A 0.5 kg block is attached to a spring (k = 12.5 N/m). The damped frequency is 0.2% lower than the natural frequency, (a) What is the damping constant? (b) How does the amplitude vary with time? (c) Determine the critical damping constant?

Answer:

Acceleration of the car, [tex]a=8\ m/s^2[/tex]

Explanation:

Given that,

Initial speed of the car, u = 0 (at rest)

Final speed of the car, v = 40 m/s

Distance covered, s = 100 m

We need to find the acceleration of the car. It can be calculated using third equation of motion as :

[tex]v^2-u^2=2as[/tex]

a is the acceleration of the car

[tex]a=\dfrac{v^2}{2s}[/tex]

[tex]a=\dfrac{(40)^2}{2\times 100}[/tex]

[tex]a=8\ m/s^2[/tex]

So, the acceleration of the car is [tex]8\ m/s^2[/tex]. Hence, this is the required solution

Answer:

The correct option is 'D': Equal and opposite force.

Explanation:

We know from Newton's third law of motion

"For every action there exists an equal and opposite reaction".

The statement is further explained as

If there exists a force from Body 1 to Body 2 then Body 2 must also exert an equal and opposite force on the Body 1.

In simple terms we can say that the action and the reaction forces exist in pairs and are always present if any one among them is present.

Answer:

(1) [tex]46.86^\circ[/tex]

(2) Diagram has been attached in the solution.

Explanation:

This question is from projectile motion.

From the given question, we will discuss the motion of the basket ball only in the vertical direction from which we will be able to find out the angle of the initial velocity with the horizontal with which it should be shoot to enter the hoop.

Part (1):

Let us assume:

Using the equation of motion for constant acceleration, we have

[tex]y_f-y_i=u_yt+\dfrac{1}{2}a_yt^2\\\Rightarrow 3.05-2.1=u\sin \theta (0.8) +\dfrac{1}{2}\times (-9.8)(0.8)^2\\\Rightarrow 0.95=7\times \sin \theta (0.8) -3.136\\\Rightarrow 0.95=5.6\sin \theta -3.136\\\Rightarrow 5.6\sin \theta= 0.95+3.136\\\Rightarrow 5.6\sin \theta= 4.086\\\Rightarrow \sin \theta= \dfrac{4.086}{5.6}\\\Rightarrow \sin \theta=0.729\\\Rightarrow \theta=\sin^{-1}(0.729)\\\Rightarrow \theta=46.86^\circ[/tex]

Hence, the angle of the shoot of the basket ball with the horizontal is [tex]46.86^\circ[/tex] such that it reaches the hoop on time.

Part (2):

For this part, a diagram has been attached.

Answer:

the radii of curvature is 30 cm.

Explanation:

given,

object is place at = 45 cm

image appears at = 90 cm

focal length = ?

refractive index = 1.5

radii of curvature = ?

[tex]\dfrac{1}{f} = \dfrac{1}{u} +\dfrac{1}{v}[/tex]

[tex]\dfrac{1}{f} = \dfrac{1}{45} +\dfrac{1}{90}[/tex]

f = 30 cm

using lens formula

[tex]\dfrac{1}{f} = (n-1)(\dfrac{1}{R_1} -\dfrac{1}{R_2})[/tex]

[tex]R_1 = R\ and\ R_2 = -R[/tex]

[tex]\dfrac{1}{f} = (n-1)(\dfrac{1}{R} +\dfrac{1}{R})[/tex]

[tex]R = (n -1)\ f [/tex]

[tex]R = 2(1.5 -1)\ 30[/tex]

R = 30 cm

hence, the radii of curvature is 30 cm.

Answer:

1) -13570.3 Btu

2) 2.25

Explanation:

The formula for potential energy is:

[tex]E_p = m*g*h[/tex]

Where m is the mass in slugs, g is the gravity and h is the change in height. Soliving the equation:

[tex]E_p = 2000lb*\frac{1 slug}{32.174 lb} *32.174 \frac{ft}{s^2} *(0ft-5280ft) = -10560000 lbf*ft * \frac{1 Btu}{778.17 lbf*ft} = -13570.3 Btu[/tex]

The car would lose -13570.3 Btu in potential energy.

2) The kinetic energy is given by:

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

Then:

[tex]\frac{E_k_2}{E_k_1} = \frac{\frac{1}{2}mv_2^2}{\frac{1}{2}mv_1^2}  = \frac{\frac{1}{2}m(1.5v_1)^2}{\frac{1}{2}mv_1^2}=1.5^2\frac{v_1^2}{v_1^2} = 2.25[/tex]

Planets like Mars exhibit direct and retrograde motion, resulting in complex paths across the night sky, which differ from the Sun's steady eastward movement along the ecliptic through the zodiac signs. Earth's faster orbit leads to the phenomenon of retrograde motion when observing other planets.

The motion of planets like Mars among the stars of the zodiac is different from the motion of the Sun because planets exhibit both direct motion and retrograde motion in their paths across the sky, as observed from Earth. Planets periodically appear to move in the opposite direction in the sky, known as retrograde motion, while the Sun moves steadily along the ecliptic through the zodiac signs. This distinct planetary motion is due to the orbital speed and path of Earth relative to other planets, leading to the phenomenon where Earth, moving faster along its orbit, occasionally overtakes other planets like Mars, changing how these planets appear to move against the backdrop of stars.

Additionally, planets can have stationary points where they appear to pause before they switch from direct to retrograde motion or vice versa. These motions are perceived from Earth because of the combination of the planets' own orbital movement around the Sun and Earth's simultaneous revolution. By contrast, the Sun follows a consistent eastward path among the fixed stars, circling the zodiac once per year, spending about a month in each zodiac sign.

Answer:

carrier power is 16.1 kW

Explanation:

given data

power transmit P = 20 kW

modulation index m = 0.7

to find out

carrier power

solution

we will apply here power transmit equation that is express as

[tex]P = 1 + \frac{m^2}{2} * carrier power[/tex]   ...............1

put here all value in equation 1  we get carrier power

[tex]P = 1 + \frac{m^2}{2} * carrier power[/tex]

[tex]20 = 1 + \frac{0.7^2}{2} * carrier power[/tex]

solve it we get

carrier power = 16.1

so carrier power is 16.1 kW

Answer:

speed of sound = 321 m/s

Rule of thumb for kilometers: 1 km every 3 seconds

Explanation:

Hi!

If a thunder originates at a place a distance D from you, and it takes time T for its sound to reach you, then:

[tex]V = \frac{D}{T}[/tex]

Whre V is the speed of sound. Time T is the time elapsed between the moment you see the flash (becuase of the assumption of tha it takes no time for the light to reach you) and the moment you hear the thunder. Then

[tex]V = \frac{1mile}{5\;s} =\frac{1609.34 \;m}{5\;s} = 321\frac{m}{s}[/tex]

To calculate the rule in kilometers:

[tex]T = \frac{1\;km}{321*10^{-3} \frac{km}{s}} \approx 3\; s[/tex]

The estimated speed of sound derived from the rule of thumb is approximately 322 m/s. For the rule in kilometers, each five seconds would represent about 1.609 kilometers.

The rule-of-thumb that every five seconds between a lightning flash and the accompanying thunder equates to one mile of distance is based on the speed of sound. To convert this to meters per second we must know that 1 mile approximately equals 1609.34 meters. Therefore, if light travels nearly instantaneously, and one mile is covered in five seconds, the speed of sound can be estimated as about 1609.34 m/5 s or approximately 322 m/s.

For the rule in kilometers, we need to convert miles into kilometers. As 1 mile is approximately 1.609 kilometers, the rule of thumb in kilometers would be that each five seconds would represent 1.609 kilometers.

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The time for projectile motion is determined completely by the vertical motion. A ball thrown vertically upwards with an initial velocity of 20.00 m/s spends approximately 4 seconds in the air and reaches its greatest height after approximately 2 seconds. The time at which the ascending ball reaches a height of 15 m above the ground can be found by setting y = 15 m and solving for t.

The time for projectile motion is determined completely by the vertical motion. Thus, any projectile that has an initial vertical velocity of 20.00 m/s and lands 10.0 m above its starting altitude spends approximately 4 seconds in the air. The greatest height reached by the ball can be determined by using the equation for vertical motion: y = yo + voyt - 1/2gt^2. Plugging in the given values and solving for t, we find that the ball reaches its greatest height after approximately 2 seconds, and the time at which the ascending ball reaches a height of 15 m above the ground can be found by setting y = 15 m and solving for t.

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Answer:

The bones are 16925 years old

Explanation:

We have to use the radioactive decay law and know that the half life of carbon-14 is [tex]t_{\frac{1}{2}}=5730 \, years[/tex]. From this information we can know the decay rate of the carbon 14,

[tex]\lambda=\frac{ln(2)}{t_{\frac{1}{2}}}=1.21\times 10^{-4} s^{-1}[/tex]

Now to know the age of the bones we must directly use the radioactive decay law:

[tex]N(t)=N_0e^{-\lambda t}=0.129N_0[/tex]

Where the rightmost part of the equation comes from the statement that the activity found is just 12.9% of the activity that would be found in a similar live animal. This means that the number of carbon-14 atoms is just 12.9% of what it was at the moment the saber-toothed tiger died.

Solving for t we have:

[tex]t=-\frac{ln(0.129)}{\lambda}=16925 \, years[/tex]

Answer:

The gravitational force is significantly constant.

Explanation:

The gravitational force is expressed as:

F= G*M*m/d^2

G= Gravitational constant

M= in this case it is the mass of the planet

m= mass of the rock

d= distance of the rocks from the ground (suppose both at the same height for better comparison)

At the same time, we know that the force that each rock experiences is equal to the product of the mass due to acceleration:

F=m*a

We can match both expressions for F:

G*M*m/d^2 = m*a

we simplify m, and we obtain that the acceleration is independent of the mass of the attracted bodies:

a= G*M/d^2

Answer:

The the decibel level be when three sleep deprived students are typing furiously in the same room is 64.77 dB.

Explanation:

Given that,

Decibel level = 60.0 dB

Using formula of decibel level

[tex]\beta= 10 log\dfrac{I}{I_{0}}[/tex]

For one student,

[tex]60.0=10 log\dfrac{I}{I_{0}}[/tex]

For 3 students,

[tex]\beta'=10 log\dfrac{3I}{I_{0}}[/tex]

[tex]\beta'=10log 3+10 log\dfrac{I}{I_{0}}[/tex]

[tex]\beta'=4.77+60.0[/tex]

[tex]beta'=64.77\ dB[/tex]

Hence, The the decibel level be when three sleep deprived students are typing furiously in the same room is 64.77 dB.

Answer:

equation of  motion for Bill is

[tex]y(t) = 4.9t^2[/tex]

equation of  motion for Ted is

[tex]y(t) = 2 + (-4.2)(t) + 4.9t^2[/tex]

Explanation:

Taking downward position positive and upward position negative

g = 9.8 m/s^2

equation of  motion for Bill is

[tex]y(t) = y_0 +v_0 t +\frac{1}{2}gt^2[/tex]

[tex]y(t) = 0 + 0(t) +\frac{1}{2}gt^2[/tex]

[tex]y(t) = \frac{1}{2}\times (9.8t)^2[/tex]

[tex]y(t) = 4.9t^2[/tex]

equation of  motion for Ted is

[tex]y_0 = 2m -1m = 2m[/tex]

[tex]y_0 = -4.2 m/s[/tex]

[tex]y(t) = y_0 +v_0 t +\frac{1}{2}gt^2[/tex]

[tex]y(t) = 2 + (-4.2)(t) +\frac{1}{2}gt^2[/tex]

[tex]y(t) = 2 + (-4.2)(t) +\frac{1}{2}\times (9.8t)^2[/tex]

[tex]y(t) = 2 + (-4.2)(t) + 4.9t^2[/tex]

Answer:

Answer:

For Bill:

[tex]x(t)=3-(4.9*t^{2})[/tex]

For Ted:

[tex]x(t)=1+(4.2*t)+(-4.9*t^{2} )[/tex]

Explanation:

For Bill:

[tex]Initial position=x_{0}=3[/tex]

[tex]Initial velocity=v_{0}=0[/tex]

Now using [tex]2^{nd}[/tex] equation of motion,we have

[tex]x-x_{0}=(v_{0} *t)+(1/2*g*t^{2})[/tex]

[tex]x_{0} =3[/tex]  ,[tex]v_{0}=0[/tex]

Thus,equation becomes

[tex]x-3=1/2*g*t^{2}[/tex]

[tex]x=3+(0.5*g*t^{2})[/tex]

Taking acceleration upward positive and downward negative.

[tex]g=-10[/tex] [tex]m/s^{2}[/tex]

[tex]x(t)=3-4.9*t^{2}[/tex]    for bill

For Ted

[tex]x_{0} =1[/tex]

[tex]v_{0}=4.2[/tex] [tex]m/s[/tex]

Using the same equation

[tex]x-x_{0}=(v_{0} *t)+(1/2*g*t^{2})[/tex]

[tex]x_{0}=1[/tex] [tex]m[/tex]

[tex]v_{0}=4.2[/tex] [tex]m/s[/tex]

Substitute values

[tex]x-1=(4.2*t)+(1/2*g*t^{2})[/tex]

[tex]g=-10[/tex] [tex]m/s^{2}[/tex]

Thus equation becomes

[tex]x(t)=1+(4.2*t)+(-4.9*t^{2})[/tex]   for Ted

Answer:

temperature change is 262.06°K

Explanation:

given data

mass = 0.07 kg

velocity = 258 m/s

to find out

what is its temperature change

solution

we know here

heat change Q is is equal to kinetic energy that is

KE = 0.5 × m× v²   ...........1

here m is mass and v is velocity

KE = 0.5 × 0.07 × 258²

KE = 2329.74 J

and we know

Q = mC∆t     .................2

here m is mass and ∆t is change in temperature and C is 127J/kg-K

so put here all value

2329.74 = 0.07 × 127 × ∆t

∆t = 262.06

so temperature change is 262.06°K

Final answer:

The temperature change of a 0.07-kg lead bullet that had been traveling at 258 m/s before coming to a stop is found to be approximately 260°C.

Explanation:

Calculating Temperature Change of a Lead Bullet

To calculate the temperature change of a 0.07-kg lead bullet that comes to a stop after traveling at 258 m/s, we must first determine how much kinetic energy the bullet had before stopping. The kinetic energy (KE) of the bullet can be calculated using the formula KE = 1/2 [tex]mv^2[/tex], where m is the mass of the bullet and v is its velocity. Plugging in the values gives us:

KE = 1/2 * 0.07 kg * (258 [tex]m/s)^2[/tex]

KE = 0.035 kg * 66564 [tex]m^2/s^2[/tex]

KE = 2330.74 J

This kinetic energy is converted to heat energy, which we will denote as Q. The temperature change (\ΔT) is then given by the formula Q = mc\ΔT, where m is the mass of the bullet and c is the specific heat capacity of lead.

Since c for lead is approximately 128 J/kg°C, we can rearrange the formula to solve for \Deltat:

ΔT = Q / (mc)

ΔT = 2330.74 J / (0.07 kg * 128 J/kg°C)

ΔT = 2330.74 J / 8.96 J/°C

ΔT = 259.96°C

Therefore, the temperature change of the bullet is approximately 260°C.

The rate at which the area of the triangle formed by the wall, the ground, and the ladder is changing, at the instant the bottom of the ladder is 36 feet from the wall, is -216 square feet per second.

This question revolves around the concept of related rates in calculus. The area of the triangle formed by the ground, the wall, and the ladder is given by the formula ½ * base * height. In this situation, the base is x, which represents the distance from the wall to the base of the ladder, and the height is the vertical reach of the ladder onto the wall, represented by y (which can be found using the Pythagorean theorem). Differentiating this equation with respect to time gives us dA/dt = ½ (x dy/dt + y dx/dt).

Given that dx/dt is 8 feet per second, and x=36 feet, to find dy/dt, we use the Pythagorean relationship (39² = 36² + y²), and solve for y to get y = 15 feet. Therefore, substituting all these values into the equation we find:** dA/dt = -216 sq.ft/sec**.

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The area of the triangle is changing at a rate of ( -285.6 ) square feet per second at the instant the bottom of the ladder is 36 feet from the wall. The negative sign indicates that the area is decreasing.

Let ( x ) be the distance of the bottom of the ladder from the wall, and ( y) be the height of the ladder on the wall. We are given that [tex]\( \frac{dx}{dt} = 8 \)[/tex] feet per second, and we want to find [tex]\( \frac{dA}{dt} \) when \( x = 36 \) feet.[/tex]

Using the Pythagorean theorem, we have [tex]\( x^2 + y^2 = 39^2 \).[/tex]Differentiating both sides with respect to time ( t ), we get:

[tex]\[ 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0 \][/tex]

We can solve for [tex]\( \frac{dy}{dt} \):[/tex]

[tex]\[ 2y\frac{dy}{dt} = -2x\frac{dx}{dt} \][/tex]

[tex]\[ \frac{dy}{dt} = -\frac{x}{y}\frac{dx}{dt} \][/tex]

Now, we can express the area ( A ) in terms of ( x ) and ( y ):

[tex]\[ A = \frac{1}{2}xy \][/tex]

Differentiating ( A ) with respect to time ( t ), we get:

[tex]\[ \frac{dA}{dt} = \frac{1}{2}\left(x\frac{dy}{dt} + y\frac{dx}{dt}\right) \][/tex]

Substituting [tex]\( \frac{dy}{dt} \)[/tex] from above, we have:

[tex]\[ \frac{dA}{dt} = \frac{1}{2}\left(x\left(-\frac{x}{y}\frac{dx}{dt}\right) + y\frac{dx}{dt}\right) \][/tex]

[tex]\[ \frac{dA}{dt} = \frac{1}{2}\left(-\frac{x^2}{y}\frac{dx}{dt} + y\frac{dx}{dt}\right) \][/tex]

[tex]\[ \frac{dA}{dt} = \frac{1}{2}\left(y - \frac{x^2}{y}\right)\frac{dx}{dt} \][/tex]

We know [tex]\( \frac{dx}{dt} = 8 \)[/tex] feet per second, and we need to find [tex]\( \frac{dA}{dt} \)[/tex]when ( x = 36 ) feet. To find ( y ) at this instant, we use the Pythagorean theorem:

[tex]\[ 36^2 + y^2 = 39^2 \][/tex]

[tex]\[ y^2 = 39^2 - 36^2 \][/tex]

[tex]\[ y^2 = 1521 - 1296 \][/tex]

[tex]\[ y^2 = 225 \][/tex]

[tex]\[ y = 15 \][/tex]

Now we can find [tex]\( \frac{dA}{dt} \):[/tex]

[tex]\[ \frac{dA}{dt} = \frac{1}{2}\left(15 - \frac{36^2}{15}\right) \times 8 \][/tex]

[tex]\[ \frac{dA}{dt} = 4\left(15 - \frac{1296}{15}\right) \][/tex]

[tex]\[ \frac{dA}{dt} = 4\left(15 - 86.4\right) \][/tex]

[tex]\[ \frac{dA}{dt} = 4\left(-71.4\right) \][/tex]

[tex]\[ \frac{dA}{dt} = -285.6 \][/tex]

So, the area of the triangle is changing at a rate of ( -285.6 ) square feet per second at the instant the bottom of the ladder is 36 feet from the wall. The negative sign indicates that the area is decreasing.

Answer:

a) greatest voltage = 29.25 V

b) power = 16 W

Explanation:

The total resistance R of the three resistors in series is:

[tex]R = (6.7 + 30.4 + 16.3) \Omega = 53.4 \Omega[/tex]  

a) The greatest current I is the one that will burn the resistor with lower power rating, which is 9.12 W:

[tex]P_{max} = I_{max}^2 R = I_{max}^2 30.4\Omega = 9.12W\\I_{max} = 0.54 A[/tex]

The voltage is:

[tex]V_{max}=IR = 0.54*53.4V= 29.25 V[/tex]

b) When the current is 0.54 A, the power is:

[tex]P = RI^2=53.4*0.3 W = 16W[/tex]

Answer:

Size of image: 1.8 cm inverted

Magnification of camera: -0.1

Explanation:

Given:

Assume:

Using sign convention, we have

[tex]u = -200\ cm\\v = 20\ cm\\h_o = 18\ cm[/tex]

The hole in the pinhole camera behaves as a convex lens. The other face opposite to the hole face behaves like a film where the image is formed to be seen. So, using the formula of magnification, we have

[tex]m = \dfrac{h_i}{h_o}=\dfrac{v}{u}\\\Rightarrow  \dfrac{h_i}{h_o}=\dfrac{v}{u}\\\Rightarrow  h_i=\dfrac{v}{u}h_o\\\Rightarrow  h_i=\dfrac{20}{-200}\times 18\\\Rightarrow  h_i=-1.8 cm[/tex]

This means the image of the bird measures 1.8 cm in length where negative sign in the calculation represents that the image formed is inverted.

Hence, the image of the bird on the film is 1.8 cm large.

Now, again using the formula of magnification, we have

[tex]m = \dfrac{h_i}{h_o}\\\Rightarrow m = \dfrac{-1.8}{18}\\\Rightarrow m =-0.1[/tex]

Hence, the magnification of the camera is -0.1.

In a pinhole camera, image size = object size * (camera depth / object distance). For a chicken 18 cm tall standing 2 meters away, the image size will be 0.198 cm. The magnification, or size of the image compared to the object, will be 0.011 or 1.1%

The size of the image formed by a pinhole camera is given by the formula: image size = object size * (camera depth / object distance). Substituting the values in this formula, image size = 18 cm * (22 cm / 2 m) = 0.198 cm. This tells us that the fierce chicken's image will be about 0.198 cm high on the film.

Next, the magnification of the camera is the ratio of the image size to the object size. So the magnification is 0.198 cm / 18 cm = 0.011. In other words, the image on the film is about 1.1% the size of the actual object.

Note:

These calculations are based on the simple model of a pinhole camera where only rays of light traveling in straight lines are considered. The actual picture may differ due to factors like the scattering of light.

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Answer:

mean = 54

standard deviation = 6.24

Explanation:

given data

mean a = 72

standard deviation σx = 12

average b = 95

standard deviation σy = 4

final grade Z = 70%  = 0.70

project y = 30%  = 0.30

to find out

mean and standard deviation

solution

final grade equation will be here

final grade = 0.70x + 0.30 y   .....a

here x is statics grade and y is class project

so

mean will be 0.70x + 0.30 y

mean = 0.70 (a) + 0.30 (b)

mean = 0.70 (12) + 0.30 (95)

mean = 54

and

standard deviation = 0.70x + 0.30 y

standard deviation = 0.70(σx) + 0.30 (σy)

standard deviation = 0.70(12) + 0.30 (4)

standard deviation = 6.24

Final answer:

Alcohol and mercury thermometers may not read the same at 60°C despite being calibrated at 0°C and 100°C because they have different rates of thermal expansion, with mercury being more linear and alcohol having a higher expansion coefficient.

Explanation:

Alcohol and mercury thermometers are calibrated at two fixed points: the freezing point of water (0°C) and the boiling point of water (100°C). However, these substances expand at different rates over the temperature range, which means the expansion is not linear. While both may be divided into 100 equal divisions between these two points, an alcohol thermometer and a mercury thermometer may not give the same reading at a temperature such as 60°C because their rates of thermal expansion differ. Mercury has a more linear expansion with temperature compared to alcohol, which has a higher coefficient of expansion, causing its physical expansion for an equal temperature increment to be larger. This inconsistency is often overlooked because it is somewhat small over the range of temperatures we normally encounter, but it becomes significant at more extreme temperatures.

Answer:

The hydro static force on the back of the dam is [tex]1.96\times10^{11}\ N[/tex]

Explanation:

Given that,

Width b= 1000 m

Depth d= 200 m

We need to calculate the average pressure

Using formula of  average pressure

[tex]P_{avg}=\rho\times g\times d_{avg}[/tex]

Put the value into the formula

[tex]P_{avg}=1000\times9.8\times100[/tex]

[tex]P_{avg}=980000\ Pa[/tex]

We need to calculate  the hydro static force on the back of the dam

Using formula of force

[tex]F = P_{avg}\times A[/tex]

Put the value into the formula

[tex]F = 980000\times1000\times200[/tex]

[tex]F=1.96\times10^{11}\ N[/tex]

Hence, The hydro static force on the back of the dam is [tex]1.96\times10^{11}\ N[/tex]

First, let's make some convertions:

[tex]268kg*\frac{1 lbm}{0.454kg}= 590.84 lbm[/tex]

[tex]133.7 ft^3*\frac{1m^3}{35.3147ft^3} = 3.78m^3[/tex]

a) weight of the fuel:

Newtons: The weight in newtons is equal to the mass in kilograms times the gravity in m/s^2.

[tex]W = m*g = 268kg*9.81m/s^2=2629.08 N[/tex]

lbf: The weight inlbf is equal to the mass in slugs times the gravity in ft/s^2.

[tex]W= m*g = 590.84 lbm *\frac{1 slug}{32.174lbm} *32.174ft/s^2 = 590.84 lbf[/tex]

b) density:

The density is the mass in kg of the fuel divided by its volume in m^3:

[tex]d = \frac{m}{v} =\frac{268kg}{3.78m^3} =70.9 kg/m^3[/tex]

c) specific volume:

The specific volume is the volume in ft^3 of the fuel divided by its mass in lbm:

[tex]v_{sp} = \frac{v}{m} =\frac{133.7 ft^3}{590.84 lbm} = 0.226 ft^3/lbm[/tex]

Answer:

upward acceleration is 211.04 m/s²

Explanation:

given data

velocity = 14.6 m/s

distance = 0.505 m

to find out

upward acceleration

solution

we have given distance and velocity and ball is going upward

so acceleration will be calculated by  velocity formula that is

v² - u² = 2×a×s   ............1

here v is velocity and u is initial velocity and s is distance and a is acceleration

and u = o because starting velocity zero

put here all there value in equation 1

v² - u² = 2×a×s

14.6² - 0 = 2×a×0.505

solve it we get a

a = 211.04

so upward acceleration is 211.04 m/s²

Answer:

[tex]r=\frac{1}{75}[/tex] Ω≈0.01333Ω

Explanation:

You can check the internal resistance model in the image I attached you.

Now using Kirchhoff's voltage law:

ε=[tex]Ir+IR[/tex]  (1)

Where:

[tex]IR=V_R=LoadVoltage=10V[/tex]

ε=12V

[tex]I=150A[/tex]

Evaluating the data provide in (1)

[tex]r=\frac{12-10}{150} =\frac{1}{75}[/tex]≈0.01333Ω

Answer:

option (d) 7.1 kN

Explanation:

Given:

Mass of the car, m = 1600 kg

Acceleration of the car, a = 1.5 m/s²

Coefficient of kinetic friction = 0.3

let the tension be 'T'

Now,

ma = T - f .................(1)

where f is the frictional force

also,

f = 0.3 × mg

where g is the acceleration due to the gravity

thus,

f = 0.3 × 1600 × 9.81 =

therefore,

equation 1 becomes

1600 × 1.5 = T - 4708.8

or

T = 2400 + 4708.8

or

T = 7108.8 N

or

T = 7.108 kN

Hence,

The correct answer is option (d) 7.1 kN

Answer:

398 m/s

Explanation:

The acceleration is given by:

a = K/(L - x)^2

K = 100 m^3/s^2

L = 0.169 m

This acceleration will result in a force:

F = m * a

F = m * K/(L - x)^2

This force will perform a work:

W = F * L

The ball will advance only until x = L - D/2

[tex]W = m * K \int\limits^{L - D/2}_0 {\frac{1}{(L - x)^2}} \, dx[/tex]

This work will be converted to kinetic energy

W = Ek

Ek = 1/2 * m * v^2

[tex]1/2 * m * v^2 = m * K \int\limits^{L - D/2}_0 {\frac{1}{(L - x)^2}} \, dx[/tex]

[tex]1/2 * v^2 = K \int\limits^{L - D/2}_0 {\frac{1}{(L - x)^2}} \, dx[/tex]

First we solve thr integral:

[tex]K \int\limits^{L - D/2}_0 {\frac{1}{(L - x)^2}} \, dx[/tex]

We use the replacement

u = L - x

du = -dx

And the limits

When x = L - D/2, u = D2/2, and when x = 0, u = L

[tex]-K \int\limits^{D/2}_L {u^-2}} \, du[/tex]

K / u evaluated between L and D/2

2*K / D - K / L

Then

1/2 * v^2 = 2*K / D - K / L

1/2 * v^2 = K * (2/D - 1/L)

v^2 = 2*K*(2/D - 1/L)

[tex]v = \sqrt{2*K*(2/D - 1/L)}[/tex]

[tex]v = \sqrt{2*100*(2/0.0025 - 1/0.169)} = 398 m/s[/tex]

Explanation:

Speed of the marathon runner, v = 9.51 mi/hr

Distance covered by the runner, d = 26.220 mile

Let t is the time taken by the marathon runner. We know that the speed of the runner is given by total distance divided by total time taken. Mathematically, it is given by :

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

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

[tex]t=\dfrac{26.220\ mi}{9.51\ mi/hr}[/tex]

t = 2.75 hours

Since, 1 hour = 60 minutes

t = 165 minutes

Since, 1 minute = 60 seconds

t = 9900 seconds

Hence, this is the required solution.

Answer:

a)  Y component of the vector =15.54 m

b) Vector magnitude = 21.6 m

Explanation:

The given vector makes 44 degree angle with Y axis, as given. This is same as 90 -44 = 46 degrees with the horizontal or X axis.

b) X component of the given vector = [tex]A_{x}[/tex] = A cos 46 =15

⇒ A = 15/cos 16 = 21.6 m = Total vector magnitude.

a) Y component of the vector = 21.6 sin 46 = 15.54 m

b) A = 21.6 m

The y-component of vector A (Ay) is approximately 10.3 m, assuming a positive y-direction due to the angle being in the second quadrant. The magnitude of vector A, found using the Pythagorean theorem with its components, is approximately 18.0 m.

To find the y-component of vector A (Ay), we use the trigonometric relationship Ay = A sin(θ), where A is the magnitude of the vector and θ is the angle it makes relative to the x-axis. However, since the angle given is clockwise from the y-axis, we must first convert it to the counterclockwise angle from the x-axis, which would be θ = 90° + 44.0° = 134.0°.

Given that Ax = -15.0 m, and assuming θ is measured counterclockwise, we can find the magnitude of A using the Pythagorean theorem: A = √(Ax² + Ay²). To find Ay, we rearrange the equation for the y-component to Ay = A sin(θ) and solve for Ay using the magnitude we just found. Our direction angle θ is in the second quadrant, meaning Ay should be positive.

Since we only have Ax, we first find A as follows: A = √((-15.0 m)² + Ay²). To determine Ay, we need to look at signs and quadrants. Ax is negative, and because the angle is measured clockwise from the y-axis, Ay must be positive. Therefore, we can infer that Ay is approximately 10.3 m by implication of the Pythagorean theorem, knowing that the other possible value is negative, which does not fit the quadrant.

To find the magnitude of A, we use the derived components: A = √((-15.0 m)² + (10.3 m)²), yielding approximately 18.0m.

Answer:

[tex]V=23.3kV[/tex]

Explanation:

Definition of the capacitance C, where a voltage V is applied and a charge Q is stored:

[tex]Q=C*V[/tex]

We solve to find V:

[tex]V=Q/C=0.14*10^{-3}C/6*10^{-9}F)=2.33*10^{4}V=23.3kV[/tex]

Answer:

The initial horizontal velocity was 21 m/s

Explanation:

Please, see the figure for a better understanding of the problem.

The equation for the position of an object moving in a parabolic trajectory is as follows:

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

Where:

r = vector position at time t

x0 = initial horizontal position

v0 = initial velocity

t = time

α = launching angle

y0 = initial vertical position

g = acceleration due to gravity

Notice that at time t = 2.4 s the vector "r" is the one in the figure. We know that the x-component of that vector is 50 m. Then using the equation for the x-component of the vector "r", we can calculate the initial velocity:

x = x0 + v0 · t · cos α

Let´s place the center of the frame of reference at the point of the kick so that x0 = 0.

x =  v0 · t · cos α

x/t = v0 · cos α

Notice in the figure that v0 · cos 30° = v0x which is the initial horizontal velocity. Remember trigonometry of right triangles:

cos α = adjacent / hypotenuse = v0x / v0

Then:

50 m/ 2.4 s = v0 · cos 30° = v0x

v0x = 21 m/s

Answer:

(a) 508.37 m

(b) 47.53 s

(c) 21.165 m

(d) 19.365 m

Explanation:

initial velocity, u = 77 km/h = 21.39 m/s

acceleration, a = - 0.45 m/s^2

(a) final velocity, v = 0

Let the distance traveled is s.

Use third equation of motion

[tex]v^{2}=u^{2}+2as[/tex]

[tex]0^{2}=21.39^{2}-2 \times 0.45 \times s[/tex]

s = 508.37 m

(b) Let t be the time taken to stop.

Use first equation of motion

v = u + at

0 = 21.39 - 0.45 t

t = 47.53 s

(c) Use the formula for the distance traveled in nth second

[tex]s_{n^{th}=u+\frac{1}{2}a\left ( 2n-1 \right )}[/tex]

where n be the number of second, a be the acceleration, u be the initial velocity.

put n = 1, u = 21.39 m/s , a = - 0.45m/s^2

[tex]s_{n^{th}=21.39-\frac{1}{2}\times 0.45\left ( 2\times 1-1 \right )}[/tex]

[tex]s_{n^{th}=21.165m[/tex]

(d)  Use the formula for the distance traveled in nth second

[tex]s_{n^{th}=u+\frac{1}{2}a\left ( 2n-1 \right )}[/tex]

where n be the number of second, a be the acceleration, u be the initial velocity.

put n = 5, u = 21.39 m/s , a = - 0.45m/s^2

[tex]s_{n^{th}=21.39-\frac{1}{2}\times 0.45\left ( 2\times 5-1 \right )}[/tex]

[tex]s_{n^{th}=19.365m[/tex]