A particle moves in a straight line with an initial velocity of 35 m/s and a constant acceleration of 38 m/s2. If at t = 0, x = 0, what is the particle's position (in m) at t = 6 s?

Answer

given,

width of trapezoidal channel, b = 6 ft

depth of water, d = 3 ft

discharge,Q = 250 ft³/s

now, we have to calculate Froude number

[tex]F_r = \dfrac{Q}{A\sqrt{gD}}[/tex]

Where D is the hydraulic radius

 [tex]D = \dfrac{A}{P}[/tex]

[tex]F_r = \dfrac{Q}{A\sqrt{g\times \dfrac{A}{P}}}[/tex]

P is the width of the channel

P = b + 2 z d

P = 6 + 2 x 1 x 3

P = 13 ft

A = d(b + z d) = 3 (6 + 3) = 27 ft²

g = 32.2 ft/s²

now,

[tex]F_r = \dfrac{Q}{A\sqrt{g\times \dfrac{A}{P}}}[/tex]

[tex]F_r = \dfrac{250}{27\sqrt{32.2\times \dfrac{27}{13}}}[/tex]

 F_r = 1.087

 F_r > 1

Froude number is greater than 1 so, the flow is Super critical flow.

Answer:

[tex]\vec{v} = (77.68~{\rm ft/s})\^i + (7.01~{\rm ft/s})\^j[/tex]

Explanation:

The x- and y- components of the velocity vector can be written as following:

[tex]\vec{v}_x = ||\vec{v}||\cos(\theta)\^i[/tex]

[tex]\vec{v}_y = ||\vec{v}||\sin(\theta)\^j[/tex]

Since the angle θ and the magnitude of the velocity is given, the vector representation can be written as follows:

[tex]\vec{v} = 78\cos(0.09)\^i + 78\sin(0.09)\^j\\\vec{v} = (77.68~{\rm ft/s})\^i + (7.01~{\rm ft/s})\^j[/tex]

Answer:

324.066096 m.

Explanation:

Given that

height of the tower ,h= 324 m

The increase in temperature ,ΔT = 17°C

We know that coefficient of thermal expansion for steel ,α= 12 x 10⁻⁶ C⁻¹

The increase in height is given as

Δ h = α h ΔT

Now by putting the values in the above equation we get

Δ h= 12 x 10⁻⁶ x 324 x 17 m

Δ h=66096 x 10⁻⁶ m

Δ h=0.066096 m

Therefore the height of the tower become 324.066096 m.

Due to thermal expansion, the Eiffel Tower, made of steel, would increase in height by approximately 0.066 meters or 6.6 cm over a day when the temperature increases by 17°C.

The height of the Eiffel Tower increases due to the phenomenon of thermal expansion, which is an increase in volume, including height, in response to an increase in temperature. The amount of expansion can be calculated using this formula: ΔL = α * L_original * ΔT. Let's apply the given values:

So ΔL = 0.000012 * 324 * 17, which equates to approximately 0.066048 m. Therefore, the Eiffel Tower would increase in height by about 0.066 meters (or 6.6 cm) over the course of a day when the temperature increases by 17°C.

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Answer:

[tex]h=9.30m[/tex]

Explanation:

We have an uniformly accelerated motion, due to the gravitational acceleration. So, we use the kinematic equations, since the ball is throw directly upward, g is negative:

[tex]h=v_0t-\frac{gt^2}{2}[/tex]

First, we need to calculate the time taken by the ball to reach the maximum height, in this point its final speed is zero:

[tex]v_f=v_0-gt\\\\\frac{0-v_0}{-g}=t\\t=\frac{v_0}{g}\\t=\frac{13.5\frac{m}{s}}{9.8\frac{m}{s^2}}\\t=1.38s[/tex]

Now, we can calculate h:

[tex]h=v_0t-\frac{gt^2}{2}\\h=13.5\frac{m}{s}(1.38s)-\frac{9.8\frac{m}{s^2}(1.38s)^2}{2}\\h=9.30m[/tex]

Answer:

Force = 3481.1 N.

Explanation:

Below is an attachment containing the solution.

Answer:

27.3 m/s

Explanation:

We are given that

Distance travel by ball=x=60 m

[tex]\theta=26^{\circ}[/tex]

We have to find the initial speed([tex]v_0)[/tex] of the ball.

[tex]x=v_0cos\theta t[/tex]

Using the formula

[tex]60=v_0cos 26 t[/tex]

[tex]t=\frac{60}{v_ocos 26}=\frac{60}{v_0\times 0.899}=\frac{66.7}{v_0}[/tex]

The value of y at point of foot  of the vertical distance

y=0

[tex]y=v_0sin\theta t-\frac{1}{2}gt^2[/tex]

Using [tex]g=9.8m/s^2[/tex]

Using the formula

[tex]0=v_0sin 26\times \frac{66.7}{v_0}-4.9\times (\frac{66.7}{v_0})^2[/tex]

[tex]4.9\times \frac{(66.7)^2}{v^2_0}=0.44\times 66.7[/tex]

[tex]v^2_0=\frac{4.9\times (66.7)^2}{0.44\times 66.7}[/tex]

[tex]v^2_0=742.8[/tex]

[tex]v_0=\sqrt{742.8}=27.3 m/s[/tex]

Hence, the initial speed of the ball=27.3 m/s

Answer:

27.3 m/s

Explanation:

Horizontal range, R = 60 m

angle of projection, θ = 26°

Let the velocity of projection is vo.

Use the formula of range of the projectile

[tex]R = \frac{u^{2}Sin2\theta} {g}[/tex]

[tex]60 = \frac{v_{0}^{2}Sin52}{9.8}[/tex]

vo = 27.3 m/s

Thus, the velocity of projection is 27.3 m/s.

The rate constant at 75°C is calculated using the two-point form of the Arrhenius equation. The original conditions, the new temperature, and the activation energy are substituted into the equation and solved for the new rate constant, k2. The result is k2 = 0.048, or 4.8 x 10^-2 s^-1.

For calculating the rate constant at a different temperature, we can use the Arrhenius equation: k = Ae^(-Ea/RT) where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant and T is the temperature in Kelvin.

To find the new temperature constant, we can transform the Arrhenius equation into the two-point form: ln(k2/k1) = (-Ea/R)(1/T2 - 1/T1).

Given:
k1 = 7.8 × 10−3 s−1,  T1=25°C = 25 + 273 = 298K
Ea = 33.6 kJ/mol = 33,600 J/mol,  R = 8.314 J/(mol·K)
T2 = 75°C = 75 + 273 = 348K

Substituting these values and solving for k2 (rate constant at 75°C), you get k2 = 0.048 or 4.8 x 10^-2 s^-1.

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Answer:

given,

reaction time.t_r = 0.50 s

deceleration of the car = 6 m/s²

initial speed,v = 20 m/s

distance at which the car stop = ?

distance travel by the car in reaction time

 d= v x t_r

 d = 20 x 0.5 = 10 m

using equation of motion

distance travel to stop the car

v² = u² + 2 a s

0² = 20² - 2 x 6 x s

 12 s = 400

 s = 33.33 m

Total distance travel by the car

D = 10 + 33.33

D = 43.33 m

Hence, the car stops before to avoid collision.

Answer:

Trigonal planar

Explanation:

Trigonal planar - it is referred to the molecular shape of atom in which three bonds exist around any central atom. As there is no lone pair at the center hence all three atoms have taken the form of a triangle. All three atom lies at same plane and known as peripheral atoms

The angle between all the three atoms is 120 degree

Answer:

F=480.491 N

Explanation:

Given that

mass ,m = 22 kg

Angular speed ω = 40 rev/min

[tex]\omega=\dfrac{2\pi \times 40}{60}\ rad/s[/tex]

ω =4.18 rad/s

The radius  r= 1.25 m

We know that centripetal force is given as

F=m ω² r

Now by putting the values in the above equation we get

[tex]F=22\times 4.18^2\times 1.25\ N[/tex]

F=480.491 N

Therefore the centripetal force on the child will be 480.491 N.

Answer:

[tex]\theta_2=sin^{-1}(\dfrac{n_1\ sin\theta_1}{n_2})[/tex]

Explanation:

Given that,

The ray makes an angle of 32 degrees with respect to the normal of the surface.

The refractive index of air, [tex]n_1=1[/tex]

The refractive index of water, [tex]n_2=1.33[/tex]

Snell's law is given by :

[tex]n_1\ sin\theta_1=n_2\ sin\theta_2[/tex]

[tex]sin\theta_2=\dfrac{n_1\ sin\theta_1}{n_2}[/tex]

[tex]\theta_2=sin^{-1}(\dfrac{n_1\ sin\theta_1}{n_2})[/tex]

So, option (4) is correct. Hence, this is the required solution.    

The answer is option d.

The correct expression for the angle (relative to the normal to the surface) for the ray in the water is d) θ2 = asin (sin(θ1).n1/n2), based on Snell's law of refraction.

The question addresses the refraction of light, specifically the change in angle as light moves from air to water. According to Snell's law, which is used to calculate the angle of refraction, the correct expression in your options is d) θ2 = asin (sin(θ1).n1/n2). Here's a step by step process:

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Answer:

The work done is zero.

Solution:

As per the question:

Charge, [tex]q = 1\mu C = 1\times 10^{- 6}\ C[/tex]

Distance moved, d = 5 m

Voltage, V = 6V

Now, we know that an equipotential surface is one where the potential is same everywhere on the surface.

Suppose the the voltage at a distance d = 5 m is V'

Thus

V' = 6 V, (since the surface is equipotential)

Work done in moving a charge is given by:

[tex]W = q\Delta V[/tex]

[tex]W = q(V - V')[/tex]

[tex]W = (1\times 10^{- 6})(V - V')[/tex]

[tex]W = (1\times 10^{- 6})(6 - 6) = 0[/tex]

Thus the work done in moving a charge on an equipotential surface comes out to be zero as the potential difference is zero.

Final answer:

The work required to move a 1 microcoulomb charge by a distance of 5 meters along an equipotential line of 6V is zero because there's no change in potential energy.

Explanation:

The question relates to determining the amount of work needed to move a charge along an equipotential line. When a charge moves along an equipotential, the potential energy of the charge does not change because the voltage (potential difference) across its path remains zero. In other words, the work done on the charge is zero since work is defined as the change in potential energy, which is given by the formula W = qV, where W is work, q is charge in coulombs, and V is potential difference in volts. For movement along an equipotential line, V = 0, hence, Work = 0 Joules.

Answer:

a) 66.4 relative to the west in the south-west direction

b) 5.455 hours

Explanation:

a)If the wind is blowing east-ward at a speed of 40km/h, then the west component of the geese velocity must be 40km/h in order to counter balance it. Geese should be flying south-west at an angle of

[tex]cos(\alpha) = 40 / 100 = 0.4[/tex]

[tex]\alpha = cos^{-1}(0.4) = 1.16 rad = 180\frac{1.16}{\pi} = 66.4^0[/tex] relative to the West

b) The south-component of the geese velocity is

[tex]100sin(\alpha) = 100sin(66.4^0) = 91.65 km/h[/tex]

The time it would take for the geese to cover 500km at this rate is

t = 500 / 91.65 = 5.455 hours

Final answer:

Astronomers utilize spectroscopy to analyze the spectrum of a star, identifying unique absorption lines corresponding to different elements, which reveals the star's composition. Spectral lines' broadening indicates temperature and pressure, and shifts in these lines help measure a star's motion, including radial and rotational velocities.

Explanation:

Understanding Stellar Spectroscopy

Astronomers use spectroscopy as a powerful tool to determine various characteristics of stars, including their composition and temperature. When light from a star passes through a prism or diffraction grating, it spreads out into a spectrum of colors. This spectrum contains dark lines known as absorption lines, which are unique to the elements present in the star's atmosphere, as different chemical elements absorb light at specific wavelengths. Therefore, by analyzing these lines, astronomers can identify the elements that make up a star.

Analyzing the broadening of spectral lines can inform us about a star's temperature and pressure. Warmer temperatures and higher pressures in a star's atmosphere tend to broaden the spectral lines. Additionally, the pressure can give clues about the star's size, as stars with lower atmospheric pressure tend to be larger, or giant stars.

Motions of the Stars are also revealed through spectroscopy. The Doppler effect causes spectral lines to shift towards the red end of the spectrum if the star is moving away from us (redshift) or towards the blue end if it is approaching (blueshift). This allows astronomers to measure the star's radial velocity. Spectral line broadening can also indicate the star's rotational velocity, while proper motion is deduced from the movement of the lines over time across the spectrum.

Answer:

(A) Spring constant will be 126.58 N/m

(B) Amplitude will be equal to 0.177 m

Explanation:

We have given mass of the block m = 200 gram = 0.2 kg

Time period T = 0.250 sec

Total energy is given TE = 2 J

(A) For mass spring system time period is equal to [tex]T=2\pi \sqrt{\frac{m}{K}}[/tex]

So [tex]0.250=2\times 3.14 \sqrt{\frac{0.2}{K}}[/tex]

[tex]0.0398=\sqrt{\frac{0.2}{K}}[/tex]

Now squaring both side

[tex]0.00158=\frac{0.2}{K}[/tex]

K = 126.58 N/m

So the spring constant of the spring will be 126.58 N/m

(B) Total energy is equal to [tex]TE=\frac{1}{2}KA^2[/tex], here K is spring constant and A is amplitude

So [tex]2=\frac{1}{2}\times 126.58\times A^2[/tex]

[tex]A^2=0.0316[/tex]

A = 0.177 m

So the amplitude of the wave will be equal to 0.177 m

Answer:

t= 22.9ºC

Explanation:

Assuming no heat exchange outside the container, before reaching to a condition of thermal equilibrium, defined by a common final temperature, the body at a higher temperature (water at 35ºC) must give heat to the body at a lower temperature (the ice), as follows:

Qw = c*m*Δt = 4190 (J/kg.ºC)*0.557 kg*(35ºC-t) (1)

This heat must be the same gained by the ice, which must traverse three phases before arriving at a final common temperature t:

1) The heat needed to reach in solid state to 0º, as ice:

Qi =ci*m*(0ºC-(-30ºC) = 0.0575kg*2090(J/kg.ºC)*30ºC = 3605.25 J

2) The heat needed to melt all the ice, at 0ºC:

Qf = cfw*m = 3.33*10⁵ J/kg*0.0575 kg = 19147.5 J

3) Finally, the heat gained by the mass of ice (in liquid state) in order to climb from 0º to a final common temperature t:

Qiw = c*m*Δt = 4190 (J/kg.ºC)*0.0575 kg*(t-0ºC)

So, the total heat gained by the ice  is as follows:

Qti = Qi + Qf + Qiw

⇒Qti = 3605.25 J + 19147.5 J + 240.9*t = 22753 J + 240.9*t (2)

As (1) and (2) must be equal each other, we have:

22753 J + 240.9*t = 4190 (J/kg.ºC)*0.557 kg*(35ºC-t)

⇒ 22753 J + 240.9*t = 81684 J -2334*t

⇒ 2575*t = 81684 J- 22753 J = 58931 J

⇒ [tex]t= \frac{58931J}{2575 J/C} = 22.9C[/tex]

⇒ t = 22.9º C

The frequency can be defined as the inverse of the period, that is, it can be expressed as

[tex]T = \frac{1}{f}[/tex]

Here,

T = Period

f = Frequency

For each value we only need to replace the value and do the calculation:

PART A)

[tex]T = \frac{1}{f}[/tex]

[tex]T = \frac{1}{60Hz}[/tex]

T = 0.0166s

PART B)

[tex]T = \frac{1}{f}[/tex]

[tex]T = \frac{1}{8*10^6}[/tex]

[tex]T = 1.25*10^{-7} s[/tex]

PART C)

[tex]T = \frac{1}{f}[/tex]

[tex]T = \frac{1}{140*10^{3}}[/tex]

[tex]T = 7.14*10^{-6}s[/tex]

PART D)

[tex]T = \frac{1}{f}[/tex]

[tex]T = \frac{1}{2.4*10^{9}}[/tex]

[tex]T = 4.166*10^{-10}s[/tex]

Answer:

a) 15.864s

b) 392.78m

c) 29.52 m/s

Explanation:

The total distance (relative to the truck) that the (front bumper of the) car travels from 24m behind the truck's rear bumper to in front of the car is

distance from car's front bumper to the truck's rear bumper + distance from truck's rear bumper to truck's front bumper (truck's length) + distance from truck's front bumper to car's rear bumper's + distance from the car's rear bumper to the car's front bumper (car's length)

= 24 + 21 + 26 + 4.5 = 75.5 m

As they start at the same speed, we can draw the following equation of motion for the car distance relative to the truck

[tex]s = at^2/2[/tex]

[tex]75.5 = 0.6t^2/2[/tex]

[tex]t^2 = 251.67[/tex]

[tex]t = \sqrt{251.67} = 15.864s[/tex]

b) The actual distance relative to Earth that the car has traveled during this time is the distance car traveled relative to the truck plus distance truck traveled relative to Earth within this time

= 75.5 + 20*15.864 = 392.78 m

c) final speed of the car is the initial speed plus the change in speed

[tex]v = v_0 + \Delta v = v_0 + at = 20 + 15.864*0.6 = 29.52 m/s[/tex]

To pass the truck, it takes the car 33.3 seconds to accelerate and overtake the truck. The car travels a distance of 333 meters during this time. The final speed of the car is 1.80 m/s.

u is the initial velocity of the car and a is the acceleration. Since the car is initially traveling at the same speed as the truck (20.0 m/s) and accelerates at a constant rate of 0.600 m/s², the equation becomes: t = (0 - 20) / -0.600. Solving for t gives us t = 33.3 seconds. To find the distance traveled by the car during this time, we can use the equation: s = ut + (1/2)at², where s is the distance, u is the initial velocity, t is the time, and a is the acceleration. Plugging in the values, we get: s = 20(33.3) + (1/2)(-0.600)(33.3)². Solving for s gives us s = 333 meters. To find the final speed of the car, we can use the equation: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Plugging in the values, we get: v = 20 + (-0.600)(33.3). Solving for v gives us v = 1.80 m/s.

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Final answer:

The length of the phasor in a phasor diagram representing a transverse wave on a string indicates the wave's amplitude, which corresponds to the maximum displacement of the medium's particles from their equilibrium position.

Explanation:

In the phasor representation of a transverse wave on a string, the length of the phasor corresponds to the amplitude of the wave. In a phasor diagram, this amplitude represents the maximum displacement of the wave particles from the equilibrium position as the wave propagates through the medium. The phasor's length will rotate in a circular motion at a rate determined by the wave's frequency, and this motion represents the oscillatory nature of the wave at a certain point in space over time. The amplitude is a crucial parameter as it determines the energy carried by the wave, with a larger amplitude indicating a greater energy transfer.

The phasor length is particularly important when analyzing multiple wave forms together, such as voltage and current in electrical circuits, where the ratio of their lengths can denote relative magnitudes, such as resistance in the circuit. In this context, however, we focus on mechanical waves on a string, and the length of the phasor would only represent the wave amplitude, not voltage or current.

Answer:

Explanation:

Given

[tex]v_x(t)=\alpha -\beta t^2[/tex]

[tex]\alpha =4\ m/s[/tex]

[tex]\beta =2\ m/s^3[/tex]

[tex]v_x(t)=4-2t^2[/tex]

[tex]v=\frac{\mathrm{d} x}{\mathrm{d} t}[/tex]

[tex]\int dx=\int \left ( 4-2t^2\right )dt[/tex]

[tex]x=4t-\frac{2}{3}t^3[/tex]

acceleration of object

[tex]a=\frac{\mathrm{d} v}{\mathrm{d} t}[/tex]

[tex]a=-4t[/tex]

(b)For maximum positive displacement velocity must be zero at that instant

i.e.[tex]v=0[/tex]

[tex]4-2t^2=0[/tex]

[tex]t=\pm \sqrt{2}[/tex]

substitute the value of t

[tex]x=4\times \sqrt{2}-\frac{2}{3}\times 2\sqrt{2}[/tex]

[tex]x=3.77\ m[/tex]

The definitions of acceleration and velocity allow to find the results for the questions about the motion of the particle are:

    A) the function of the acceleration is: a = -4t

         and the position function is:   x = 4 t - ⅔ t³

    B) The maximum displacement is: x = 3.77 m

Given parameters

To find

    a) position and acceleration as a function of time,

    b) maximum displacement,

The acceleration of defined as the change in velocity with time.

      a = [tex]\frac{dv}{dt}[/tex]  

Let's calculate.

      a = - 2βt  

      a = - 2 2 t

      a = -4 t

The speed is defined by the variation of the position with respect to time.

       v = [tex]\frac{dx}{dt}[/tex]  

       dx = v dt

We integrate.

      ∫ dx = ∫ v dt

      x - x₀ = ∫ (α - β t²)

      x-x₀ = αt - βt³/ 3

we substitute.

      x = 4 t - ⅔ t³

B) To find the maximum displacement we use the first derivative to be zero.

     [tex]\frac{dx}{dt}[/tex]  = 0

      4 - 2t² = 0        

      t² = 2

       t = √2 = 1.414 s

Let's find the position for this time.

     x = 4 √2 - ⅔ (√2)³

     x = 3.77 m

In conclusion using the definitions of acceleration and velocity we can find the result for the questions about the motion of the particle are:

    A) the function of the acceleration is: a = -4t

        and the position function is: x = 4 t - ⅔ t³

    B) The maximum displacement is: x = 3.77 m

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Answer:

Total angular displacement will be 240 radian

Explanation:

In first case object starts from rest so initial angular speed [tex]\omega _i=0rad/sec[/tex]

Angular acceleration is given [tex]\alpha =2rad/sec^2[/tex]

Final angular speed[tex]\omega _f=24rad/sec[/tex]

From third equation of motion [tex]\omega _f^2=\omega _i^2+2\alpha \Theta[/tex]

So [tex]24^2=0^2+2\times 2\times \Theta[/tex]

[tex]\Theta =144[/tex] radian

Now in second case as the objects finally stops

So final velocity [tex]\omega _f=0rad/sec[/tex]

Angular acceleration [tex]\alpha =-3rad/sec^2[/tex]

So [tex]0^2=24^2-2\times 3\times \Theta[/tex]

[tex]\Theta =96[/tex] radian

So total angular displacement will be 96+144 = 240 radian

To find the angular displacement, we can analyze the motion of the object in two phases: the first phase of acceleration and the second phase of deceleration. We can calculate the time and angular displacement in each phase using the formulas for angular speed and angular displacement. Finally, we can add the angular displacements from both phases to find the total angular displacement.

To find the angular displacement, we need to analyze the motion of the object in two phases: the first phase of acceleration and the second phase of deceleration. In the first phase, the object starts from rest and accelerates at a rate of 2 rad/s^2 until it reaches an angular speed of 24 rad/s. We can use the formula:

Final Angular Speed = Initial Angular Speed + Angular Acceleration * Time

Using this formula, we can find the time taken in the first phase. Then, we can calculate the angular displacement during the first phase using the formula:

Angular Displacement = Initial Angular Speed * Time + 0.5 * Angular Acceleration * Time^2

In the second phase, the object decelerates at a rate of -3 rad/s^2 until it stops. We can use the same formulas to find the time and angular displacement in the second phase. Finally, we can add the angular displacements from both phases to get the total angular displacement.

The total angular displacement will be the sum of the angular displacements in the two phases.

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Answer:

38.6 mi/h

Explanation:

7.4 mi/h = 7.4mi/h * (1/60)hour/min * (1/60) min/s = 0.00206 mi/s

Let v (mi/s) be your original speed, then the time t it takes to go 1 mi/s is

t = 1/v

Since you increase v by 0.00206 mi/s, your time decreases by 15 s, this means

t - 15 = 1/(v+0.00206)

We can substitute t = 1/v to solve for v

[tex]\frac{1}{v} - 15 = \frac{1}{v + 0.00206}[/tex]

We can multiply both sides of the equation with v(v+0.00206)

v+0.00206 - 15v(v+0.00206) = v

[tex]-15v^2 - 0.0308v + 0.00206 = 0[/tex]

[tex]v= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

[tex]v= \frac{0.03083\pm \sqrt{(-0.03083)^2 - 4*(-15)*(0.00205)}}{2*(-15)}[/tex]

[tex]v= \frac{0.03083\pm0.35}{-30}[/tex]

v = -0.01278 or v = 0.01 0724 mi/s

Since v can only be positive we will pick v = 0.010724 mi/s or 0.010724*3600 = 38.6 mi/h

Final answer:

To find the transformed vector representations in a rotated coordinate system, the angle of vector A is adjusted by the rotation angle, and the components are calculated using trigonometric functions. Vector B's components in the rotated system are found using a rotation matrix.

Explanation:

The provided question pertains to transforming the representation of vectors in a rotated coordinate system in the subject of physics. The coordinate system is rotated counterclockwise, and the goal is to find the new representations of vectors A and B in unit-vector notation on the x'y' system. Given the initial magnitude and direction angle of vector A and the Cartesian components of vector B on the xy coordinate system, we can calculate their components on the rotated x'y' coordinate system.

The original vector A has a magnitude of 14.4 m and an angle of 51.6° from the positive x-axis. After rotation by 20°, the new angle becomes 51.6° - 20.0° = 31.6° from the new x'-axis. Using the formulas Ax' = A cos θ' and Ay' = A sin θ', where θ' is the new angle, we can find the rotated components of A.

The vector B is already given in Cartesian coordinates as ( 14.3 m )i + (8.52 m )j. To find the components of B in the rotated system, we use a rotation matrix, giving us new components Bx' and By'.

In conclusion, to find the transformed vectors in the rotated system, we apply the rotation to both the magnitude and angle of A, and use a rotation matrix for the components of B.

Answer:

a)[tex]a=5 i+2t j - 6\ t^2k[/tex]

b)[tex]a=\dfrac{1}{24.83}(5i+4j-24k)\ m/s^2[/tex]

Explanation:

Given that

v(t) = 5 t i + t² j - 2 t³ k

We know that acceleration a is given as

[tex]a=\dfrac{dv}{dt}[/tex]

[tex]\dfrac{dv}{dt}=5 i+2t j - 6\ t^2k[/tex]

[tex]a=5 i+2t j - 6\ t^2k[/tex]

Therefore the acceleration function a will be

[tex]a=5 i+2t j - 6\ t^2k[/tex]

The acceleration at t = 2 s

a= 5 i + 2 x 2 j - 6 x 2² k  m/s²

a=5 i + 4 j -24 k m/s²

The magnitude of the acceleration will be

[tex]a=\sqrt{5^2+4^2+24^2}\ m/s^2[/tex]

a= 24.83 m/s²

The direction of the acceleration a is given as

[tex]a=\dfrac{1}{24.83}(5i+4j-24k)\ m/s^2[/tex]

a)[tex]a=5 i+2t j - 6\ t^2k[/tex]

b)[tex]a=\dfrac{1}{24.83}(5i+4j-24k)\ m/s^2[/tex]

To solve this problem we will apply the concepts related to the electric field based on the laws of Coulomb. Said electric field is equivalent to the product between the Coulomb constant and the rate of change of the charge and the squared distance. Mathematically this is,

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

Here,

k = Coulomb's constant

q = Charge

r = Distance

Replacing we have that

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

[tex]1.08*10^2 = \frac{(9*10^{9})q}{(6.4*10^{6})^2}[/tex]

Solving for q,

[tex]q = 491520 C[/tex]

Therefore the net charge of the Earth under the previous condition is 491520 C

Answer:

The angle is 2.33°.

Explanation:

Given that,

Speed of ball = 12 m/s

Acceleration = 0.4 m/s²

We need to calculate the time

Using formula of time of flight

[tex]t=\dfrac{2u}{g}[/tex]

[tex]t=\dfrac{2v\cos\theta}{g}[/tex]

Put the value into the formula

[tex]t=\dfrac{2\times12\cos\theta}{9.8}[/tex]

[tex]t=2.44\cos\theta[/tex]

We need to calculate the angle

Using equation of motion along vertical direction

[tex]s=ut-\dfrac{1}{2}at^2[/tex]

[tex]s=v\sin\theta\times t-\dfrac{1}{2}at^2[/tex]

Put the value in the equation

[tex]0=12\sin\theta\times2.44\cos\theta-\dfrac{1}{2}\times0.4\times(2.44\cos\theta)^2[/tex]

[tex]2\times12\sin\theta\times2.44=0.4\times(2.44)^2\cos\theta[/tex]

[tex]\tan\theta=\dfrac{0.4\times2.44}{2\times12}[/tex]

[tex]\theta=\tan^{-1}(0.04066)[/tex]

[tex]\theta=2.33^{\circ}[/tex]

Hence, The angle is 2.33°.

Answer:

261.307 m

Explanation:

b = Base of triangle = 450 m

p = Perpendicular of the triangle

[tex]\theta[/tex] = Angle of the triangle = [tex]30^{\circ}[/tex]

From trigonometry

[tex]tan\theta=\dfrac{p}{b}[/tex]

[tex]\Rightarrow p=btan\theta[/tex]

[tex]\Rightarrow p=450\times tan30[/tex]

[tex]\Rightarrow p=259.807\ m[/tex]

Height of the building = 1.5+259.807 = 261.307 m

Answer:

on increasing pressure, temperature will also increase.

Explanation:

Considering the ideal gas equation as:

[tex]PV=nRT[/tex]

where,  

P is the pressure

V is the volume

n is the number of moles

T is the temperature  

R is Gas constant having value = 0.0821 L.atm/K.mol

Thus, at constant volume and number of moles, Pressure of the gas is directly proportional to the temperature of  the gas.

P ∝ T

Also,  

Also, using Gay-Lussac's law,

[tex]\frac {P_1}{T_1}=\frac {P_2}{T_2}[/tex]

Thus, on increasing pressure, temperature will also increase.

Answer:

David will not  be able to catch the bill .

Explanation:

Reaction time = .2 s .

During this period bill will fall vertically between the fingers.

Distance of fall = 1/2 x g x t²

= .5 x 9.8 x 0.2²

= 19.6 cm or 20 cm .

Generally the bill has size of the order of 25 cm . From central point it requires a fall of 12.5 cm for the bill to escape the catch . Since fall is of 20 cm , that means bill will fall below the level of fingers in .2 s .

So David will not be able to catch the bill.

Answer:

a point of destructive interference.

Explanation:

the wavelength of the sound:

λ= v/f

v= velocity of sound =340 m/s

f= frequency of sound wave= 1800 Hz

L_1 = 4 m

then speaker is at the distance of

[tex]L_2 = sqrt(4^2+2^2)[/tex]

= 2√5 m  

ΔL = L_2-L_1

x = ΔL/λ

Now, if this result is an integer, the waves will add up  at the point. If it is nearly an integer + 0.5, the waves will have a destructive interference at the point. If it is neither of them , then  point is "something in between".

[tex]x= \frac{2\sqrt{5}-4 }{\frac{340}{1800} } =2.4995[/tex]

which is  close to 2.5, an integer + 0.5. So it's a point of destructive interference.

The result is within an integer value of +0.5, thus its a point of destructive interference.

The given parameters;

The resultant distance between the speakers is calculated as follows;

[tex]L = \sqrt{2^2 + 4^2} \\\\L = 4.47 \ m[/tex]

The wavelength of the sound wave is calculated as;

[tex]v = f\lambda\\\\\lambda = \frac{v}{f} \\\\\lambda = \frac{340}{1800} \\\\\lambda = 0.188 \ m[/tex]

Now, determine if the point is constructive interference, perfect destructive interference, or something in between?

[tex]x = \frac{\Delta L}{\lambda} \\\\x = \frac{4.47 - 4}{0.188} \\\\x = 2.5[/tex]

The result is within an integer value of +0.5, thus its a point of destructive interference.

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