The wavelength of light is 5000 angstrom. Express it in nm and m.

Answers

Answer 1

Answer:

1 angstrom = 0.1nm

5000 angstrom = 5000/1 × 0.1nm

= 500nm

[tex]1 \: angstrom = 1 \times {10}^{ - 10} m[/tex]

5000 angstrom = 5000 × 1 × 10^-10

= 5 × 10^-7 m

Hope this helps you


Related Questions

Un tubo de acero de 40000 kilómetros forma un anillo que se ajusta bien a la circunferencia de la tierra. Imagine que las personas junto a él respiran para calentarlo con su aliento y aumentar su temperatura 1 grado Celsius. El tubo se hace más largo. También ya no queda ajustado. ¿A qué distancia sube sobre sobre el nivel del suelo? (solo tomar en cuenta la expansión radial al centro de la tierra, y aplicar la fórmula geométrica que relaciona la circunferencia C con el radio r: C= 2πr).

Answers

Answer:

82.76m

Explanation:

In order to find the distance of the steel ring to the ground, when its temperature has raised by 1°C, you first calculate the radius of the steel tube before its temperature increases.

You use the formula for the circumference of the steel ring:

[tex]C=2\pi r[/tex]    (1)

C: circumference of the ring = 40000 km = 4*10^7m (you assume the circumference is the length of the steel tube)

you solve for r in the equation (1):

[tex]r=\frac{C}{2\pi}=\frac{4*10^7m}{2\pi}=6,366,197.724m[/tex]

Next, you use the following formula to calculate the change in the length of the tube, when its temperature increases by 1°C:

[tex]L=Lo[1+\alpha \Delta T][/tex]         (2)

L: final length of the tube = ?

Lo: initial length of the tube = 4*10^7m

ΔT = change in the temperature of the steel tube = 1°C

α: thermal coefficient expansion of steel = 13*10^-6 /°C

You replace the values of the parameters in the equation (2):

[tex]L=(4*10^7m)(1+(13*10^{-6}/ \°C)(1\°C))=40,000,520m[/tex]

With the new length of the tube, you can calculate the radius of a ring formed with the tube. You again solve the equation (1) for r:

[tex]r'=\frac{C}{2\pi}=\frac{40,000,520m}{2\pi}=6,366,280.484m[/tex]

Finally, you compare both r and r' radius:

r' - r = 6,366,280.484m - 6,366,197.724m = 82.76m

Hence, the distance to the ring from the ground is 82.76m

Explanation:

Un tubo de acero de 40000 kilómetros forma un anillo que se ajusta bien a la circunferencia de la tierra. Imagine que las personas junto a él respiran para calentarlo con su aliento y aumentar su temperatura 1 grado Celsius. El tubo se hace más largo. También ya no queda ajustado. ¿A qué distancia sube sobre sobre el nivel del suelo? (solo tomar en cuenta la expansión radial al centro de la tierra, y aplicar la fórmula geométrica que relaciona la circunferencia C con el radio r: C= 2πr).

a child rides her bicycle in her level drive way. what form of energy does she possess A. Elastic B. Nuclear C. Potential D. Kinetic

Answers

Answer:

Kinetic Energy

Explanation:

Kinetic energy is energy due to motion.

Answer:

D

Explanation:

Elastic energy is energy stored in a object when there is a strain or compression on the object.

Nuclear energy is the energy found in the nucleus of an atom.

Potential energy is energy that an object stores because of its position to other objects.

Kinetic energy is the energy that an object has due to motion.

The child is riding her bicycle, therefore the child is in motion. So, the correct answer must be D. Kinetic energy

The skier starts from rest. The total distance travelled by the skier during the descent is 2800 m. The average resistive force on the skier is 220 N. Calculate the work done against the resistive force

Answers

Answer:

Explanation:

Force equal to resistive force will be applied for movement . So force applied

F = 220 N .

displacement = 2800 m

work done against resistive force

= force x displacement

= 220 x 2800 J

= 6.16 x 10⁵ J .

Magnetic field lines exit out of the . Magnetic field lines enter into the . Magnetic field lines travel around a bar magnet in

Answers

Answer:

Magnetic field lines exit out of the North pole . Magnetic field lines enter into the South pole. Magnetic field lines travel around a bar magnet in closed loops.

Explanation:

Magnetic field lines shows the direction of a magnetic force and how it acts, it gives the direction of the magnetic field at that point in time.

For a bar magnetic, the magnetic field lines runs from the north pole to the south pole, i.e. it exits the north pole and enters into the south pole. This magnetic field lines also go through the magnet forming closed loops without ends.

Answer:

Magnetic field lines exit out of the

✔ north pole

.

Magnetic field lines enter into the

✔ south pole

.

Magnetic field lines travel around a bar magnet in

✔ a closed loop

.

Explanation:

Guys I'm in kind of a PICKLE!!!!!! I know people say it a lot but I will give Brainiest to the best explained answer. Determine the net force charge acting at q1 (+ 2.0 × 10^-5C), caused by q2 (-4.0 × 10-5 C) and q3 (-4.0 × 10^-5C). They create a right angles triangle, where q1 is at the 90° corner

Determine the net electric field acting at q1

Answers

Answer:

E≅1.2×10^7 N/C

Explanation:

First off I'd like to say that I'm taking "net electric field" to mean that they don't want this answer to be put into vector component form and instead want magnitudes. Sometimes the wording of these questions throws me off, so sorry ahead of time if that's what they want from you!

Edit: I ended up adding it anyways ;P

Since we are observing the net electric field acting at q1, we need to use the formula:  [tex]E=k\frac{q}{r^{2} }[/tex]

And since we are observing the effects of multiple charges at once...

E=ΣE, which just means wee need to add all the observed electric fields together:

ΣE= [tex]k\frac{q2}{r^{2} } +k\frac{q3}{r^{2} }[/tex]

Since we are observing [static] electric fields here, we don't actually need q1's charge. (Though if you wanted to find the net force you would.) Now, before we start plugging values in, let's acknowledge what we know. We know that:

q2=q3they are the same distance from q1

These are actually really nice to have, because now we can simplify our expression to:

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

Now let's plug in our values and get an answer out.

E= 2(8.99×10^9)(4×10^-5)/(0.24)

Plugging all that in, I get:

E≅1.2×10^7 N/C

If you end up needing the net force, F=(q1)(E). That is, you just multiply the electric field by the value of q1. And again, if your teacher wants the answer in vector component form, then the answer will look different.

Let me know what doesn't make sense, or if I got something wrong. Good luck with AP Phy.!

Edit: I put the component form for my answer in the attachment. I also noticed a small calculator related error in my original answer. I updated that to match the new one.

Can someone please help me with this ? Neon gas in a container was heated from 20°C to 120°C. It's new volume is 150ml. What was the original volume ?

Answers

Answer:

110 mL

Explanation:

Ideal gas law:

PV = nRT

Assuming the container isn't rigid, and the pressure is constant, then:

V/T = V/T

Plug in values (remember to use absolute temperature).

V / 293 K = 150 mL / 393 K

V = 110 mL

Two identical resistors are connected in parallel across a 26-V battery, which supplies them with a total power of 7.1 W. While the battery is still connected, one of the resistors is heated so that its resistance doubles. The resistance of the other resistor remains unchanged. Find (a) the initial resistance of each resistor, and (b) the total power delivered to the resistors after one resistor has been heated.

Answers

Answer:

A) R = 190.42 Ω

B) P = 5.325 W

Explanation:

We are given;

Total power;P_tot = 7.1 W

Voltage;V = 26 V

A)We are told that while the battery is still connected, one of the resistors is heated, so that its resistance doubles.

Thus, the power is doubled.

Now, formula for power is;

P = IV

Thus, since power is doubled, we have;

P = 2(IV)

Now, formula for current is; I = V/R

So, P = 2V²/R

Making R the subject, we have;

R = 2V²/P

In this question, P is p_total = 7.1 W

Thus;

R = (2 × 26²)/7.1

R = 190.42 Ω

B) Now, the resistance of the resistors are R and 2R.

Formula for power in this context is;

P = V²/R

Thus,

Total power delivered to the resistors is;

P = V²/R + V²/2R

P = 3V²/2R

P = (3 × 26²)/(2 × 190.42)

P = 5.325 W

What can happen if a body moves through speed of light

Answers

As waves travel into the denser medium, they slow down and wavelength decreases.

Explanation:

The denser the medium the slower the waves (speed of light) travels.

◦•●◉✿When the body approaches the speed of light, the body's length appears to contract in the direction of travel, and its mass appears to increase from the point of view of a stationary observer. Only photons move to light velocity. They don´t have length.✿◉●•◦

A car speeds over a hill past point A, as shown in the figure. What is the maximum speed the car can have at point A such that its tires will not leave the track? Round to one decimal place and include units. Image:

Answers

Answer:

11.8 m/s

Explanation:

At the top of the hill, there are two forces on the car: weight force pulling down (towards the center of the circle), and normal force pushing up (away from the center of the circle).

Sum of forces in the centripetal direction:

∑F = ma

mg − N = m v²/r

At the maximum speed, the normal force is 0.

mg = m v²/r

g = v²/r

v = √(gr)

v = √(9.8 m/s² × 14.2 m)

v = 11.8 m/s

What type of image does a concave lens form? A. real image B. magnified image C. virtual image D. reverse image

Answers

the answer to this question is C. virtual image

Answer part (d) please

Answers

Answer:

MARK me brainliest please and follow my page

Explanation:

All you have to do to get the average speed is to calculate the total distance covered and divide it by the total time taken

= 16/18 = 0.88m/s

Average speed = (distance covered) / (time to cover the distance)

For the full 18 seconds described by the graph . . .

Average speed = (16 meters) / (18 seconds)

Average speed = (16 / 18) m/s

Average speed =  0.89 m/s

What kind of substance can you pour from one container into another without a change in volume

Answers

Liquid is the answer
Please mark brainliest please

Answer:

Liquids

Explanation:

Liquids take up the shape of the container it is poured into but will never change its volume.

Un depósito de gran superficie se llena de agua hasta una altura de 0,3 m. En el fondo del depósito hay un orificio de 5 cm2 de sección por el que sale el agua con un chorro continuo. A) ¿Qué cantidad de líquido saldrá del depósito expresada en m3/s?

Answers

Answer:

a) El caudal de salida del chorro es [tex]1.213\times 10^{-3}\,\frac{m^{3}}{s}[/tex].

Explanation:

a) Asúmase que el tanque se encuentra a presión atmósferica y que la sima del tanque tiene una altura de 0 metros. La rapidez de salida del chorro del depósito se determined a partir del Principio de Bernoulli, cuya línea de corriente entre la cima y la sima del tanque queda descrita por la siguiente ecuación:

[tex]\Delta z = \frac{v_{out}^{2}}{2\cdot g}[/tex]

Donde:

[tex]\Delta z[/tex] - Diferencia de altura, medida en metros.

[tex]g[/tex] - Constante gravitacional, medida en metros por segundo al cuadrado.

[tex]v_{out}[/tex] - Rapidez de salida del chorro, medida en metros por segundo.

Se despeja la rapidez de salida del chorro:

[tex]v_{out} = \sqrt{2\cdot g \cdot \Delta z}[/tex]

Si [tex]g = 9.807\,\frac{m}{s^{2}}[/tex] y [tex]\Delta z = 0.3\,m[/tex], entonces la rapidez de salida del chorro es:

[tex]v_{out} = \sqrt{2\cdot \left(9.807\,\frac{m}{s^{2}} \right)\cdot (0.3\,m)}[/tex]

[tex]v_{out} \approx 2.426\,\frac{m}{s}[/tex]

Ahora, la cantidad de líquido que sale del depósito por unidad de tiempo se obtiene al multiplicar la rapidez de salida del chorro por el área transversal del orificio. Esto es:

[tex]\dot V_{out} = v_{out}\cdot A_{t}[/tex]

Donde:

[tex]v_{out}[/tex] - Rapidez de salida del chorro, medida en metros por segundo.

[tex]A_{t}[/tex] - Área transversal del orificio, medido en metros cuadrados.

[tex]\dot V_{out}[/tex] - Caudal de salida del chorro, medido en metros cúbicos por segundo.

Dado que [tex]v_{out} = 2.426\,\frac{m}{s}[/tex] y [tex]A_{t} = 5\,cm^{2}[/tex], el caudal de salida del chorro es:

[tex]\dot V_{out} = \left(2.426\,\frac{m}{s} \right)\cdot (5\,cm^{2})\cdot \left(\frac{1}{10000}\,\frac{m^{2}}{cm^{2}} \right)[/tex]

[tex]\dot V_{out} = 1.213\times 10^{-3}\,\frac{m^{3}}{s}[/tex]

El caudal de salida del chorro es [tex]1.213\times 10^{-3}\,\frac{m^{3}}{s}[/tex].

Which types of materials have a molecular structure that makes them good conductors? Explain how understanding this relationship can help people design more efficient technologies

Answers

Answer:

Most conductors are made from elements metallic elements

Metallic elements are those elements grouped as alkali metals, alkaline earth metals, basic metals, rare earth elements, actinides, and basic metals

Conductors of electricity carry electricity by means of the movable charged particles present in the conducting material

The movable charged particle are the electrons which are most mobile in metals because of their crystalline structure and available valence electron, which are the electrons in the outermost orbit of an atom and hence freest to move about within the material mass

As such the availability of free electrons determine conductivity of materials

Explanation:

Understanding the relationship between electrical conductivity and the structure of atoms and molecules will contribute to the design of efficiency and reliability of technologies deployed to remote locations such as the Moon, by ensuring the best possible output from the electric control inputs and the redesigning of existing electrical installations using efficient amount of manufacturing materials, thereby saving the environment and costs.

Free electo

Electric conductors possess movable electrically charged particles, referred to as "electrons" in metals. When an electric charge is applied to a metal at certain points, the electrons will move and allow electricity to pass through. Materials with high electron mobility are good conductors and materials with low electron mobility are not good conductors, instead referred to as "insulators."

Silver is the best conductor of electricity because it contains a higher number of movable atoms (free electrons). For a material to be a good conductor, the electricity passed through it must be able to move the electrons; the more free electrons in a metal, the greater its conductivity. However, silver is more expensive than other materials and is not normally used unless it is required for specialized equipment like satellites or circuit boards. Copper is less conductive than silver but is cheaper and commonly used as an effective conductor in household appliances. Most wires are copper-plated and electromagnet cores are normally wrapped with copper wire. Copper is also easy to solder and wrap into wires, so it is often used when a large amount of conductive material is required.

A student throws a 120 g snowball at 7.5 m/s at the side of the schoolhouse, where it hits and sticks. What is the magnitude of the average force on the wall if the duration of the collision is 0.15 s

Answers

Answer:

The magnitude of the average force on the wall during the collision is 6 N.

Explanation:

Given;

mass of snowball, m = 120 g = 0.12 kg

velocity of the snowball, v = 7.5 m/s

duration of the collision between the snowball and the wall, t = 0.15 s

Magnitude of the average force can be calculated by applying Newton's second law of motion;

F = ma

where;

a is acceleration = v / t

a = 7.5 / 0.15

a = 50 m/s²

F = ma

F = 0.12 x 50

F = 6 N

Therefore, the magnitude of the average force on the wall during the collision is 6 N.

How many atoms of oxygen in the chemical formula 2 Ca(ClO2)2?

Answers

Answer:

8

Explanation:

Ca(ClO2)2  -    2*2 = 4 Oxygen atoms

2 Ca(ClO2)2  - 2*4 = 8 Oxygen atoms

What is the wavelength of a wave that has a speed of 3 km/s and a frequency of 12 Hz? A. 36 km B. 3.6 km C. 0.25 km D. 4 km

Answers

Answer:

c. 0.25km

Explanation:

v=f x wavelength

3000 = 12 x wavelength

wavelength = 3000/12 = 250m

250m to km

To convert m to km, we divide by 1000

250/1000 =0.25km

wavelength = 0.25km

10 advantages of friction

Answers

Answer:

1. it helps to change the direction.

2. it helps us to walk on ground.

3. it helps the vechils to break while moving.

4. helps in changing one form of enegry to another form. eg when we rub our hands we feel heat energy.

5. it opposites the force.

6. it helps us to change shape of objects.eg we roll the dough to make it roti.

7. it changes the state of body from rest motion.eg when we push any obj from inclined plane it moves.

i all know is just 7..

A pendulum at position A is released and swings through position B to position Con the other side.
B
1. Describe the total mechanical energy at each of the following positions. (3)
A.
B.
C

Answers

Explanation:

Given the conditions A,B and C when the pendulum is released, at point A the initial velocity of the pendulum is zero(0), the potential energy stored is maximum(P.E= max),

the conditions can be summarized bellow

point A

initial velocity= 0

final velocity=0

P.E= Max

K.E= 0

point B

initial velocity= maximum

final velocity=maximum

P.E=K.E

point C

initial velocity= min

final velocity=min

P.E= 0

K.E= max

Rick is moving a wheelbarrow full of bricks out to the curb. The bricks in the wheelbarrow weigh more than Rick is able to carry, yet he is able to move the bricks. How is this possible?

Answers

Answer is given below

Explanation:

This is happen because here when Rick walks with full loaded wheelbarriow of bricks, he able to move it because Rick lifts the wheelbarrow handle So, most of the weight of full loaded wheelbarrow's load goes on that's wheel and due to friction force between wheel and surface it can easy to move He uses force to rotate the wheel, much more than the force applied to the rim of the wheel on the axis of rotation or torque

coma Narrows Bridge across the Puget Sound collapsed. The bridge was a suspension bridge. The wind blowing through the narrows matched the natural frequency of the bridge. This resulted in a large movement of roadway, which eventually caused the bridge to fail. What characteristic of waves caused the bridge to collapse

Answers

Answer:

amplitude

Explanation:

Amplitude is the characteristic of waves which caused the bridge to collapse.   Amplitude of a wave is the maximum amount of displacement of a particle occurs in the medium from its rest position. When the frequency of a wave reaches the natural frequency of the bridge, the oscillation of the bridge produce an amplitude where it causing the destruction of the bridge which is called Resonance. So we can say that amplitude is the characteristic of waves which is responsible for the collapse of the bridge.

Answer: C.interference, because constructive interference occurred when the wind frequency matched the natural frequency of the bridge

Explanation:

can all alpha beta and gamma radiation treat cancer

Answers

Answer:

No

Explanation:

Only a few of alpha beta and gamma radiation can treat cancer

A ball has a mass of 0.25 kg and is moving to the right at 1.0 m/s. It hits a ball of mass 0.15 kg that is initially at rest. After the collision, the 0.15 kg ball moves off to the right with a velocity of 0.75 m/s. What is the final velocity of the 0.25 kg ball? 0.42 m/s to the right 0.42 m/s to the left 0.55 m/s to the right 0.55 m/s to the left

Answers

Answer:

C-0.55 m/s to the right

Explanation:

edg

Answer:

0.55 to the right

Explanation:

Write the importance of sound in our life.

Answers

It communicates messages in many different ways, soothes the mind and also helps relieve stress from our everyday lives. It is a sound that is in a controlled and tuned type of matter, relying on frequency and pitch of the voice. Not only is basic sound important, but I believe music is crucial to human life as well.

Will give brainliest for the person who answers first
A 75W television is on for 400s, how much energy is transferred, leave your answer in Joules

Answers

Energy = power x time
So, 75W x 400s = 30,00J

Answer:

30,000 J

Explanation:

Energy (Joules) = Power (Watt) × Time (seconds)

Energy (J) = 75 × 400

Energy (J) = 30,000

30,000 Joules of energy will be transferred.

1. Find the energy required to melt 255g of ice at 0°C into water at 0°C

Answers

Answer:

E = 85170 J (/ 85.2 kJ)

Explanation:

Take the latent heat of fusion of water be 334J / g.

From the equation E = ml,

E = energy required (unknown),

mass m = 255g,

latent heat of fusion l = 334J / g,

E = 255 x 334

E = 85170 J (/ 85.2 kJ)

state the observations made in this experiment ​

Answers

Answer:

the water is going to boil and the mercury ill melt and shoot the cork out the bottom of the tube

Explanation:

On a horizontal frictionless surface a mass M is attached to two light elastic strings both having length l and both made of the same material. The mass is displaced by a small displacement Δy such that equal tensions T exist in the two strings, as shown in the figure. The mass is released and begins to oscillate back and forth. Assume that the displacement is small enough so that the tensions do not change appreciably. (a) Show that the restoring force on the mass can be given by F = -(2T∆y)/l (for small angles) (b) Derive an expression for the frequency of oscillation.

Answers

Answer:

ω = √(2T / (mL))

Explanation:

(a) Draw a free body diagram of the mass.  There are two tension forces, one pulling down and left, the other pulling down and right.

The x-components of the tension forces cancel each other out, so the net force is in the y direction:

∑F = -2T sin θ, where θ is the angle from the horizontal.

For small angles, sin θ ≈ tan θ.

∑F = -2T tan θ

∑F = -2T (Δy / L)

(b) For a spring, the restoring force is F = -kx, and the frequency is ω = √(k/m).  (This is derived by solving a second order differential equation.)

In this case, k = 2T/L, so the frequency is:

ω = √((2T/L) / m)

ω = √(2T / (mL))

Un contenedor de 1800 N está en reposo sobre un plano inclinado a 28°, el coeficiente de fricción entre el contenedor y el plano es de 0.4. Calcule la fuerza P paralela al plano y dirigida hacia arriba de éste que hará que el contenedor se mueva hacia arriba con una velocidad constante.

Answers

Answer:

F = 1480.77N

Explanation:

In order to calculate the required force to push the container with a constant velocity, you take into account the the sum of force on the container is equal to zero. Furthermore, you have for an incline the following sum of forces:

[tex]F-Wsin\alpha-F_r=0\\\\F-Wsin\alpha-N\mu cos\alpha=0\\\\F-Wsin\alpha-W\mu cos\alpha=0[/tex]     (1)

F: required force = ?

W: weight of the container = 1800N

N: normal force = weigth

α: angle of the incline = 28°

g: gravitational acceleration = 9.8m/s^2

μ: coefficient of friction = 0.4

You solve the equation (1) for F and replace the values of the other parameters:

[tex]F=W(sin\alpha+\mu cos\alpha)\\\\F=(1800N)(sin28\°+(0.4)cos28\°)=1480.77N[/tex]

The required force to push the container for the incline with a constant velocity is 1480.77N

A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of Ï3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

Answers

Answer:

t = 5π/18 + 2nπ/3 or π/6 + 2nπ/3 where n is a natural number

Explanation:

To solve the problem/ we first write the differential equation governing the motion. So,

[tex]m\frac{d^{2} x}{dt^{2} } = -kx \\ m\frac{d^{2} x}{dt^{2} } + kx = 0\\\frac{d^{2} x}{dt^{2} } + \frac{k}{m} x = 0[/tex]

with m = 1 slug and k = 9 lb/ft, the equation becomes

[tex]\frac{d^{2} x}{dt^{2} } + \frac{9}{1} x = 0\\\frac{d^{2} x}{dt^{2} } + 9 x = 0[/tex]

The characteristic equation is

D² + 9 = 0

D = ±√-9 = ±3i

The general solution of the above equation is thus

x(t) = c₁cos3t + c₂sin3t

Now, our initial conditions are

x(0) = -1 ft and x'(0) = -√3 ft/s

differentiating x(t), we have

x'(t) = -3c₁sin3t + 3c₂cos3t

So,

x(0) = c₁cos(3 × 0) + c₂sin(3 × 0)

x(0) = c₁cos(0) + c₂sin(0)

x(0) = c₁ × (1) + c₂ × 0

x(0) = c₁ + 0

x(0) = c₁ = -1

Also,

x'(0) = -3c₁sin(3 × 0) + 3c₂cos(3 × 0)

x'(0) = -3c₁sin(0) + 3c₂cos(0)

x'(0) = -3c₁ × 0 + 3c₂ × 1

x'(0) = 0 + 3c₂

x'(0) = 3c₂ = -√3

c₂ = -√3/3

So,

x(t) = -cos3t - (√3/3)sin3t

Now, we convert x(t) into the form x(t) = Asin(ωt + Φ)

where A = √c₁² + c₂² = √[(-1)² + (-√3/3)²] = √(1 + 1/3) = √4/3 = 2/√3 = 2√3/3 and Ф = tan⁻¹(c₁/c₂) = tan⁻¹(-1/-√3/3) = tan⁻¹(3/√3) = tan⁻¹(√3) = π/3.

Since tanФ > 0, Ф is in the third quadrant. So, Ф = π/3 + π = 4π/3

x(t) = (2√3/3)sin(3t + 4π/3)

So, the velocity  v(t) = x'(t) = (2√3)cos(3t + 4π/3)

We now find the times when v(t) = 3 ft/s

So (2√3)cos(3t + 4π/3) = 3

cos(3t + 4π/3) = 3/2√3

cos(3t + 4π/3) = √3/2

(3t + 4π/3) = cos⁻¹(√3/2)

3t + 4π/3 = ±π/6 + 2kπ    where k is an integer

3t  = ±π/6 + 2kπ - 4π/3

t  = ±π/18 + 2kπ/3 - 4π/9

t = π/18 + 2kπ/3 - 4π/9 or -π/18 + 2kπ/3 - 4π/9

t = π/18 - 4π/9 + 2kπ/3  or -π/18 - 4π/9 + 2kπ/3

t = -7π/18 + 2kπ/3 or -π/2 + 2kπ/3

Since t is not less than 0, the values of k ≤ 0 are not included

So when k = 1,

t = 5π/18 and π/6. So,

t = 5π/18 + 2nπ/3 or π/6 + 2nπ/3 where n is a natural number

The time interval at which the mass will head downward at the velocity of 3 ft/s is t = 5π/18 + 2nπ/3.

Given data:

The mass suspended from spring is, m = 1 slug.

The spring constant is, k = 9 lb/ft.

The magnitude of upward velocity is, v = 3 ft/s.

The magnitude of downward velocity is, v' = 3 ft/s.

The given problem can be resolved by framing a differential equation that governs the motion of spring. The differential equation governing the motion of spring is,

[tex]m \dfrac{d^{2}x}{dt^{2}}=-kx\\\\\\\dfrac{d^{2}x}{dt^{2}}+\dfrac{k}{m}x=0[/tex]

With m = 1 slug and k = 9 lb/ft, the equation becomes

[tex]\dfrac{d^{2}x}{dt^{2}}+\dfrac{9}{1}x=0\\\\\\\dfrac{d^{2}x}{dt^{2}}+9x=0[/tex]  

Now, the characteristic equation is,

D² + 9 = 0

D = ±√-9 = ±3i

And the general solution of the above equation is,

x(t) = c₁cos3t + c₂sin3t

Now, our initial conditions are

x(0) = -1 ft and x'(0) = -√3 ft/s

differentiating x(t), we have

x'(t) = -3c₁sin3t + 3c₂cos3t

So,

x(0) = c₁cos(3 × 0) + c₂sin(3 × 0)

x(0) = c₁cos(0) + c₂sin(0)

x(0) = c₁ × (1) + c₂ × 0

x(0) = c₁ + 0

x(0) = c₁ = -1

Also,

x'(0) = -3c₁sin(3 × 0) + 3c₂cos(3 × 0)

x'(0) = -3c₁sin(0) + 3c₂cos(0)

x'(0) = -3c₁ × 0 + 3c₂ × 1

x'(0) = 0 + 3c₂

x'(0) = 3c₂ = -√3

c₂ = -√3/3

So,

x(t) = -cos3t - (√3/3)sin3t

Now, we convert x(t) into the form x(t) = Asin(ωt + Φ)

where A = √c₁² + c₂²

              = √[(-1)² + (-√3/3)²]

              = √(1 + 1/3) = √4/3 = 2/√3 = 2√3/3

and Ф = tan⁻¹(c₁/c₂) = tan⁻¹(-1/-√3/3) = tan⁻¹(3/√3) = tan⁻¹(√3) = π/3.

Since tanФ > 0, Ф is in the third quadrant. So, Ф = π/3 + π = 4π/3

x(t) = (2√3/3)sin(3t + 4π/3)

So, the velocity v(t) = x'(t) = (2√3)cos(3t + 4π/3)

We now find the times when v(t) = 3 ft/s

So (2√3)cos(3t + 4π/3) = 3

cos(3t + 4π/3) = 3/2√3

cos(3t + 4π/3) = √3/2

(3t + 4π/3) = cos⁻¹(√3/2)

3t + 4π/3 = ±π/6 + 2kπ    

where k is an integer

3t  = ±π/6 + 2kπ - 4π/3

t  = ±π/18 + 2kπ/3 - 4π/9

t = π/18 + 2kπ/3 - 4π/9 or -π/18 + 2kπ/3 - 4π/9

t = π/18 - 4π/9 + 2kπ/3  or -π/18 - 4π/9 + 2kπ/3

t = -7π/18 + 2kπ/3 or -π/2 + 2kπ/3

Since t is not less than 0, the values of k ≤ 0 are not included

So when k = 1,

t = 5π/18 and π/6.

t = 5π/18 + 2nπ/3

here, n is a natural number.

Thus, we can conclude that the time interval at which the mass will head downward at the velocity of 3 ft/s is t = 5π/18 + 2nπ/3.

Learn more about the differential equation here:

https://brainly.com/question/14620493  

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