A stone is thrown vertically upward with a speed of 12m/s from the edge of a cliff 70 m high (a) How much later it reaches the bottom of the cliff? (b) What is its speed just before hitting? And (c) what total distance did it travel? (Ans: 5.2 sec, 38.94 m/s, 84.7 m)

Answer:

[tex] v_f = 13m/s + 0.75 \frac{m}{s^2} * 16 s= 13 m/s +12m/s = 25 m/s[/tex]

Explanation:

For this case we know that the initial velocity is given by:

[tex] v_i = 7 \frac{m}{s}[/tex]

The final velocity on this case is given by:

[tex] v_f = 13 \frac{m}{s}[/tex]

And we know that it takes 8 seconds to go from 7m/s to 13m/s. We can use the following kinematic formula in order to find the acceleration during the first interval:

[tex] v_f = v_i +at[/tex]

If we solve for the acceleration we got:

[tex] a = \frac{v_f -v_i}{t} = \frac{13 m/s -7 m/s}{8 s}= 0.75 \frac{m}{s^2}[/tex]

So for the other traject we assume that the acceleration is constant and the train travels for 16 s. The initial velocity on this case would be 13m/s from the first interval and we can find the final velocity with the following formula:

[tex] v_f = v_ i +a t[/tex]

And if we replace we got:

[tex] v_f = 13m/s + 0.75 \frac{m}{s^2} * 16 s= 13 m/s +12m/s = 25 m/s[/tex]

Answer:

[tex]v=1.92m/s[/tex]

Explanation:

Given data

Length h=2.0m

Angle α=25°

To find

Speed of bob

Solution

From conservation of energy we know that:

[tex]P.E=K.E\\mgh=(1/2)mv^{2}\\ gh=(1/2)v^{2}\\v^{2}=\frac{gh}{0.5}\\ v=\sqrt{\frac{gh}{0.5}}\\ v=\sqrt{\frac{(9.8m/s^{2} )(2.0-2.0Cos(25^{o} ))}{0.5}}\\v=1.92m/s[/tex]

Given values:

As we know,

The conservation of energy:

→ [tex]Potential \ energy = Kinetic \ energy[/tex]

or,

→ [tex]mgh = \frac{1}{2} mv^2[/tex]

or,

→    [tex]v^2 = \frac{gh}{0.5}[/tex]

          [tex]= \sqrt{\frac{gh}{0.5} }[/tex]

By substituting the values, we get

          [tex]= \sqrt{\frac{(9.8)(2.0-2.0 Cos (25^{\circ}))}{0.5} }[/tex]

          [tex]= 1.92 \ m/s[/tex]

Thus the above answer is right.

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The angle of incidence, when there is perfect transmission is the polarization angle. The expression for the polarization is

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

Where

[tex]\theta[/tex] = Polarization angle

[tex]\phi[/tex] = Angle with the vertical axis

We have that the angle is

[tex]\theta = 90-19.5[/tex]

[tex]\theta = 70.5\°[/tex]

The ratio of the intensities depends on the cosine of the polarization angle. The polarization angle found from the wearer’s leaning angle can be used to find the fraction of the reflected ray intensity that will pass through the sunglasses.

Applying the Malus Law we have that

[tex]\frac{I}{I_0} = cos^2 \theta[/tex]

Here,

[tex]I[/tex] = Final intensity

[tex]I_0[/tex] = Initial intensity

Replacing we have,

[tex]\frac{I}{I_0} = cos^2 (70.5)[/tex]

[tex]\frac{I}{I_0} = 0.11[/tex]

Therefore the fraction of the reflected light intensity which passes through the sunglasses is 0.11

By applying Malus's Law, a person leaning at 19.5° while wearing polarized sunglasses with a vertical axis lets approximately 88.9% of the reflected, polarized light pass through.

                  I = Io cos²(θ)

                  I = Io cos²(19.5°)

                  cos(19.5°) ≈ 0.943

                  (0.943)² ≈ 0.889

Therefore, the fraction of the reflected light intensity that will pass through the sunglasses is approximately 0.889 or 88.9%.

Answer:

all areas of knowledge that wish to understand the physical, chemical and biological process must know electrostatics

Explanation:

Electrostatic interactions, have many rare manifestations in nature, which causes many reasons to study them.

- Lightning is a very striking form of electricity

- The biological processes are governed by currents of inanes and potential differences

- The transfer of nutrients and fertilizers to plants is with ion exchange, electrostatic forces

- all modern electronics is based on electricity

- the electric charge in very dry places, creates high currents that can create fires or kill people

In summary all areas of knowledge that wish to understand the physical, chemical and biological process must know electrostatics

A non-physicist might be interested in electrostatic interactions due to their real-world applications, such as photocopiers and air filters, as well as their significance in biological systems.

A non-physicist might be interested in electrostatic interactions for a couple of reasons. First, electrostatic interactions play a significant role in several real-world applications, such as photocopiers, laser printers, ink-jet printers, and electrostatic air filters. Understanding these interactions can help in understanding how these technologies work and how they can be improved. Second, electrostatic interactions are also important in biological systems, where they affect the interactions between molecules. This knowledge is relevant in fields like medicine and biology.

Answer:

a. The apparent depth of the object is less than the real depth.

Explanation:

When we observe for any object lying at the bottom of the pool from the edge of the pool then we are actually viewing an object from an optically rarer medium into an optically denser medium.

The schematic shows the apparent view of the object due to the bending of the rays coming form the object to our eyes.

The rays when coming from a denser medium to a rarer medium they bend away from the normal of the interface.

The correct option is a. The apparent depth of the object is less than the real depth because of the refraction of light at the water-air interface.

Light bends away from the normal as it exits the water, making the object appear shallower than it actually is. When light travels from water to air, it bends away from the normal because water is denser than air.

This bending makes the object appear to be at a shallower depth than it actually is. The apparent depth of the object is therefore less than the real depth of the object.

To summarize, the correct statement is: The apparent depth of the object is less than the real depth. Option a is correct.

Answer:

4

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

[tex]m_1[/tex] = Mass of Earth

[tex]m_2[/tex] = Mass of Moon

r = Distance between Earth and Moon

Old gravitational force

[tex]F_o=\dfrac{Gm_1m_2}{r^2}[/tex]

New gravitational force

[tex]F_n=\dfrac{Gm_1m_2}{(\dfrac{1}{2}r)^2}[/tex]

Dividing the equations

[tex]\dfrac{F_n}{F_o}=\dfrac{\dfrac{Gm_1m_2}{(\dfrac{1}{2}r)^2}}{\dfrac{Gm_1m_2}{r^2}}\\\Rightarrow \dfrac{F_n}{F_o}=\dfrac{\dfrac{Gm_1m_2}{\dfrac{1}{4}r^2}}{\dfrac{Gm_1m_2}{r^2}}\\\Rightarrow \dfrac{F_n}{F_o}=4[/tex]

The ratio is [tex]\dfrac{F_n}{F_o}=4[/tex]

The new force would be 4 times the old force

If the distance between the Earth and Moon were halved, the force of gravity between them would be quadrupled.

According to Newton's law of universal gravitation, the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this relationship is expressed as:

F= G*m*M/r^2

If the distance between the Earth and Moon were halved, the force of gravity between them would be quadrupled. This is because the force of gravity is inversely proportional to the square of the distance between two objects. So, if the distance is divided by 2, the new force would be multiplied by (2^2) = 4.

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The correct options are a) a. false b) b. false c) a.True d) a.True e) a.True f) a.True g) b. false.

(a) The given statement is incorrect, An electric field is the gradient of the potential. when the Potential is constant The electric field is zero. The correct option is b. false

(b)  If an electric field of space is zero, it does not imply that there is no charge . From Gauss law in its divergence form, It implies there are an equal number of opposite charges present. No statement can be said about the charges present in that region. The correct option is b. false.

(c) The statement is correct because the Intensity of an electric field is line integral of the electric potential. The correct otption is a. True

(d) Electric field lines always point from high potential to low potential. The correct option is a. True

(e) When the electric field strength is not zero, an electric potential is zero at all points on the equatorial line of the electric dipole. The correct option is a. True

(f) The statement is correct as conductors allow the free flow of charge within themselves. The correct statement is a. True

(g) It is incorrect as dielectric breakdown occurs when a charge buildup exceeds the electrical limit or dielectric strength of a material. The correct option is b. False

The options are a) false b) false c) True d) True e) True f) True g) false.

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Final answer:

The relationship between electric fields and electric potential is complex. Zero electric field does not guarantee zero electric potential, and equipotential surfaces arise when there is no change in potential, forming naturally around conductors in electrostatic conditions.

Explanation:

Addressing the statements related to electric fields and electric potential:

False: If the electric field is zero in some region of space, it does not imply that the electric potential is also zero; the potential could have a nonzero constant value.

True: If the electric potential is uniformly zero, the electric field must also be zero since a nonzero field would indicate a change in potential.

False: Zero electric potential at a point does not necessarily mean that the electric field is zero at that point.

True: Electric field lines always point from regions of higher to lower potential.

True: The value of the electric potential can be chosen to be zero at any convenient reference point.

True: In electrostatics, the surface of a conductor is an equipotential surface because the electric field within a conductor must be zero.

False: The breakdown voltage of air varies depending on conditions such as air density and humidity, and is generally accepted to be approximately 3×10⁶ V/m but the exact value can differ.

Answer:

T = 295.57 s

Explanation:

given,

mass of the rocket = 200 Kg

mass of the fuel = 100 Kg

acceleration = 35 m/s²

time, t = 35 s

time taken by the rocket to hit the ground, = ?

Final speed of the rocket when fuel is empty

using equation of motion

v = u + a t

v = 0 + 35 x 35

v = 1225 m/s

height of the rocket where fuel is empty

v² = u² + 2 a s

1225² = 0 + 2 x 35 x h₁

h₁ = 21437.5 m

After 35 s the rocket will be moving upward till the final velocity becomes zero.

Now, using equation of motion to find the height after 35 s

v² = u² + 2 g h₂

0² = 1225² + 2 x (-9.8) h₂

h₂ = 76562.5 m

total height = h₁ + h₂

          = 76562.5 m + 21437.5 m = 98000 m

now, time taken by before the rocket hit the ground

using equation of motion

[tex]s = u t +\dfrac{1}{2}at^2[/tex]

[tex]-13500 = 1225 t -\dfrac{1}{2}\times 9.8 \times t^2[/tex]

negative sign is used because the distance travel by the rocket is downward.

4.9 t² - 1225 t - 13500 = 0

[tex]t = \dfrac{-(-1225)\pm \sqrt{1225^2 - 4\times 4.9 \times (-13500)}}{2\times 4.9}[/tex]

t = 260.57 s

neglecting the negative sign

total time the rocket was in air

T = t₁ + t₂

T = 35 + 260.57

T = 295.57 s

Time for which rocket was in air is equal to 295.57 s.

Answer:

(a) The time the book is in the air stays the same.

(b) The horizontal distance the book travels doubles.

(c) The speed of the book just before it reaches the floor increases but not doubles.

Explanation:

The following kinematics equations will be used to solve this question:

[tex]v_y = v_{y_0} + a_yt\\y - y_0 = v_{y_0}t + \frac{1}{2}a_yt^2\\v_y^2 = v_{y_0}^2 + 2a_y(y - y_0)[/tex]

(a) Initially, the y-component of the velocity is zero. So, the x-component of the velocity is doubled.

We will use the second equation for both cases:

[tex]0 - y_0 = v_{y_0}t - \frac{1}{2}(g)t^2 = 0 - \frac{1}{2}gt^2\\0 - y_0 = v_{y_0}t_2 -\frac{1}{2}gt_2^2 = 0 - \frac{1}{2}gt_2^2\\t = t_2[/tex]

Since, the initial velocity in the y-direction is zero, and the height of the table is constant. The time it takes from the edge of the table to the floor is the same.

(b) For the horizontal distance the book travels, we should use the second equation again, and keep in mind that the acceleration in the x-direction is zero.

[tex]x - 0 = v_{x_0}t + \frac{1}{2}a_xt^2 = v_0t\\x_2 = 2v_0t_2 = 2v_0t[/tex]

Hence, the horizontal distance doubles.

(c) The vertical velocity does not change, since the initial velocity in the y-direction is zero. The horizontal velocity does not change along the motion, since the acceleration in the x-direction is zero. However, since the initial velocity in the x-direction doubles, the final velocity in the x-direction doubles as well.

[tex]v_x = v_{x_0} + a_xt = v_{x_0}[/tex]

However, this does not mean that final speed of the book will double. Because the speed of the object is calculated as follows:

[tex]v_{\rm final_1} = \sqrt{v_x^2 + v_y^2} = \sqrt{v_0^2 + v_y^2}\\v_{\rm final_2} = \sqrt{v_{x_2}^2 + v_y^2} = \sqrt{(2v_0)^2 + v_y^2}[/tex]

As can be seen from above, the final speed increases but not doubles.

Doubling the initial horizontal velocity of a book in projectile motion would result in the same time in the air, double the distance covered, and a higher (but not doubled) speed when it hits the floor.

This question explores the concepts of projectile motion in physics. When we double the initial horizontal velocity of the book from v0 to 2v0, the following changes occur:

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Using the concept of Newton's second law, as the elevator starts from rest and travels upward, the scale shows a value of 94.04 kg

Acceleration is the time derivative of velocity. Therefore acceleration can be given by;

[tex]a=a(t)=\frac{dv(t)}{dt} = \frac{d}{dt} (3t+0.2t^2) = 3+0.4t[/tex]

When t = 3s;

[tex]a= 3+(0.4\times 4)=4.6\,m/s^2[/tex]

As the elevator is moving upward the net force on the weighing scale is given using Newton's second law of motion;

[tex]F_{net}=m(a+g) =(64\,kg)\times (9.8\,m/s^2 +4.6\,m/s^2)=921.6\,N[/tex]

A weighing scale is usually caliberated to show the value in kilograms.

Therefore the weighing scale will show the reading;

[tex]m=\frac{F_{net}}{g} =\frac{921.6\,N}{9.8\,m/s^2}= 94.04\,kg[/tex]

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The apparent weight you would experience in the accelerating elevator is 925.12N. This higher reading is due to the addition of the upward acceleration of the elevator to the standard pull of gravity.

To solve this question, we first need to find the elevator's acceleration. The acceleration is the derivative of the velocity: v(t)=(3.0m/s²)t+(0.20m/s³)t² with respect to time t. The derivative is a(t) = 3.0m/s² +2×(0.20m/s³)×t.

Substitute t=4.0s into a(t) to get a(4.0s) = 3.0m/s² + 2×0.2×4.0 = 4.6m/s². This is the acceleration at t=4.0s.

The apparent weight, or reading on the scale, is given by the equation: F=m(g+a), where m is the mass (64 kg), g is the acceleration due to gravity (9.8 m/s²), and a is the acceleration of the elevator. Thus the apparent weight is F=64(9.8+4.6)=925.12 N.

This is your weight when the elevator is accelerating upward; you would feel heavier due to the addition of the elevator's upward acceleration to the natural gravitational pull.

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The aqueous solution with the highest concentration of particles will have the lowest freezing point.

The freezing point of a solution is determined by the concentration of solute particles in the solution. The greater the concentration of particles, the lower the freezing point. In this case, the aqueous solution with the lowest freezing point will have the highest concentration of particles.

Out of the options given, the aqueous solution with the highest concentration of particles is 0.60 m aqueous sucrose. Therefore, aqueous sucrose (0.60 m) will have the lowest freezing point.

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Answer:

E along –x; F along +x

Explanation:

The electric field at point P due to the negative point charge is directed along the -x axis, and the force experienced by the electron at point P due to this field is also along the -x axis.

The electric field at point P due to the negative point charge at x = 0 is directed along the -x axis. This is because electric field lines always point away from positive charges and toward negative charges. Since the negative charge is located at x = 0, the electric field at point P, which is on the positive x-axis, points in the opposite direction, i.e., along the -x axis.

The force experienced by the electron at point P due to this electric field will be in the same direction as the electric field, i.e., along the -x axis. Like charges repel each other, so the negative point charge will exert a repulsive force on the electron.

Answer:

800

Explanation:

Focal length of telescope, F = 200cm

Focal length of eyepiece, f = 0.25

The magnifying power of a telescope is given as the ratio of the focal length of the objective of the telescope to the focal length of the lens. Mathematically:

M = F/f

Therefore, when F = 200cm and f = 0.25cm:

M = 200/0.25

M = 800

Answer:

-48cm

Explanation:

the following data are given

focal length of telescope=200cm,

focal length of the eyepiece=0.25cm

From the genera formula used to find the magnifying power which is expressed as

[tex]M=-\frac{fx_{o}}{f_{e}}[1+\frac{f_{e}}{d}][/tex]

where

[tex]f_{e} = focal length of thr eye piece\\ f_{o} =focal length of the telescope\\[/tex]

and d=least distance of distinct vision=25cm

if we substitute values into the formula, we arrive at

[tex]M=-\frac{fx_{o}}{f_{e}}[1+\frac{f_{e}}{d}]\\M=-\frac{200cm}{0.25cm}[1+\frac{0.25cm}{25cm}]\\M=-800cm[1+0.01]\\M=-800cm(1.01)\\M=-808cm \\M=-808cm[/tex]

hence from the answer, we can conclude that the magnifying power of the telescope is -808cm

Answer:

17.97m/s

Explanation:

Density of air (ρ)air=1.23 kg/m3, and

Air speed (V) =20 m/sec, pressure gradient along the streamline, ∂p/∂x = 100N/m^3.

The equation of motion along the stream line directions:

considering the momentum balance along the streamline.

γsinθ-∂p/∂x=ρV(∂V/∂x)

Neglecting the effect of gravity , then γ=ρg=0

So, ∂p/∂x= -ρV(∂V/∂x)

∂V/∂x= - 100/(20X1.23)= -4.0650407/S

Also δV/δx=∂V/∂x

∂V/∂x=-4.0650407/S and δx=0.5 m

δV = (-4.0650407/S) *(0.5m)

δV = -2.0325203 m/S

So net air speed will be V+δV= -2.0325203+20 ≅17.96748 m/s

Approximately, V+δV=17.97m/s.

Using Bernoulli's equation and the given pressure gradient, we can calculate the airspeed at a point 0.5 m further along the streamline to be approximately 45.83 m/s.

To solve this problem, we can use Bernoulli's equation, which is a principle in fluid dynamics that states the total mechanical energy in a fluid system is constant if no energy is added or removed by work or heat transfer. It is formulated as p1 + 1/2 ρ v1² + ρgh1 = p2 + 1/2 ρ v2² + ρgh2.

In this case, we assume potential energy (ρgh) terms to be zero because there is no change in height, and the fluid (air) is incompressible. We are also given that the pressure gradient is 100 N/m³ which is effectively the change in pressure (Δp = p2 - p1), and the change in the distance along the streamline Δs = 0.5 m.

So we are left with: Δp + 1/2 ρ v1² = 1/2 ρ v2². Since ρ also cancels out, we are left with v2² = v1² + 2 Δp/ρ Δs. Plugging in given values we get v2 = √( 20² + 2*100*0.5) = √2100 = 45.83 m/s, assuming air density (ρ) is approximately 1.225 kg/m³ at sea level and at 15 °C.

Therefore, the airspeed at a point 0.5 m further along the streamline is approximately 45.83 m/s.

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Answer:

a) 8.83*10⁵ m/s  b) 2.80*10⁶ m/s

Explanation:

a) Assuming no other forces acting on the proton, the acceleration on it is produced by the electric field.

By definition, the  force due to the electric field is as follows:

F = q*E = e*E (1)

where e is the elementary charge, the charge carried by only one proton, and is e = 1.6*10⁻¹⁹ C.

According to Newton's 2nd law, this force is at the same time, the product of the mass of the proton, times the acceleration a:

F = mp*a (2)

From (1) and(2), being left sides equal, right sides must be equal too:

[tex]F = e*E = mp*a[/tex]

Solving for a:

[tex]a = \frac{e*E}{mp} =\frac{1.6e-19C*1.36e5N/C}{1.67e-27kg} =1.3e13 m/s2[/tex]

⇒ a = 1.3*10¹³ m/s²

As we have the value of a (which is constant due to the field is uniform), the displacement x, and we know that the initial velocity is 0, in order to get the value of the speed, we can use the following kinematic equation:

[tex]vf^{2} -vo^{2} = 2*a*x[/tex]

Replacing by v₀ = 0, a= 1.3*10¹³ m/s² and  x = 0.03 m, we can find vf as follows:

[tex]vf =\sqrt{2*(1.3e13 m/s2)*0.03m} = 8.83e5 m/s[/tex]

⇒ vf = 8.83*10⁵ m/s

b) We can just repeat the equation from above, replacing x=0.03 m by x=0.3 m, as follows:

[tex]vf =\sqrt{2*(1.3e13 m/s2)*0.3m} = 2.80e6 m/s[/tex]

⇒ vf = 2.80*10⁶ m/s

To find the speed of the proton after traveling a certain distance in a uniform electric field, we can use the equations of motion and the equation for force. By plugging in the known values and solving for acceleration, we can then use the equation for speed to find the desired velocity.

To find the speed of the proton, we can use the equation:

F = ma

where F is the force, m is the mass, and a is the acceleration.

The force on the proton is given by:

F = qE

where q is the charge of the proton and E is the electric field strength.

Plugging in the values, we get:

qE = ma

Solving for acceleration, we have:

a = qE/m

For the first part of the question, with a distance of 3.00 cm, we can use the equation:

v^2 = u^2 + 2as

where v is the final velocity, u is the initial velocity (which is 0), a is the acceleration, and s is the distance.

Plugging in the known values, we have:

v^2 = 0 + 2 * (qE/m) * s

Simplifying:

v = sqrt(2 * (qE/m) * s)

For the second part of the question, with a distance of 30.0 cm, we can use the same equation:

v^2 = u^2 + 2as

Plugging in the known values:

v^2 = 0 + 2 * (qE/m) * s

Simplifying:

v = sqrt(2 * (qE/m) * s)

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Incomplete question as we have not told to find any quantity so I have chosen  some quantities to find.So complete question is here

A 2.00-kg object is attached to a spring and placed on a frictionless, horizontal surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations.

Find (a) the force constant of the spring

 (b) the frequency of the oscillations

 (c) the maximum speed of the object.

Answer:

(a) [tex]k=100N/m[/tex]

(b) [tex]f=1.126Hz[/tex]

(c) [tex]v_{max}=1.41m/s[/tex]

Explanation:

Given data

Mass of object m=2.00 kg

Horizontal Force F=20.0 N

Distance of the object from equilibrium A=0.2 m

To find

(a) Force constant of the spring k

(b) Frequency F of the oscillations

(c) The Maximum Speed V of the object

Solution

For (a) force constant of the spring k

From Hooke's Law we know that:

[tex]F=kx\\k=F/x\\where\\x=A(amplitude)\\So\\k=F/A\\k=(20N/0.2000m)\\k=100N/m[/tex]

For (b) frequency F of the oscillations

The frequency of the motion for an object in simple harmonic motion is expressed as:

[tex]f=\frac{1}{2\pi } \sqrt{\frac{k}{m} }\\ f=\frac{1}{2\pi } \sqrt{\frac{100N/m}{2.0kg} }\\f=1.126Hz[/tex]

For (c) maximum speed V of the object

For an object in simple harmonic motion the maximum values of the magnitude velocity is given as:

[tex]v_{max}=wA\\ where\\w=\sqrt{\frac{k}{m} }\\ so\\v_{max}=\sqrt{\frac{k}{m} }A\\v_{max}=\sqrt{\frac{100N/m}{2.0kg} }(0.200m)\\v_{max}=1.41m/s[/tex]

To solve this problem we will apply the concepts related to the cinematic equations of angular motion. On these equations, angular acceleration is defined as the squared difference of angular velocity over twice the radial displacement. This is mathematically:

[tex]\alpha = \frac{\omega^2-\omega_0^2}{2\theta}[/tex]

Our values are,

[tex]\text{Initial angular velocity} = \omega_0 =6.36 rad/s[/tex]

[tex]\text{Final angular velocity} = \omega =0[/tex]

[tex]\text{Angular displacement} = \theta = 14.7rev = 29.4\pi rad[/tex]

Replacing,

[tex]\alpha = \frac{- 6.36^2}{29.4\pi}[/tex]

[tex]\alpha = -0.43rad/s^2[/tex]

Therefore the angular acceleration is [tex]-0.43rad/s^2[/tex]

Answer:

(a). D(t) = (√2gh)t - 1/2gt²

(b). D(t) = tk [(√2gh) + g(tk - 2t) / 2]

Explanation:

we would be using the equation of motion to determine the distance D

from the question;

H-boris = h₀ + V₀ (t - tk) - 1/2g(t -tk)²

H-arabella = h₀ + V₀t - 1/2gt²

(a). the vertical displacement is given;

D(t) = H-arabella - H-boris

D(t) = h₀ + V₀t -1/2gt₂ - (h₀ + V₀(t-tk) -1/2g(t-tk)₂)

D(t) = h₀ + V₀t - 1/2gt₂ - h₀ where 0 ∠ t ∠ tk

this gives D(t) = V₀t -1/2gt²

where V₀ = √2gh

∴ D(t) = (√2gh)t - 1/2gt²

(b). We already know vertical displacement as;

D(t) = H-arabella - H-boris

D(t) = h₀ + V₀t -1/2gt₂ - (h₀ + V₀(t-tk) -1/2g(t-tk)₂)

= V₀tk - 1/2gt² + 1/2g(t -tk)²

  = V₀tk + 1/2gtk² - gttk

  = √2gh tk + 1/2gt² - gttk

this gives D(t) = tk [(√2gh) + g(tk - 2t) / 2]

cheers i hope this helps.

Answer:

12J

Explanation:

Kinetic Energy before collision = 1/2mv1^2 = 1/2×2×4^2 = 16J

Velocity after collision (v2) = m1v1/m1+m2 = 2×4/2+2 = 8/4 = 2m/s

Kinetic Energy after collision = 1/2mv2^2 = 1/2×2×2^2 = 4J

Kinetic Energy lost = 16J - 4J = 12J

Answer:

Lost in kinetic energy = 12 J

Explanation:

From the law of conservation of momentum,

Total momentum before collision = Total momentum after collision

mu+m'u' = V(m+m')..................................... Equation 1

Where m = mass of first body, u = initial of the first body, m' = mass of the second body, u' = initial velocity of the second body, V = common velocity.

Making V the subject of the equation,

V = mu+m'u'/(m+m')........................... Equation 2

Where m = 2.0 kg, m' = 2.0 kg, u = 4.0 m/s, u' = 0 m/s ( stationary).

Substitute into equation 2

V = (2×4 + 2×0)/(2+2)

V = 8/4

V = 2 m/s.

Total kinetic energy before collision = 1/2mu² = 1/2(2)(2)² = 16 J.

Total Kinetic energy after collision = 1/2V²(m+m') = 1/2(2²)(4) = 4 J.

Thus

Lost in kinetic energy = 16 - 4

Lost in kinetic energy = 12 J

To solve this problem we will apply the concept of Newton's second law with which we will obtain the strength of the proton. We know the mass and the acceleration is given in the statement. Subsequently said Force by equilibrium can be matched the electrostatic force of Coulomb, defined as the product between the charge and the electric field. Our values are

[tex]m = 1.67*10^{-27}kg[/tex]

[tex]a = 10*10^{11} m/s^2[/tex]

Applying the Newton's second law,

[tex]F = ma[/tex]

[tex]F = (1.67*10^{-27}kg)( 10*10^{11} m/s^2)[/tex]

[tex]F = 1.67*10^{-15}N[/tex]

By the Coulomb's equation for electrostatic Force we have that

[tex]F = qE[/tex]

Remember that the charge of a proton is [tex]1.6*10^{-19}C[/tex]

Replacing we have,

[tex]1.67*10^{-15} = 1.6*10^{-19} E[/tex]

[tex]E = 10437.5 N/C[/tex]

Therefore the magnitude of the electric field at the proton's location is [tex]10437.5 N/C[/tex]

The magnitude of the electric field at the proton's location is 10,437.5 N/C.

The given parameters:

The magnitude of the electric field at the proton's location is calculated as follows;

F = ma

F = qE

qE = ma

[tex]E = \frac{ma}{q} \\\\E = \frac{1.67 \times 10^{-27} \times 10\times 10^{11}}{1.6 \times 10^{-19}} \\\\E = 10,437.5 \ N/C[/tex]

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Answer:

[tex]E = 6.38\times 10^6~N/C[/tex]

Explanation:

The question states that we can approximate the line as an infinite wire. In that case, the electric field can be found by Gauss' Law.

We should draw an imaginary cylindrical surface with an arbitrary height, h, around the wire. The radius of the cylinder should be equal to 65.1 cm.

Gauss' Law:

[tex]\int \vec{E}d\vec{a} = \frac{Q_{enc}}{\epsilon_0}[/tex]

The integral in the left-hand side is not to be taken, because we know the area of the cylinder. The enclosed charge in the right-hand side is equal to the charge of the portion of the wire inside the imaginary surface.

The charge density of the wire is

[tex]\lambda = \frac{Q}{L} = \frac{230 \times 10^{-3}}{1000} = 2.3 \times 10^{-4}[/tex]

The charge enclosed by the imaginary surface is

[tex]Q_{enc} = \lambda h = 2.3\times 10^{-4}h[/tex]

Finally, Gauss' Law yields

[tex]E2\pi rh = \frac{\lambda h}{\epsilon_0}\\E = \frac{\lambda}{2\pi \epsilon_0r} = \frac{2.3 \times 10^{-4}}{2\pi\epsilon_0(65.1\times 10^{-2})} = 6.38\times 10^6~N/C[/tex]

To solve this problem we will apply the principle of conservation of energy and the definition of kinematic energy as half the product between mass and squared velocity. So,

[tex]KE_i = KE_f[/tex]

[tex]KE_f = \frac{1}{2} mv^2[/tex]

Here,

m = Mass

V = Velocity

Replacing,

[tex]KE_f = \frac{1}{2} (12000)(11)^2[/tex]

[tex]KE_f = 72600J[/tex]

Therefore the  final kinetic energy of the two car system is 72.6kJ

The final kinetic energy of two car system is 72,600 Joules.

To understand more, check below explanation.

The energy is neither be created nor be destroyed only transfer from one form to another form.

So that,

      Initial kinetic energy = Final kinetic energy

It is given that, mass,m = 1200kg , speed, v = 11m/s.

As we know that,

        Kinetic energy[tex]=\frac{1}{2}*m*v^{2}[/tex]

Substute above values in above formula.

        [tex]K.E=\frac{1}{2}*1200*(11)^{2} \\\\K.E=600*121=72,600J[/tex]    

Hence, the final kinetic energy of two car system is 72,600 Joules.

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Answer: 3.35 m/s

Explanation:

This problem is related to the Conservation of Momentum principle, which states the following:

"If two objects or bodies are in a closed system and both collide, the total momentum of these two objects before the collision will be the same as the total momentum of these same two objects after the collision".

Of course, the momentum of each object may change after the collision, however, the total momentum of the system does not change.  In addition, according to this law it is established that the initial momentum [tex]p_{o}[/tex] must be equal to the final momentum [tex]p_{f}[/tex]:

[tex]p_{o}=p_{f}[/tex] (1)

Where:

[tex]p_{o}=m_{1}V_{1}+m_{2}V_{2}[/tex] (2)

[tex]p_{f}=m_{1}U_{1}+m_{2}U_{2}[/tex] (3)

Being:

[tex]m_{1}=282 kg[/tex] the mass of the first car

[tex]V_{1}=3.70 m/s[/tex] the initial velocity of the first car

[tex]m_{2}=155 kg[/tex] the mass of the second car

[tex]V_{2}=-1.38 m/s[/tex] the initial velocity of the second car

[tex]U_{1}=1.10 m/s[/tex] the final velocity of the first car

[tex]U_{2}[/tex] the final velocity of the second car

Substituting (2) and (3) in (1):

[tex]m_{1}V_{1}+m_{2}V_{2}=m_{1}U_{1}+m_{2}U_{2}[/tex] (4)

Isolating [tex]U_{2}[/tex]:

[tex]U_{2}=\frac{m_{1}(V_{1}-U_{1})+m_{2}V_{2}}{m_{2}}[/tex] (5)

Solving:

[tex]U_{2}=\frac{282 kg(3.70 m/s-1.10 m/s)+(155 kg)(-1.38 m/s)}{155 kg}[/tex] (6)

Finally:

[tex]U_{2}=3.35 m/s[/tex]

Answer:

3.35

Explanation:

Answer:

The coyote crashes 66 m from the base of the cliff.

Explanation:

Hi there!

The equation of the position vector of the Coyote is the following:

r = (x0 + v0 · t + 1/2 · a · t², y0 + 1/2 · g · t²)

Where:

r = postion vector of the Coyote at time t.

x0 = initial horizontal position.

v0 = initial horizontal velocity.

t = time.

a = horizontal acceleration.

y0 = initial vertical position.

g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).

Let's place the origin of the frame of reference at the edge of the cliff so that x0 and y0 = 0.

When the Coyote reaches the ground, the vertical component of its position vector (r1 in the figure) will be equal to -29 m. When the vertical component of the position vector is -29 m, the horizontal component will be equal to the horizontal distance traveled by the Coyote (r1x in the figure). So, let's find the time at which the y-component of the position vector is -29 m:

y = y0 + 1/2 · g · t²  (y0 = 0)

-29 m = -1/2 · 9.8 m/s² · t²

t² = -29 m / -4.9 m/s²

t = 2.4 s

Now, let's find the x-component of the vector r1 in the figure:

x = x0 + v0 · t + 1/2 · a · t²     (x0 = 0)

x = 24 m/s · 2.4 s + 1/2 · 3 m/s² · (2.4 s)²  

x = 66 m

The coyote crashes 66 m from the base of the cliff.

The problem involves determining the distance Wile E. Coyote travels horizontally before he crashes into the ground. It involves the use of motion equations to first calculate the time it takes for the coyote to hit the ground and then the distance it travels horizontally. The calculated distance is roughly 65.4 meters.

In this problem, we are asked to calculate the distance that Wile E. Coyote travels along the ground before he crashes.

To solve this, we need to use the laws of motion. First, since the coyote falls with a constant acceleration due to gravity, we can use the equation of motion to determine the time it takes for him to hit the ground:

29 = 0.5 * g * t2

By plugging in the acceleration due to gravity (g=9.8m/s2), we determine that t is approximately 2.44 seconds.

Next, we have to calculate how far the coyote travels horizontally. He starts with a velocity of 24m/s, but he also accelerates due to the rocket. So the distance he travels is given by:

x = v0t + 0.5*a*t2

Substituting the known values (v0 = 24m/s, a = 3m/s2, t = 2.44s), we find that the coyote travels roughly 65.4 meters along the ground before he crashes into it.

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Answer:

The height of the oil column above the tank top is 8 cm.

Explanation:

By applying Bernoulli's equation between point A and B as shown in the attached diagram

[tex]P_{atm}+\rho_{oil}g(h+0.12)=P_{atm}+\rho_{water}g(0.06+0.12)[/tex]

Here

               [tex]P_{atm}+\rho_{oil}g(h+0.12)=P_{atm}+\rho_{water}g(0.06+0.12)\\898\times (h+0.12)=998 \times (0.06+0.12)\\(h+0.12)=\frac{998}{898} \times (0.18)\\h+0.12=0.20\\h=0.20-0.12\\h=0.08m \approx 8 cm[/tex]

So the height of the oil column above the tank top is 8 cm.

Answer:

Explanation:

Point charge is Placed at the center of a spherical Gaussian surface and according to Gauss law electric flux through Gaussian surface is given by [tex]\frac{1}{\epsilon _0}[/tex] times the charge enclosed

where [tex]\epsilon _0=[/tex]permittivity of free space

So unless we take charge outside of the surface flux will not change for the Gaussian surface

So Statement A,B,C and E are false

Option D is correct because we take the charge outside of the Gaussian surface                    

The electric flux depends on the charge enclosed by the Gaussian surface and the location of the charges. If the volume of the surface changes, but the charge remains inside, the flux remains the same. If the charge is moved off-center or outside the surface, or if a second charge is added outside the surface, the flux will change.

The electric flux depends on the charge enclosed by the Gaussian surface. So, if the sphere is replaced by a cube of the same volume with the charge at the center, the electric flux will not change. However, if the cube is one-tenth the volume of the original sphere, the electric flux will decrease by a factor of 10, since the charge is still at the center.

If the point charge is moved off center, but still inside the original sphere, the electric flux will not change, as long as the charge remains enclosed by the Gaussian surface.

If the point charge is moved to just outside the sphere, the electric flux will increase, as some electric field lines are now passing through the Gaussian surface.

If a second point charge is placed just outside the sphere, the electric flux through the Gaussian surface will increase even more, as the electric field lines from both charges are now passing through the surface.

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Answer: The heart pumps 124.2 billion cm³ of blood in a lifetime

Explanation:

as an adult the pulse rate average must be around 72 beats per minute.

The heart beats about 103,680 times in a day.

There are 365 days in a year

number of heart beat in a year = 365 days x 103,680 = 37,843,200 beats in a year

For every the heart pumps 50cm³ of blood,

Hence,

Amount of blood pump in a year = 50 x 37,843,200 = 1,892,160,000cm³ of blood pumped in a year.

Using the estimated lifespan average an individual is 69 years

So in a life time,

The human heart pumps = 1,892,160,000 x 69 years = 124,200,000,000

If the heart pumps 50cm³ of blood per beat, the heart pumps a total of 130,559,040,000 cm³ (130.6 billion cm³) of blood in a LIFETIME.

The human heart beats approximately 108,000 times per day, amounting to nearly 3 billion beats in a 75-year lifespan and pumps about 2.6 million gallons of blood. To estimate the volume in cubic meters, we use the flow rate of 5 L/min over a 75-year period and convert liters to cubic meters.

Estimating Human Heart Beat and Blood Volume Pumped Over a Lifetime

The vital importance of the heart is evident in its tireless work throughout a person's life. Assuming an average heart rate of 75 beats per minute, we can estimate that a human heart beats about 108,000 times in one day, which amounts to more than 39 million times in one year, and nearly 3 billion times during a 75-year lifespan. When it comes to the volume of blood pumped, with each contraction pumping approximately 70 mL of blood, the heart pumps roughly 5.25 liters of blood per minute. This translates to about 14,000 liters per day, and over a year, the heart would pump approximately 10,000,000 liters, or roughly 2.6 million gallons of blood through an extensive network of vessels.

Answer:

(a) 8 m/s

(b) 5 s

Explanation:

(a)

Using newton's equation of motion,

v² = u²+2gs ..................... Equation 1

Where v = final velocity, u = initial velocity, g = acceleration due to gravity at the surface of moon, s = height reached.

make u the subject of the equation,

u² = v²-2gs

u = √(v²-2gs)................ Equation 2

Note: As the stone is thrown up, v = 0 m/s, g is negative

Given: v = 0 m/s, s = 20 m, g = -1.6 m/s²

Substitute into equation 2

u = √(0-2×20×[-1.6])

u = √64

u = 8 m/s.

(b)

Using,

v = u+ gt

Where t = time of flight to reach the maximum height.

Make t the subject of the equation,

t = (v-u)/g................................... Equation 3

Given: v =  0 m/s, u = 8 m/s, g = - 1.6 m/s²

Substitute into equation 3

t = (0-8)/-1.6

t = -8/-1.6

t = 5 seconds.

The flexural strength of the ZrO2 block is 7680 lb/in^2. The flexural modulus of the ZrO2 block is 33546.74 lb/in^2.

To calculate the flexural strength of the ZrO2 block, we need to use the formula:

Flexural strength = (3F * L) / (2 * b * h^2)

where F is the applied force, L is the distance between supports, b is the width of the block, and h is the thickness of the block. Substituting the given values, we have:

Flexural strength = (3 * 400 lb * 8 in.) / (2 * 0.50 in. * (0.25 in.)^2) = 7680 lb/in^2

To calculate the flexural modulus, we can use the formula:

Flexural modulus = (F * L^3) / (4 * b * h^3 * y)

where y is the deflection of the block. Substituting the given values, we have:

Flexural modulus = (400 lb * (4 in.)^3) / (4 * 0.50 in. * (0.25 in.)^3 * 0.037 in.) = 33546.74 lb/in^2

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