A car is driven east for a distance of 50 km, then north for 28 km, and then in a direction 28° east of north for 21 km. Determine (a) the magnitude of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.

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:

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:

D = 25 miles

Explanation:

To solve this problem, we just need to know how much time it took both bicyclists to collide and that will be the same amount of time that the bee flew at 25miles per hour. With those values we could calculate the distance it traveled.

Since both bicyclists collide, we know that Xa=Xb, so:

Xa = V*t = 10*t     and    Xb = 20 - V*t = 20 - 10*t

10*t = 20 - 10*t      Solving for t:

t = 1 hour  Now we can calculate the distance for the bee:

D = Vbee * t = 25 * 1 = 25 miles

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:

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:

4.555 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: 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:

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]

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:

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

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:

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:

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:

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]

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:

(a). The potential on the negative plate is 42.32 V.

(b). The equivalent capacitance of the two capacitors is 0.69 μF.

Explanation:

Given that,

Charge = 10.1 μC

Capacitor C₁ = 1.10 μF

Capacitor C₂ = 1.92 μF

Capacitor C₃ = 1.10 μF

Potential V₁ = 51.5 V

Let V₁ and V₂ be the potentials on the two plates of the capacitor.

(a). We need to calculate the potential on the negative plate of the 1.10 μF capacitor

Using formula of potential difference

[tex]V_{1}=\dfrac{Q}{C_{1}}[/tex]

Put the value into the formula

[tex]V_{1}=\dfrac{10.1 \times10^{-6}}{1.10\times10^{-6}}[/tex]

[tex]V_{1}=9.18\ V[/tex]

The potential on the second plate

[tex]V_{2}=V-V_{1}[/tex]

[tex]V_{2}=51.5 -9.18[/tex]

[tex]V_{2}=42.32\ v[/tex]

(b). We need to calculate the equivalent capacitance of the two capacitors

Using formula of equivalent capacitance

[tex]C=\dfrac{C_{1}\timesC_{2}}{C_{1}+C_{2}}[/tex]

Put the value into the formula

[tex]C=\dfrac{1.10\times10^{-6}\times1.92\times10^{-6}}{(1.10+1.92)\times10^{-6}}[/tex]

[tex]C=6.99\times10^{-7}\ F[/tex]

[tex]C=0.69\ \mu F[/tex]

Hence, (a). The potential on the negative plate is 42.32 V.

(b). The equivalent capacitance of the two capacitors is 0.69 μF.

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:

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:

maximum flow speed is 5.278 cm/s

Explanation:

given data

temperature = 20°C

diameter  d = 3.8 cm

density of water ρ = 1 × 10³ kg/m³

to find out

maximum flow speed

solution

we have given steam line flow so it is laminar flow

and for laminar flow we know reynolds number is Re= 2000

so

Re = [tex]\frac{\rho*v*d}{\mu}[/tex]   ............1

here μ is dynamic viscosity = 0.001003 for 20°C for water

put all value in equation 1

Re = [tex]\frac{\rho*v*d}{\mu}[/tex]

2000 = [tex]\frac{1000*v*3.8*10^{-2}}{0.001003}[/tex]

solve it we get v

v = 0.0527 m/s

so maximum flow speed is 5.278 cm/s

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:

[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:

8 months needed to grow one inch in diameter by water oak and it grows 24 inches of height in a year.

Explanation:

In 2 different ways the trunk of a tree grows first the height than second in diameter usually in diameter tree grows one ring per year. We can say on counting the number of rings we can determine the age of a tree both of the growth does not occur in the same rate.

Tree grows more higher as compared to the  diameter growth and mature trees usually grows 1 inch in diameter every year.

Water oak gets 24 inches height growth every year and   1.5 inch growth in diameter annually means if we divide 1.5 inches by 12 months we gets 0.125 inches growth monthly.so we can say 8 months are needed to grow the diameter of water oak 1 inch.

Answer:

241.27 N

Explanation:

Both the negative charge will pull the positive charge towards it . Let the forces be F₁ and F₂

F₁ = [tex]\frac{9\times10^9\times8\times6.7\times10^{-18}}{(2.3\times10^{-2})^2}[/tex]

F₁ = 91.19 X 10⁻⁵ N

F₂=[tex]\frac{9\times10^9\times8\times13.4\times10^{-18}}{(2.3\times10^{-2})^2}[/tex]

F₂ = 182.38 X 10⁻⁵ N

F₁ and F₂ act at 60 degree so their resultant will be calculated as follows

R² = (91.19)² +(182.38)² + 2 X 91.19 X 182.38 Cos 60

R² = 58209.30

R = 241.27 N

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]

The electric field strength between the disks can be found using the formula E = (k × q₁ × q₂) / (q × r²), where k is the Coulomb constant, q₁ and q₂ are the charges on the disks, and r is the distance between them. To calculate the launch speed of the proton, we can use the conservation of energy principle and equate the potential energy gained by the proton with its kinetic energy.

Part A:

To find the electric field strength between the disks, we can use the formula:

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

The electric force between the disks can be calculated using the formula:

Electric force (F) = (k × q₁ × q₂) / r₂

where k is the Coulomb constant, q₁ and q₂ are the charges on the disks, and r is the distance between them.

Substituting the given values, we get:

E = (k × q₁ × q₂) / (q × r₂)

Now, we can calculate the electric field strength.

Part B:

To calculate the launch speed of the proton, we can use the conservation of energy principle. The potential energy gained by the proton when moved from negative disk to positive disk can be equated to its kinetic energy.

Potential energy (PE) = Kinetic energy (KE)

PE = qV (where V is the potential difference between the disks)

KE = (1/2)mv2 (where m is the mass of the proton, v is the launch speed)

Equating the two equations, we get:

qV = (1/2)mv2

Now, we can solve for the launch speed.

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The electric field strength between the disks is around 79402.08N/C. The proton must be launched at a speed of roughly 2.86*10^6 m/s to just barely reach the positive disk.

This problem involves concepts from electrodynamics and quantum mechanics. To answer Part A, we can use the formula E=σ/ε0 where σ is the surface charge density and ε0 is the permittivity of free space. The surface charge density σ=Q/A where Q is charge and A is the area of the disk, A = π*(d/2)^2. Plugging this in, we get E=~79402.08 N/C.

For Part B, the energy that a proton must have to reach the other plate is equal to the work done by the electric field in bringing the proton from the negative to the positive plate, which is W=qE*d, where q is the charge of the proton and d is the distance between the plates. Equating this to kinetic energy,1/2mv^2, we get v = sqrt((2*q*E*d)/m). Substituting appropriate values gives us v=~2.86*10^6 m/s.

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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:

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:

a) 6.95 m/s

b) 1.42 seconds

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration due to gravity = 9.81 m/s²

[tex]v^2-u^2=2as\\\Rightarrow -u^2=2as-v^2\\\Rightarrow -u^2=2\times -9.81\times 2.46-0^2\\\Rightarrow u=\sqrt{2\times 9.81\times 2.46}\\\Rightarrow u=6.95\ m/s[/tex]

a) The vertical speed when it leaves the ground. is 6.95 m/s

[tex]v=u+at\\\Rightarrow t=\frac{v-u}{a}\\\Rightarrow t=\frac{0-6.95}{-9.81}\\\Rightarrow t=0.71\ s[/tex]

Time taken to reach the maximum height is 0.71 seconds

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow 2.46=0t+\frac{1}{2}\times 9.81\times t^2\\\Rightarrow t=\sqrt{\frac{2.46\times 2}{9.81}}\\\Rightarrow t=0.71\ s[/tex]

Time taken to reach the ground from the maximum height is 0.71 seconds

b) Time it stayed in the air is 0.71+0.71 = 1.42 seconds

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:

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.