Two 2.5-cm-diameter disks face each other, 1.5 mm apart. They are charged to ± 14 nC. Part A.
What is the electric field strength between the disks?
Express your answer in newtons per coulomb.

Part B.
A proton is shot from the negative disk toward the positive disk. What launch speed must the proton have to just barely reach the positive disk?
Express your answer in meters per second.

Answer:

Acceleration, [tex]a=-31.29\ m/s^2[/tex]

Explanation:

It is given that,

Initial speed of the aircraft, u = 140 mi/h = 62.58 m/s

Finally, it stops, v = 0

Time taken, t = 2 s

Let a is the acceleration of the aircraft. We know that the rate of change of velocity is called acceleration of the object. It is given by :

[tex]a=\dfrac{v-u}{t}[/tex]

[tex]a=\dfrac{0-62.58}{2}[/tex]

[tex]a=-31.29\ m/s^2[/tex]

So, the acceleration of the aircraft is [tex]31.29\ m/s^2[/tex] and the car is decelerating. Hence, this is the required solution.

Answer:

Approximately 21 km.

Explanation:

Refer to the not-to-scale diagram attached. The circle is the cross-section of the sphere that goes through the center C. Draw a line that connects the top of the building (point B) and the camera on the robot (point D.) Consider: at how many points might the line intersects the outer rim of this circle? There are three possible cases:

There's only one such line that goes through the top of the building and intersects the outer rim of the circle only once. That line is a tangent to this circle. In other words, it is perpendicular to the radius of the circle at the point A where it touches the circle.

The camera needs to be on this tangent line when the building starts to disappear. To find the length of the arc that the robot has travelled, start by finding the angle [tex]\angle \mathrm{B\hat{C}D}[/tex] which corresponds to this minor arc.

This angle comes can be split into two parts:

[tex]\angle \mathrm{B\hat{C}D} = \angle \mathrm{B\hat{C}A} + \angle \mathrm{A\hat{C}D}[/tex].

Also,

[tex]\angle \mathrm{B\hat{A}C} = \angle \mathrm{D\hat{A}C} = 90^{\circ}[/tex].

The radius of this circle is:

[tex]\displaystyle r = \frac{c}{2\pi} = \rm \frac{4\times 10^{7}\; m}{2\pi}[/tex].

The lengths of segment DC, AC, BC can all be found:

In the two right triangles [tex]\triangle\mathrm{DAC}[/tex] and [tex]\triangle \rm BAC[/tex], the value of [tex]\angle \mathrm{B\hat{C}A}[/tex] and [tex]\angle \mathrm{A\hat{C}D}[/tex] can be found using the inverse cosine function:

[tex]\displaystyle \angle \mathrm{B\hat{C}A} = \cos^{-1}{\rm \frac{AC}{BC}} [/tex]

[tex]\displaystyle \angle \mathrm{D\hat{C}A} = \cos^{-1}{\rm \frac{AC}{DC}} [/tex]

[tex]\displaystyle \angle \mathrm{B\hat{C}D} = \cos^{-1}{\rm \frac{AC}{BC}} + \cos^{-1}{\rm \frac{AC}{DC}}[/tex].

The length of the minor arc will be:

[tex]\displaystyle r \theta = \frac{4\times 10^{7}\; \rm m}{2\pi} \cdot (\cos^{-1}{\rm \frac{AC}{BC}} + \cos^{-1}{\rm \frac{AC}{DC}}) \approx 20667 \; m \approx 21 \; km[/tex].

Answer:

Part A:

[tex]\rm 2.8\times 10^3\ Volts.[/tex]

Part B:

[tex]\rm 5.6\times 10^{-5}\ J.[/tex]

Explanation:

Part A:

The potential energy at a point due to a charge is defined as

[tex]\rm U=qV[/tex].

where,

V = electric potential at that point.

Therefore,

[tex]\rm V=\dfrac{U}{q}=\dfrac{42\times 10^{-6}}{15\times 10^{-9}}=2.8\times 10^3\ Volts.[/tex]

Part B:

Now, if the charge at that point is replaced with [tex]\rm q_1 = 20\ nC = 20\times 10^{-9}\ C.[/tex], then the electric potential energy at that point is given by

[tex]\rm U=q_1V = 20\times 10^{-9}\times 2.8\times 10^3=5.6\times 10^{-5}\ J.[/tex]

Electric potential energy is the energy which is required to move a unit charge from a point to another point in the electric field.

Electric potential energy is the energy which is required to move a unit charge from a point to another point in the electric field.

It can be given as,

[tex]U=qV[/tex]

Here, [tex]q[/tex] is the charge and [tex]V[/tex] is the electric potential.

Thus the electric potential can be given as,

[tex]V=\dfrac{U}{q}[/tex]

Given information-

The value of electric potential energy is 42 μJ.

As the value of electric potential energy is 42 μJ and charge is 15 nC. Then using the above formula,  the electric potential can be given as,

[tex]V=\dfrac{42\times10^{-6}}{15\times10^{-9}}\\V=2.8\times10^3\rm volts[/tex]

Thus the value of electric potential is [tex]2.8\times10^3[/tex] V.

As the value of electric potential is [tex]2.8\times10^3[/tex] V and charge is 20 nC. Then using the above formula,  the electric potential can be given as,

[tex]U=({2.8\times10^{3})}\times({20\times10^{-9}})\\V=5.6\times10^{-5}\rm J[/tex]

Thus the value of electric potential energy is [tex]5.6\times10^{-5}[/tex] J.

Hence,

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Answer:

3. v(t) = (0.05 m/s 3 )t 2

Explanation:

Instantaneous acceleration  = 0.1 t

dv /dt = 0.1 t

dv = 0.1 t dt

Integrating on both sides

v(t)  = ( 0.1 /2) t²

= .05 t²

Answer: a) 278 * 10^-12 F and 19.4 * 10^-9 C

b) 17.44 * 10^3 N/C and c) C=k*C0 and V=70/k

Explanation: In order to solve this problem we have to use the expression of the capacitor of parallel plates as:

C=A*ε0/d where A is the area of the plates and d the distance between them

C=Π r^2*ε0/d

C=π*0.2^2*8.85*10^-12/0.004=278 * 10^-12F= 278 pF

then

ΔV= Q/C

so Q= ΔV*C=70V*278 pF=19.4 nC

The electric field between the plates is given by:

E= Q/(A*ε0)=19.4 nC/(π*0.2^2*8.85*10^-12)=17.44 *10^3 N/C

If it is introduced a dielectric between the plates, then the new C is increased a factor k while the potential between the plates decreases a factor 1/k.

Answer:

minimum amount of energy required is 8.673 MJ

Explanation:

given data

mass m1 = 80 kg

additional mass m2 = 20 kg

height = 8850 m

to find out

minimum amount of energy required

solution

we know here total mass is m = m1 + m2

m = 80 + 20

m = 100 kg

so now apply energy equation for climb high that is

energy = m×g×h   ....................1

here m is mass and g is 9.8 and h is height

put here all value in equation 1

energy = m×g×h

energy = 100 ×9.8×8850

energy = 8673000 J

so minimum amount of energy required is 8.673 MJ

Final answer:

To climb Mt. Everest at 8850 m high, an 80-kg climber carrying a 20-kg pack requires a minimum of 867,300 Joules (867.3 kJ) of energy, based on the gravitational potential energy calculation.

Explanation:

The question relates to calculating the minimum amount of energy required for an 80-kg climber carrying a 20-kg pack to climb Mt. Everest, 8850 m high. To calculate the energy, we use the formula for gravitational potential energy, which is Potential Energy (PE) = mass (m) × gravity (g) × height (h). The mass of the climber and the pack combined is 100 kg (80 + 20), the acceleration due to gravity (g) is approximately 9.8 m/s2, and the height (h) is 8850 m. Therefore, the required energy can be calculated as PE = 100 kg × 9.8 m/s2 × 8850 m.

The calculation gives us PE = 867,300 Joules or 867.3 kJ. This is the minimum amount of energy required for the climber and his pack to reach the summit of Mt. Everest, assuming no energy is lost to friction or other forces.

Answer:

500 m

Explanation:

t = Time taken

u = Initial velocity = 50 m/s

v = Final velocity = 0

s = Displacement

a = Acceleration = -2.5 m/s²

Equation of motion

[tex]v=u+at\\\Rightarrow 0=50-2.5\times t\\\Rightarrow \frac{-50}{-2.5}=t\\\Rightarrow t=20\ s[/tex]

Time taken by the train to stop is 20 seconds

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow s=50\times 20+\frac{1}{2}\times -2.5\times 20^2\\\Rightarrow s=500\ m[/tex]

∴ The engineer applied the brakes 500 m from the station

Answer:

Work done, W = 0.0219 J

Explanation:

Given that,

Force constant of the spring, k = 290 N/m

Compression in the spring, x = 12.3 mm = 0.0123 m

We need to find the work done to compress a spring. The work done in this way is given by :

[tex]W=\dfrac{1}{2}kx^2[/tex]

[tex]W=\dfrac{1}{2}\times 290\times (0.0123)^2[/tex]

W = 0.0219 J

So, the work done by the spring is 0.0219 joules. Hence, this is the required solution.

Final answer:

The correct answer is b) 1.78 J, which is calculated using the formula W = 1/2kx² for the work done on a spring.

Explanation:

The work done to compress a spring is given by the formula W = 1/2kx², where k is the spring constant and x is the displacement in meters. For a spring with a force constant of 290.0 N/m compressed by 12.3 mm (which should be converted to meters by dividing by 1000, thus 0.0123 m), the work done can be calculated as follows:

W = 1/2 × 290.0 N/m × (0.0123 m)² = 1/2 × 290.0 × 0.00015129 = 1.78 J.

Therefore, the correct answer is b) 1.78 J.

Answer:

4.56 seconds

63.25 m

Explanation:

t = Time taken for the car to stop

u = Initial velocity = 60 km/h = 100 km/h = 100000/3600 = 27.78 m/s

v = Final velocity = 0

s = Displacement

a = Acceleration = -6.1 m/s²

Time taken by the car to stop

[tex]v=u+at\\\Rightarrow t=\frac{v-u}{a}\\\Rightarrow t=\frac{0-27.78}{-6.1}\\\Rightarrow t=4.56\ s[/tex]

Time taken by the car to stop is 4.56 seconds

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow s=27.78\times 4.56+\frac{1}{2}\times -6.1\times 4.56^2\\\Rightarrow s=63.25\ m[/tex]

The total stopping distance would be 63.25 m

Answer:

The intensity of Sound 2 up to two significant digits is  [tex]77 W/m^{2}[/tex]

Solution:

As per the question:

Intensity of Sound 1, [tex]I_{a} = 47.0 W/m^{2}[/tex]

Intensity of Sound 2, [tex]I_{b} = 2.6 dB + I_{a}(in dB)[/tex]

Now,

The intensity of sound in decibel (dB) is:

[tex]I_{dB} = 10log_{10}\frac{I}{I_{c}}[/tex]

where

[tex]I_{c} = 1\times 10^{- 12} W/m^{2}[/tex] = threshold or critical sound intensity

Now,

Intensity of Sound 1, [tex]I_{a}[/tex] in dB is given by:

[tex]I_{a} = 10log_{10}\frac{47.0}{1\times 10^{- 12}} = 136.72 dB[/tex]

Therefore,

[tex]I_{b} = 2.6 dB + I_{a}(dB) = 2.6 + 136.27 = 138.87 dB[/tex]

Now,

The Intensity, [tex]I_{b}[/tex] in [tex]W/m^{2}[/tex] is given by:

[tex]I_{b}dB = 10log_{10}\frac{I_{b}}{I_{c}}[/tex]

[tex]138.87 = 10log_{10}\frac{I_{b}}{1\times 10^{- 12}}[/tex]

[tex]\frac{138.87}{10} = log_{10}\frac{I_{b}}{1\times 10^{- 12}}[/tex]

[tex]I_{b} = 10^{13.887}\times 1\times 10^{- 12}} = 7.709\times 1\times 10^{- 12}}[/tex]

[tex]I_{b} = 77.09 W/m^{2}[/tex]

The iron vat will be 0.01056 meters (or 10.56 millimeters) longer when it contains boiling water at 1 atm pressure.

Given data:

To calculate the change in length of the iron vat when it is heated from room temperature (20°C) to boiling water temperature, we can use the linear thermal expansion formula:

ΔL = α * L * ΔT

Where:

ΔL = Change in length

α = Coefficient of linear expansion of iron (given)

L = Original length of the vat

ΔT = Change in temperature

Given:

Original length, L = 11 m

Coefficient of linear expansion of iron, α = 12 x 10^-6 /°C (approximate value for iron)

Change in temperature, ΔT = Boiling point of water (100°C) - Room temperature (20°C) = 80°C

Now, plug in the values and calculate:

[tex]\Delta L = (12 * 10^{-6} ) * (11 m) * (80 ^\circ C)[/tex]

ΔL ≈ 0.01056 m

Hence, the iron vat will be 0.01056 meters longer.

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The iron vat will be approximately 1.056 cm longer when it contains boiling water at 1 atm pressure compared to at room temperature due to linear thermal expansion. This is calculated using the relevant formula and the coefficients for iron.

The subject of this question is linear thermal expansion, related to physics. When the iron vat is heated by boiling water, it expands. The length of the expansion can be calculated using the formula ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of linear expansion for iron (12 ×10^-6/°C), L is the original length, and ΔT is the change in temperature. Considering the original length L as 11 m and ΔT as the difference between 100°C (boiling point of water at 1 atm pressure) and 20°C (room temperature), which is 80°C. So, plug in these values we have ΔL = 12 ×10^-6/°C * 11m * 80°C = 0.01056 m or about 1.056 cm. Therefore, the iron vat is about 1.056 cm longer when it contains boiling water at 1 atm pressure compared to at room temperature.

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Answer:

[tex]L = 2.746 cm[/tex]

Explanation:

As we know that thermal expansion coefficient of aluminium is given as

[tex]\alpha = 24 \times 10^{-6} per ^oC[/tex]

now we also know that after thermal expansion the final length is given as

[tex]L = L_o(1 + \alpha \Delta T)[/tex]

here we know that

[tex]L_o = 2.739 cm[/tex]

[tex]\alpha = 24 \times 10^{-6}[/tex]

[tex]\Delta T = 108 - 0= 108^oC[/tex]

now we will have

[tex]L = 2.739(1 + 24 \times 10^{-6} (108))[/tex]

[tex]L = 2.746 cm[/tex]

Answer:

(A) Yes, the driver hits the deer with a speed of 14.66 m/s.

(B) 22.21 m/s

Explanation:

Assume:

Part (A):

Before the brakes were applied, the driver moves with a constant velocity of 25 m/s for 1.5 s.

Let us find out distance traveled travel by the driver in this time interval.

[tex]x = 25\times 1.5 = 37.5\ m[/tex]

After the brakes were applied, we have

[tex]u = 25\ m/s\\v = 0\ m/s\\a = -10\ m/s^2\\[/tex]

Using the formula for constant acceleration motion, we have

[tex]v^2=u^2+2as\\\Rightarrow s = \dfrac{v^2-u^2}{2a}\\\Rightarrow s = \dfrac{(0)^2-(25)^2}{2\times (-10)}\\\Rightarrow s = 31.25\ m[/tex]

This means the driver moves a distance of 31.25 m to stop from the time he applies brakes.

So, the total distance traveled by the driver in the entire journey = 37.5 m + 31.25 m = 68.75 m.

Since, the distance traveled by the driver (68.75 m) is greater than the distance of the deer from the driver (58.0 m). So, the driver hits the deer before stopping.

For calculating the speed of the driver with which the driver hits the deer, we have to calculate the actual distance to be traveled by the driver from the instant he applies brake so that the deer does not get hit. This distance is given by:

s = 58 m - 37.5 m = 20.5 m

Now, again using the equation for constant acceleration, we have

[tex]v^2=u^2+2as\\\Rightarrow v^2=(25)^2+2(-10)(20.5)\\\Rightarrow v^2=625-410\\\Rightarrow v^2=215\\\Rightarrow v=\pm \sqrt{215}\\\Rightarrow v=\pm 14.66\ m/s\\[/tex]

Since the driver hits the deer, this means the speed must be positive.

[tex]\therefore v = 14.66 m/s[/tex]

Hence, the driver hits the deer at a speed of 14.66 m/s.

Part (B):

Let the maximum velocity of the driver so that it does not hit the deer be u.

Then,

[tex]x = u\times 1.5 = 1.5u[/tex]

After travelling this distance, the driver applies brakes till it just reaches the position of the deer and does not hit her.

This much distance is S (let).

Using the equation of constant acceleration, we have

[tex]v^2=u^2+2aS\\\Rightarrow (0)^2=u^2+2(-10)(58-x)\\\Rightarrow 0=u^2-1160+30u\\\Rightarrow u^2+30u-1160=0\\[/tex]

On solving the above quadratic equation, we have

[tex]u = 22.21\ m/s\,\,\, or u = -52.21\ m/s[/tex]

But, this speed can not be negative.

So, u = 22.21 m/s.

Hence, the driver could have traveled with a maximum speed of 22.21 m/s so that he does not hit the deer.

Answer:

40.5 W/m²

Explanation:

Intensity of unpolarized light = 432 W/m² = I₀

Intensity of light as it passes through first polarizer

I = I₀×0.5

⇒I = 432×0.5

⇒I = 216 W/m²

Intensity of light as it passes through second polarizer. So, Intensity after it passes through first polarizer is the input for the second polarizer

I = I₀×0.5 cos²30

⇒I = 216×0.75

⇒I = 162 W/m²

Intensity of light as it passes through third polarizer.

I = I₀×0.5 cos²30×cos²60

⇒I = 162×0.25

⇒I = 40.5 W/m²

Intensity of light at the end is 40.5 W/m²

Answer: 2 s

Explanation: In order to solve this problem we have to use the formule of the final speed  getting with a constant acceleration, it is given by;

Vfinal=Vo+a*t    where Vo is zero.

so then  t=Vfinal/a = (10 m/s)/5 m/s^2= 2 s

Final answer:

To calculate the time taken for a car to accelerate from 0 m/s to 10 m/s at 5 m/s², use the given equation and solve for time, resulting in 2 seconds.

Explanation:

The time taken for a car to accelerate from 0 m/s to 10 m/s at 5 m/s² can be calculated using the equation:

Final velocity = Initial velocity (u) + acceleration (a) x time (t)

Plugging in the values:
10 m/s = 0 m/s + 5 m/s² * t

10 m/s = 5t m/s²

Solving for t:
t = 10 m/s / 5 m/s² = 2 seconds

Answer:

The ball land at 3.00 m.

Explanation:

Given that,

Speed = 40 m/s

Angle = 35°

Height h = 1 m

Height of fence h'= 12 m

We need to calculate the horizontal velocity

Using formula of horizontal velocity

[tex]V_{x}=V_{i}\cos\theta[/tex]

[tex]V_{x}=40\times\cos35[/tex]

[tex]V_{x}=32.76\ m/s[/tex]

We need to calculate the time

Using formula of time

[tex]t = \dfrac{d}{v}[/tex]

[tex]t=\dfrac{130}{32.76}[/tex]

[tex]t=3.96\ sec[/tex]

We need to calculate the vertical velocity

[tex]v_{y}=v_{y}\sin\theta[/tex]

[tex]v_{y}=40\times\sin35[/tex]

[tex]v_{y}=22.94\ m/s[/tex]

We need to calculate the vertical position

Using formula of distance

[tex]y(t)=y_{0}+V_{i}t+\dfrac{1}{2}gt^2[/tex]

Put the value into the formula

[tex]y(3.96)=1+22.94\times3.96+\dfrac{1}{2}\times(-9.8)\times(3.96)^2[/tex]

[tex]y(3.96)=15.00\ m[/tex]

We need to calculate the distance

[tex]s = y-h'[/tex]

[tex]s=15.00-12[/tex]

[tex]s=3.00\ m[/tex]

Hence, The ball land at 3.00 m.

Answer:

35 mph

Explanation:

The key of this problem lies in understanding the way that projectile motion works as we are told to neglect the height of the javelin thrower and wind resistance.

When the javelin is thown, its velocity will have two components: a x component and a y component. The only acceleration that will interact with the javelin after it was thown will be the gravety, which has a -y direction. This means that the x component of the velocity will remain constant, and only the y component will be affected, and can be described with the constant acceleration motion properties.

When an object that moves in constant acceleration motion, the time neccesary for it to desaccelerate from a velocity v to 0, will be the same to accelerate the object from 0 to v. And the distance that the object will travel in both desaceleration and acceleration will be exactly the same.

So, when the javelin its thrown, it willgo up until its velocity in the y component reaches 0. Then it will go down, and it will reach reach the ground in the same amount of time it took to go up and, therefore, with the same velocity.

Answer:

Explanation:

mass of block, m = 20 kg

let teh angle of inclination of the ramp with the horizontal is θ.

Weight of the block = mg

There are two components of the weight

The component of weight of block along the plane = mg Sinθ

The component of weight of block normal to the plane = mg Cosθ

Friction force is acting opposite to the motion, i.e., along the plane and upwards, f = μN = μ mg Cosθ

Look at the diagram to find the directions of force.

Answer:

4.555 x 10^-11 m

Explanation:

Potential difference, V = 27.3 kV

Let the wavelength is λ.

The energy associated with the potential difference, E = 27.3 keV

E = 27.3 x 1000 x 1.6 x 10^-19 J = 4.368 x 10^-15 J

The energy associated with the wavelength is given by

[tex]E=\frac{hc}{\lambda }[/tex]

Where, h is Plank's constant = 6.63 x 10^-34 Js

c is velocity of light  3 x 10^8 m/ s

By substituting the values, we get

[tex]4.368\times10^{-15}=\frac{6.634\times10^{-34}\times 3\times 10^{8}}{\lambda }[/tex]

λ = 4.555 x 10^-11 m

Answer:

Electric force, [tex]F=1.29\times 10^6\ N[/tex]

Explanation:

Given that,

Charge 1, [tex]q_1=24\ C[/tex]

Charge 2, [tex]q_2=-24\ C[/tex]

Distance between charges, d = 2 km

The electric force is given by :

[tex]F=k\dfrac{q_1q_2}{d^2}[/tex]

k is the electrostatic constant

[tex]F=8.98755\times 10^9\times \dfrac{(24)^2}{(2\times 10^3)^2}[/tex]

F = 1294207.2 N

or

[tex]F=1.29\times 10^6\ N[/tex]

Hence, this is the required solution.

Answer:

The electric force on the proton is 8.2x10^-10 N

Explanation:

We use the formula to calculate the distance between two points, as follows:

r = ((x2-x1)^2 + (y2-y1)^2)^1/2, where x1 and x2 are the x coordinate, y2, y1 are the y coordinate. replacing values:

r = ((0.36-0)^2 + (0.39-0)^2)^1/2 = 0.53 nm = 5.3x10^-10 m

We will use the following expression to calculate the electrostatic force:

F = (q1*q2)/(4*pi*eo*r^2)

Here we have:

q1 = q2 = 1.6x10^-19 C, 1/4*pi*eo = 9x10^9 Nm^2C^-2

Replacing values:

F = (1.6x10^-19*1.6x10^-9*9x10^9)/((5.3x10^-10)^2) = 8.2x10^-10 N

The electric force on a proton can be calculated using Coulomb's Law and the charges and distance between the proton and electron. The charge of both particles is ±1.602 × 10^{-19} C, and the distance is calculated from their coordinates on the x and y-axis.

To find the electric force on the proton, we can use Coulomb's Law. Coulomb's Law states that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.

The formula for Coulomb's Law is:
F = k * |q1 * q2| / r2,

where F is the force, k is Coulomb's constant (approximately 8.9875 × 109 N m2/C2), q1 and q2 are the charges of the particles, and r is the distance between them.

The charge of a proton (q1) and an electron (q2) is approximately ±1.602 × 10−16 C.

The distance r can be found using the Pythagorean theorem, since we have the x and y coordinates:

r = √(x2 + y2) = √(0.362 + 0.392).

After calculating the distance r, we can plug all the values into the Coulomb's Law formula to get the magnitude of the force.

Answer:

The travel would take 6.7 years.

Explanation:

The equation for an object moving in a straight line with acceleration is:

x = x0 + v0 t + 1/2a*t²

where:

x = position at time t

x0 = initial position

v0 = initial velocity

a = acceleration

t = time

In a movement with constant speed, a = 0 and the equation for the position will be:

x = x0 + v t

where v = velocity

Let´s calculate the position from the Earth after half a year moving with an acceleration of 1.3 g = 1.3 * 9.8 m/s² = 12.74  m/s²:

Seconds in half a year:

1/2 year = 1.58 x 10⁷ s

x = 0 m + 0 m/s + 1/2 * 12.74 m/s² * (1.58 x 10⁷ s)²  = 1.59 x 10¹⁵ m

Now let´s see how much time it takes the travel to the nearest star after this half year.

The velocity will be the final velocity achived after the half-year travel with an acceleration of 12.74 m/s²

v = v0 + a t

Since the spacecraft starts from rest, v0 = 0

v = 12.74 m/s² * 1.58 x 10⁷ s = 2.01 x 10 ⁸ m/s

Using the equation for position:

x = x0 + v t

4.1 x 10¹⁶ m = 1.59 x 10¹⁵ m + 2.01 x 10 ⁸ m/s * t

(4.1 x 10¹⁶ m - 1.59 x 10¹⁵ m) / 2.01 x 10 ⁸ m/s = t

t = 2.0 x 10⁸ s * 1 year / 3.2 x 10 ⁷ s = 6.2 years.

The travel to the nearest star would take 6.2 years + 0.5 years = 6.7 years.

Answer:

They all hit at the same time

Explanation:

Let the time of flight is T.

The maximum height is H and the horizontal range is R.

The formula for the time of flight is

[tex]T=\frac{2uSin\theta }{g}[/tex] ..... (1)

Te formula for the maximum height is

[tex]H=\frac{u^{2}Sin^{2}\theta }{2g}[/tex]    .... (2)

From equation (1) and (2), we get

[tex]\frac{T^{2}}{H}=\frac{\frac{4u^{2}Sin^{2}\theta }{g^{2}}}{\frac{u^{2}Sin^{2}\theta }{2g}}[/tex]    

[tex]\frac{T^{2}}{H}=\frac{8}{g}[/tex]

[tex]T=\sqrt{\frac{8H}{g}}[/tex]

here, we observe that the time of flight depends on the maximum height and according to the question, the maximum height for all the three balls is same so the time of flight of all the three balls is also same.

Answer:

The speed of ocean currents approximately is 0.4667 m/s

Solution:

As per the question:

The ocean currents water moves at 30 W near Africa and 70 W near Puerto Rico

Time taken, t = 1 year = [tex]365\times 24\times 3600 = 3.15\times 10^{7} s[/tex]

The distance between Puerto Rico and Africa, x = [tex]1.46\times 10^{7} m[/tex]

Thus

The speed of the ocean currents there:

speed, [tex]v = \frac{d}{t} = \frac{1.47\times 10^{7}}{3.15\times 10^{7}}[/tex]

[tex]s = 0.4667 m/s[/tex]

Answer:

A) 3.11 seconds

B) 82.33 m

C)  5.32 seconds

D) 25.2 m/s

Explanation:

t = Time taken

u = Initial velocity = 12 m/s

v = Final velocity

s = Displacement

a = Acceleration due to gravity = 9.81 m/s²

[tex]v=u+at\\\Rightarrow 0=12-9.8\times t\\\Rightarrow \frac{-12}{-9.81}=t\\\Rightarrow t=1.22 \s[/tex]

Time taken to reach maximum height is 1.22 seconds

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow s=12\times 1.22+\frac{1}{2}\times -9.81\times 1.22^2\\\Rightarrow s=7.33\ m[/tex]

So, the stone would travel 7.33 m up

B) So, total height stone would fall is 7.33+75 = 82.33 m

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow 82.33=0t+\frac{1}{2}\times 9.81\times t^2\\\Rightarrow t=\sqrt{\frac{82.33\times 2}{9.81}}\\\Rightarrow t=4.1\ s[/tex]

A) Total time taken by the stone to reach the ground from the time it left the person's hand is 4.1+1.22 = 5.32 seconds.

C) When s = 82.33-50 = 33.33 m

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow 32.33=0t+\frac{1}{2}\times 9.81\times t^2\\\Rightarrow t=\sqrt{\frac{32.33\times 2}{9.81}}\\\Rightarrow t=2.57\ s[/tex]

The time it takes from the person's hand throwing the ball to the distance of 50 m from the ground is 2.57+1.22 = 3.777 seconds

D)

[tex]v=u+at\\\Rightarrow v=0+9.81\times 2.57\\\Rightarrow v=25.2\ m/s[/tex]

The stone is moving at 25.2 m/s when it is 50m from the bottom of the building

Answer:

f=171.43Hz

Explanation:

Wave frequency is the number of waves that pass a fixed point in a given amount of time.

The frequency formula is: f=v÷λ, where v is the velocity and λ is the wavelength.

Then replacing with the data of the problem,

f=[tex]\frac{120\frac{m}{s} }{0.7m}[/tex]

f=171.43[tex]\frac{1}{s}[/tex]

f=171.43 Hz (because [tex]\frac{1}{s} = Hz[/tex], 1 hertz equals 1 wave passing a fixed point in 1 second).

The electric potential at the center (C) of the circle formed by the rod is zero due to the symmetry and cancellations of charges. Calculating the potential at point P requires a detailed approach considering the distance D and the charge distribution.

The question involves calculating the electric potential at two different points with reference to a specially configured plastic rod. For part (a), the total charge along the circumference is given by the sum of Q1 and Q2, where Q2 = -6Q1. However, due to the uniform distribution and the symmetry of the setup, the electric potential at the center of the circle (point C) would effectively be zero because the contributions from the positively and negatively charged segments cancel each other out.

For part (b), calculating the electric potential at point P, which lies on the axis and is a distance D from the center, would require the application of principles of superposition and the integration of electric potential contributions from each infinitesimal segment of the charged parts of the rod. Considering the arrangement and the charges' nature, this part of the solution involves more intricate calculations and requires knowing the specific distribution of the charges along the rod.

Answer:

a) 3.65 seconds

b) 35.87 m/s

Explanation:

s = Displacement = 65.6 m

u = Initial velocity

v = Final velocity

t = Time taken

a = Acceleration due to gravity = 9.81 m/s² (downward direction is taken as positive and upward is taken as negative)

b) Equation of motion

[tex]v^2-u^2=2as\\\Rightarrow 0^2-u^2=2\times -9.81\times 65.6\\\Rightarrow u=\sqrt{2\times 9.81\times 65.6}\\\Rightarrow u=35.87\ m/s[/tex]

Initial pop up velocity is 35.87 m/s

a)

[tex]v=u+at\\\Rightarrow t=\frac{v-u}{a}\\\Rightarrow t=\frac{0-35.87}{-9.81}\\\Rightarrow t=3.65\ s[/tex]

It took 3.65 seconds to reach this height

Answer:

Explanation:

For an object to move with constant velocity, the acceleration of the object must be zero:

[tex]\vec{a} = \vec{0}[/tex].

As the net force equals acceleration multiplied by mass , this must mean:

[tex]\vec{F}_{net} = m \vec{a} = m * \vec{0} = \vec{0}[/tex].

So, the sum of the three forces must be zero:

[tex]\vec{F}_1 + \vec{F}_2 + \vec{F}_3 = \vec{0}[/tex],

this implies:

[tex]\vec{F}_3  = - \vec{F}_1 - \vec{F}_2[/tex].

To obtain this sum, its easier to work in Cartesian representation.

First we need to define an Frame of reference. Lets put the x axis unit vector [tex]\hat{i}[/tex] pointing east,  with the y axis unit vector [tex]\hat{j}[/tex] pointing south, so the positive angle is south of east. For this, we got for the first force:

[tex]\vec{F}_1 = 83.7 \ N \ (-\hat{j})[/tex],

as is pointing north, and for the second force:

[tex]\vec{F}_2 = 59.9 \ N \ (-\hat{i})[/tex],

as is pointing west.

Now, our third force will be:

[tex]\vec{F}_3  = - 83.7 \ N \ (-\hat{j}) - 59.9 \ N \ (-\hat{i})[/tex]

[tex]\vec{F}_3  =  83.7 \ N \ \hat{j}  + 59.9 \ N \ \hat{i}[/tex]

[tex]\vec{F}_3  =  (59.9 \ N , 83.7 \ N ) [/tex]

But, we need the magnitude and the direction.

To find the magnitude, we can use the Pythagorean theorem.

[tex]|\vec{R}| = \sqrt{R_x^2 + R_y^2}[/tex]

[tex]|\vec{F}_3| = \sqrt{(59.9 \ N)^2 + (83.7 \ N)^2}[/tex]

[tex]|\vec{F}_3| = 102.92 \ N[/tex]

this is the magnitude.

To find the direction, we can use:

[tex]\theta = arctan(\frac{F_{3_y}}{F_{3_x}})[/tex]

[tex]\theta = arctan(\frac{83.7 \ N }{ 59.9 \ N })[/tex]

[tex]\theta = 57 \° 24 ' 48''[/tex]

and this is the angle south of east.

Answer:

Magnitude of velocity V = 6.015 m/sec

Y component of velocity = 3.61 m/sec

Explanation:

We have given magnitude of velocity in x direction, that is horizontal component [tex]V_X=3.8m/sec[/tex]

Angle with x axis = 37°

We know that horizontal component [tex]V_X=Vcos\Theta[/tex]

So [tex]4.8=Vcos37^{\circ}[/tex]

[tex]V=6.015m/sec[/tex]

Y component of velocity is given by [tex]V_Y=Vsin\Theta =6.015sin37^{\circ}=3.61m/sec[/tex]