ m/s} + (39.24 \text{ m/s}^2)t \] \[ t = \frac{1324 \text{ m/s}}{39.24 \text{ m/s}^2} \] \[ t \approx 33.74 \text{ s} \][/tex]

(b) To find the distance traveled during this time, we use the kinematic equation:

[tex]\[ s = ut + \frac{1}{2}at^2 \]\\ where \( s \) is the distance, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time. Since \( u = 0 \) m/s: \[ s = 0 \times t + \frac{1}{2}(39.24 \text{ m/s}^2)t^2 \] \[ s = \frac{1}{2}(39.24 \text{ m/s}^2)(33.74 \text{ s})^2 \] \[ s \approx \frac{1}{2}(39.24 \text{ m/s}^2)(1137.52 \text{ s}^2) \] \[ s \approx 22320.54 \text{ m} \] \[ s \approx 22.32 \text{ km} \][/tex]

Therefore, the shortest time the pilot can take to reach Mach 4 without graying out is approximately 33.74 seconds, and the distance traveled during this period of acceleration is approximately 22.32 kilometers.

Answer:

The number of electrons that must be moved from one electrode to the other to accomplish this is 1.4 X 10⁹ electrons.

Explanation:

Step 1: calculate the charge on each electrode

Given;

Electric field strength = 2.0 X 10⁶ N/C

The distance between the electrode = 1mm = 1 X 10⁻³ m

Electric field strength (E) = Force (F)/Charge (q)

[tex]E =\frac{Kq}{r^2}[/tex]

where;

E is the electric field strength = 2.0 X 10⁶ N/C

K is coulomb's constant = 8.99 X 10⁹ Nm²/C²

r is the distance between the electrodes = 1 X 10⁻³ m

q is the charge in each electrode = ?

[tex]q = \frac{Er^2}{K} = \frac{(2X10^6)(1X10^{-3})^2}{8.99 X10 ^9}[/tex] = 0.2225 X 10⁻⁹ C

The charge on each electrode is 0.2225 X 10⁻⁹ C

Step 2: calculate the number of electrons to be moved from one electrode to the other.

1 electron contains 1.602 X 10⁻¹⁹ C

So, 0.2225 X 10⁻⁹ C will contain how many electrons ?

= (0.2225 X 10⁻⁹)/(1.602 X 10⁻¹⁹)

= 1.4 X 10⁹ electrons

Therefore, the number of electrons that must be moved from one electrode to the other to accomplish this is 1.4 X 10⁹ electrons.

Final answer:

To create a uniform electric field of 2.0 x 10^6 N/C between two electrodes 1.0 mm apart, a charge of 3.54 x 10^-6 C is required on each electrode. Approximately 2.21 x 10^13 electrons must be moved from one electrode to the other to achieve this.

Explanation:

To solve the immediate question, we first need to calculate the charge required to create a uniform electric field of strength 2.0 x 106 N/C between two rectangular electrodes that are 1.0 mm apart. The relationship between the electric field (E), the charge (Q), the permittivity of free space (ε0), and the distance (d) between the plates is given by the equation E = Q / (ε0 × A), where A is the area of one of the plates. Given the plates have dimensions of 1.0 cm * 2.0 cm, we firstly convert these into meters, giving an area of 0.0002 m2. The permittivity of free space (ε0) is 8.85 x 10-12 C2/N·m2.

Substituting the given values into the formula, we get Q = E × ε0 × A = 2.0 x 106 N/C × 8.85 x 10-12 C2/N·m2 × 0.0002 m2 = 3.54 x 10-6 C. Therefore, this is the charge required on each electrode.

To find the number of electrons that must be moved, we use the relationship between charge and number of electrons, which is given by Q = n × e, where n is the number of electrons and e is the charge of an electron (1.602 x 10-19 Coulombs). So, n = Q / e = 3.54 x 10-6 C / 1.602 x 10-19 C = 2.21 x 1013 electrons. Hence, approximately 2.21 x 1013 electrons need to be moved from one electrode to the other to create the desired electric field.

Answer:

Explanation:

a ) Height to be cleared = 5 - 1.6 = 3.4 m

Horizontal distance to be cleared = 5 m .

angle of throw = 56°

here y = 3.4 , x = 5 , θ = 56

equation of trajectory

y = x tanθ - 1/2 g ( x/ucosθ)²

3.4 = 5 tan56 - 1/2 g ( 5/ucos56)²

3.4 = 7.4 - 122.5 / .3125u²

122.5 / .3125u² = 4

u² = 98

u = 9.9 m /s

Range = u² sin 2 x 56 / g

= 9.9 x 9.9 x .927 / 9.8

= 9.27 m

horizontal distance be-yond the fence will the rock land on the ground

= 9.27 - 5

= 4.27 m

Answer:

a) [tex]u=13.3032\ m.s^{-1}[/tex]

b) [tex]s=3.7597\ m[/tex]

Explanation:

Given:

horizontal distance between the fence and the point of throwing, [tex]r=14\ m[/tex]

height of the fence from the ground, [tex]h_f=5\ m[/tex]

height of projecting the throw above the ground, [tex]h'=1.6\ m[/tex]

angle of projection of throw from the horizontal, [tex]\theta=56^{\circ}[/tex]

The horizontal component of the velocity that remains constant throughout the motion:

[tex]u_x=u\cos\theta[/tex]

Now the time taken to reach the distance of the fence:

use equation of motion,

[tex]t_f=\frac{r}{u_x}[/tex]

[tex]t_f=\frac{14}{u.\cos56}[/tex] .................................(1)

Now the time taken to reach the fence height (this height must be attained on the event of descending motion of the rock for the velocity to be minimum).

Maximum Height of the projectile:

[tex]v_y^2=u_y^2-2\times g.h_m[/tex]

[tex]h_m=\frac{u_y^2}{19.6}[/tex] ........................(4)

Now the height descended form the maximum height to reach the top of the fence:

[tex]\Delta h=h_m-h'[/tex]

[tex]\Delta h=(\frac{u_y^2}{19.6} -3.4)\ m[/tex]

time taken to descent this height from the top height:

[tex]\Delta h=u_{yt}.t_d+\frac{1}{2} \times g.t_d^2[/tex]

where:

[tex]u_{yt}=[/tex]  initial vertical velocity at the top point

[tex]t_d=[/tex] time of descend

[tex](\frac{u_y^2}{19.6} -3.4)=0+0.5\times 9.8\times t_d^2[/tex]

[tex]t_d=\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]..............................(2)

So we find the time taken by the rock to reach the top of projectile where the vertical velocity is zero:

[tex]v_y=u_y-g.t_t[/tex]

where:

[tex]u_y=[/tex] initial vertical velocity

[tex]v_y=[/tex] final vertical velocity

[tex]t_t=[/tex] time taken to reach the top height of the projectile

[tex]0=u_y-g.t_t[/tex]

[tex]t_t=\frac{u_y}{9.8}\ seconds[/tex] .................................(3)

Now the combined events of vertical and horizontal direction must take at the same time as the projectile is thrown:

So,

[tex]t_f=t_t+t_d[/tex]

[tex]\frac{14}{u.\cos56}=\frac{u_y}{9.8} +\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]\frac{14}{u.\cos56}=\frac{u\sin56}{9.8} +\sqrt{(\frac{(u.\sin56)^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]\frac{196}{u^2.\cos^2 56} +\frac{u^2\sin^2 56}{96.04} -2.857\times \tan56=\frac{u^2\sin^2 56}{96.04} -0.694[/tex]

[tex]u=13.3032\ m.s^{-1}[/tex]

Max height:

[tex]h_m=\frac{(u.\sin 56)^2}{19.6}[/tex]

[tex]h_m=\frac{(13.3032\times \sin56)^2}{19.6}[/tex]

[tex]h_m=6.2059\ m[/tex]

Now the rock hits down the ground 1.6 meters below the level of throw.

Time taken by the rock to fall the gross height [tex]h_g=h_m+h'[/tex]:

[tex]h_g=u_{yt}.t_g+\frac{1}{2} g.t_g^2[/tex]

[tex]7.8059=0+0.5\times 9.8\times t_g^2[/tex]

[tex]t_g=1.2621\ s[/tex]

Time taken to reach the the top of the fence from the top, using eq. (2):

[tex]t_d=\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]t_d=\sqrt{(\frac{(u.\sin56)^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]t_d=0.7567\ s[/tex]

Time difference between falling from top height and the time taken to reach the top of fence:

[tex]\Delta t=t_g-t_d[/tex]

[tex]\Delta t=1.2621-0.7567[/tex]

[tex]\Delta t=0.5054\ s[/tex]

b)

Now the horizontal distance covered in this time:

[tex]s=u.\cos56\times\Delta t[/tex]

[tex]s=13.3032\times \cos56\times 0.5054[/tex]

[tex]s=3.7597\ m[/tex] is the horizontal distance covered after crossing the fence.

Answer:

221.17 kJ

Explanation: Note the heat of vaporization is in kJ/mol,then to determine the number of moles of water: divide the mass by 18. Then multiply the number of moles by the molar heat of vaporization of water.

N = 97.6 ÷ 18

Q=molar heat *moles

Q = (40.79) * (97.6 ÷ 18)

This is approximately 221.17 kJ

Answer:

head straight across the river (perpendicular to the bank).

Explanation:

To cross the river in the shortest time first your velocity should be relative to the earth has to have the largest possible component to the bank

suppose,

S be the swimmer

E be the earth

W be the water

[tex]u_{x/y}[/tex]   be the velocity of X relative to Y

resultant velocity relative to E will be:

[tex]u_{S/E}=u_{S/W}+u_{W/E}[/tex]

[tex]u_{W/E}[/tex] is parallel to the bank so,

[tex]u_{S/E}[/tex] has its largest component perpendicular to the bank when [tex]u_{S/W}[/tex] is in that direction

so to cross the river in the shortest time you should straight across the current will then carry you downstream so your path relative to the earth is directed at angle downstream

Final answer:

To cross the river in the shortest time, you must swim perpendicularly to the current. In this case, swimming directly eastward with a speed of 1.5 m/s across a 60 m wide river flowing north at 1.2 m/s will take you across in 40 seconds without being carried downstream.

Explanation:

To cross the river in the shortest time, you must aim to minimize the time spent fighting the water current. The key is to swim in a direction such that your velocity relative to the water combines with the river's velocity to give a resultant path straight across the river. Since the river is flowing north with a speed of 1.2 m/s and your swimming speed relative to the water is 1.5 m/s, the shortest path across would be due east.

If you swim directly eastward, your swimming speed relative to the water ensures that you are moving across the river without being pushed downstream. Thus, the only velocity affecting your eastward crossing is your swimming speed, which is perpendicular to the current. Since the water's current is orthogonal to your motion, it does not affect the time it takes to cross. You'll cross the 60 m wide river in the shortest amount of time by moving at your maximum speed of 1.5 m/s directed perpendicularly to the current.

Considering a swimmer in the given scenario, here's an example to illustrate this concept with numbers:

Therefore, the time taken to cross the river is 40 seconds, and the path taken by the swimmer is perpendicular to the flow of the river, relative to the Earth.