The electric field 14.0 cm from the surface of a copper ball of radius 2.0 cm is directed toward the ball's center and has magnitude 9.0 ✕ 10^2 N/C. How much charge is on the surface of the ball (in C)? (Include the sign of the value in your answer.)

The average velocity between the first two signs is 9.1 m/s, between the second and third signs is 10.4 m/s, and the car's acceleration is approximately 0.26 m/s².

To solve for the average velocity and acceleration of the car passing three equally spaced traffic signs, we can use the kinematic equations for uniformly accelerated motion.

Part a: average velocity between the first two signs

The average velocity is calculated by dividing the displacement (distance between the signs) by the time interval: vavg1,2 = d /
[tex](t_2 - t_1)[/tex]. Substituting the given values, we have vavg1,2 = 25 m / (3.9 s - 1.3 s), resulting in an average velocity of 9.1 m/s.

Part b: average velocity between the second and third signs

Similarly, for the average velocity between the second and third signs: vavg2,3 = d / ([tex]t_3 - t_2[/tex]). Using the given times, we calculate vavg2,3 = 25 m / (5.5 s - 3.9 s), giving an average velocity of 10.4 m/s.

Part c: magnitude of the acceleration

To find the acceleration, we can use the velocities we just calculated, and the time between the signs. We can set up an equation for motion for the time intervals. As initial velocity v0 for the second interval is the final velocity of the first interval, we get v0 = vavg1,2 and final velocity v = vavg2,3. Then we can use the equation v = v0 + a[tex](t_3 - t_2)[/tex] to find the acceleration. Rearranging for a, we have a = [tex](v - v_0) / (t_3 - t_2)[/tex]. After calculating, we find that the acceleration is approximately 0.26 m/s².

Answer:

6.93 m/s

Explanation:

mass of marble, m = 2 g

Diameter of the bowl = 4.9 m

radius of bowl, r = half of diameter of the bowl = 2.45 m

The length of the string is 2.45 m

Now it is executing simple harmonic motion, thus the potential energy at the bottom is equal to the kinetic energy at the highest point.

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

where, m be the mass of marble, v be the velocity of marble at heighest point, g be the acceleration due to gravity and r be the radius of the path which is equal to the length of the string

By substituting the values

9.8 x 2.45 = 0.5 x v^2

v = 6.93 m/s

Thus, the speed is 6.93 m/s.

Answer:

a) 94.176 m/s

b) 452.04 m

Explanation:

t = Time taken by the quarter to reach the ground = 9.6 s

u = Initial velocity

v = Final velocity

s = Displacement

a)

[tex]v=u+at\\\Rightarrow v=0+9.81\times 9.6\\\Rightarrow v=94.176 m/s[/tex]

Speed of the quarter just before it hits the ground is 94.176 m/s

b)

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow s=0\times t+\frac{1}{2}\times 9.81\times 9.6^2\\\Rightarrow s=452.04\ m[/tex]

The building is 452.04 m high

Answer:

The current in the wire under the influence of the force is 216.033 A

Solution:

According to the question:

Length of the wire, l = 0.676 m

[tex]\theta = 57.7^{\circ}[/tex]

Magnetic field of the Earth, [tex]B_{E} = 5.29\times 10^{- 5} T[/tex]

Forces experienced by the wire, [tex]F_{m} = 6.53\times 10^{-3} N[/tex]

Also, we know that the force in a magnetic field is given by:

[tex]F_{m} = IB_{E}lsin\theta[/tex]

[tex]I = \frac{F_{m}}{B_{E}lsin\theta}[/tex]

[tex]I = \frac{6.53\times 10^{-3}}{5.29\times 10^{- 5}\times 0.676sin57.7^{\circ}[/tex]

I = 216.033 A

Answer:

It takes you 32.27 seconds to travel 121 m using the speed ramp

Explanation:

Lets explain how to solve the problem

- The speed ramp has a length of 121 m and is moving at a speed of

 2.2 m/s relative to the ground

- That means the speed of the ramp is 2.2 m/s

- You can cover the same distance in 78 seconds when walking on

  the ground

Lets find your speed on the ground

Speed = Distance ÷ Time

The distance is 121 meters

The time is 78 seconds

Your speed on the ground = 121 ÷ 78 = 1.55 m/s

If you walk at the same rate with respect to the speed ramp that

you walk on the ground

That means you walk with speed 1.55 m/s and the ramp moves by

speed 2.2 m/s

So your speed using the ramp = 2.2 + 1.55 = 3.75 m/s

Now we want to find the time you will take to travel 121 meters using

the speed ramp

Time = Distance ÷ speed

Distance = 121 meters

Speed 3.75 m/s

Time = 121 ÷ 3.75 = 32.27 seconds

It takes you 32.27 seconds to travel 121 m using the speed ramp

Answer:

5.4 m

Explanation:

mass, m = 34 kg

initial velocity, u = 9 m/s

final velocity, v = 0 m/s

coefficient of friction, μ = 0.6

Angle of inclination, θ = 10°

Let teh distance traveled before to come into rest is d.

According to the work energy theorem, the work done by all the forces is equal to the change in kinetic energy of the block.

Work done by the gravitational force = W1 = -  mg Sinθ x d  

Work done by the frictional force = W2 = - μ N = - μ mg Cosθ x d

negative sign shows that the direction of force and the direction of displacement is opposite to each other.

Total work done W = W1 + w2

W = - 34 x 9.8 x Sin 10 x d - 0.6 x 34 x 9.8 x cos 10 x d

W = - 254.86 d

Change in kinetic energy = 0.5 x m (v^2 - u^2)

                                         = 0.5 x 34 (0 - 81) = - 1377

So, W = change in KE

- 254.86 d = - 1377

d = 5.4 m

Answer:

Final volumen first process [tex]V_{2} = 98,44 cm^{3}[/tex]

Final Pressure second process [tex]P_{3} = 1,317 * 10^{10} Pa[/tex]

Explanation:

Using the Ideal Gases Law yoy have for pressure:

[tex]P_{1} = \frac{n_{1} R T_{1} }{V_{1} }[/tex]

where:

P is the pressure, in Pa

n is the nuber of moles of gas

R is the universal gas constant: 8,314 J/mol K

T is the temperature in Kelvin

V is the volumen in cubic meters

Given that the amount of material is constant in the process:

[tex]n_{1} = n_{2} = n [/tex]

In an isobaric process the pressure is constant so:

[tex]P_{1} = P_{2} [/tex]

[tex]\frac{n R T_{1} }{V_{1} } = \frac{n R T_{2} }{V_{2} }[/tex]

[tex]\frac{T_{1} }{V_{1} } = \frac{T_{2} }{V_{2} }[/tex]

[tex]V_{2} = \frac{T_{2} V_{1} }{T_{1} }[/tex]

Replacing : [tex]T_{1} =786 K, T_{2} =1209 K, V_{1} = 64 cm^{3}[/tex]

[tex]V_{2} = 98,44 cm^{3}[/tex]

Replacing on the ideal gases formula the pressure at this piont is:

[tex]P_{2} = 3,92 * 10^{9} Pa[/tex]

For Temperature the ideal gases formula is:

[tex]T = \frac{P V }{n R }[/tex]

For the second process you have that [tex]T_{2} = T_{3} [/tex]  So:

[tex]\frac{P_{2} V_{2} }{n R } = \frac{P_{3} V_{3} }{n R }[/tex]

[tex]P_{2} V_{2}  = P_{3} V_{3} [/tex]

[tex]P_{3} = \frac{P_{2} V_{2}}{V_{3}} [/tex]

[tex]P_{3} = 1,317 * 10^{10} Pa[/tex]

Answer:

Explanation:

given,

mass of neutron = 1.675 × 10⁻²⁷ kg

1 kg =  5.58 × 10³⁵ eV/c²                                        

a) mass of proton = 1.675 × 10⁻²⁷ ×  5.58 × 10³⁵ = 9.34 × 10⁸ eV/c²

b) mass of electron = 9.109 x 10⁻³¹ × 5.58 × 10³⁵ = 5.08 × 10⁵eV/c²

c) mass of hydrogen = 1.675 × 10⁻²⁷  × 5.58 × 10³⁵ = 9.34 × 10⁸ eV/c²

Answer:8769 Kg/m^3

Explanation: First of all we have to calculate the volume of the cylinder so it is equal to:

Vcylinder= π*r^2*h where r and h are the radius and heigth, respectively.

Vcylinder= π*0.0275^2*0.048=1.14*10^-4 m^3

The density is defined by ρ=mass/volume

the we have

ρ=1Kg/1.14*10^-4 m^3=8769 Kg/m^3

Answer:

[tex]9.11\times 10^{-15}.[/tex]

Explanation:

The water droplet is initially neutral, it will obtain a 40 nC of charge when a charge of  -40 nC is removed from the water droplet.

The charge on one electron, [tex]\rm e=-1.6\times 10^{-19}\ C.[/tex]

Let the N number of electrons have charge -40 nC, such that,

[tex]\rm Ne=-40\ nC\\\Rightarrow N=\dfrac{-40\ nC}{e}=\dfrac{-40\times 10^{-9}\ C}{-1.6\times 10^{-19}\ C}=2.5\times 10^{11}.[/tex]  

Now, mass of one electron = [tex]\rm 9.11\times 10^{-31}\ kg.[/tex]

Therefore, mass of N electrons = [tex]\rm N\times 9.11\times 10^{-31}=2.5\times 10^{11}\times 9.11\times 10^{-31}=2.2775\times 10^{-19}\ kg.[/tex]

It is the mass of the of the water droplet that must be removed in order to obtain a charge of 40 nC.

Let it is m times the total mass of the droplet which is [tex]25\ \rm mg = 25\times 10^{-6}\ kg.[/tex]

Then,

[tex]\rm m\times (25\times 10^{-3}\ kg) = 2.2775\times 10^{-19}\ kg.\\m=\dfrac{2.2775\times 10^{-19}\ kg}{25\times 10^{-3}\ kg}=9.11\times 10^{-15}.[/tex]

It is the required fraction of mass of the droplet.

Answer:

False.

Explanation:

Yes the magnitude of a vector is always positive , but a vector consists of

when two vectors are added their direction may be opposite to each other For example-

[tex]a=3i+3j[/tex]

[tex]b=3i-6j[/tex] ,

then their resultant

[tex]r=b+a\\r=3i+3j+3i-6j\\r=6i-3j[/tex]

This resultant vector's x and y component equal to y and x component of vector b so its magnitude will be equal to magnitude of vector b.

Therefore, the resultant magnitude not necessary equal to the magnitude of either vector.

Answer:Sometime decreases and sometime increases.

Explanation:

Given

Projectile is launched at [tex]35 ^{\circ}[/tex]

so horizontal velocity=ucos35

vertical velocity=usin35

since gravity acts in the vertical direction, therefore, it affects only the vertical component of the projectile.

So Vertical velocity first decrease up to maximum height and then starts to increase.

Answer with Explanation:

Newton's laws are applicable for inertial frames of reference which is a frame which is not accelerating when seen from the observer standing on earth.

For the person as he presses the brakes his frame is a decelerating frame of reference hence he cannot apply the newtons laws of motion as they are in their original form but if he analyses the motion he has to apply a correction known as  pseudo-force on the object he is analyzing. Pseudo Force has no basis in newton's laws but are a correction that needs to be applied if he wishes to analyse the motion from non inertial frame of reference

While as a person standing on earth outside the car since his frame is an inertial frame of reference he can apply newton's laws of motion without any correction.  

Answer:

The energy of the capacitors connected in parallel is 0.27 J

Given:

C = [tex]2.0\micro F = 2.0\times 10^{- 6} F[/tex]

C' = [tex]4.0\micro F = 4.0\times 10^{- 6} F[/tex]

Potential difference, V = 300 V

Solution:

Now, we know that the equivalent capacitance of the two parallel connected capacitors is given by:

[tex]C_{eq} = C + C' = 2.0 + 4.0 = 6.0\micro F = 6.0\times 10^{- 6} F[/tex]

The energy of the capacitor, E is given by;

[tex]E = \frac{1}{2}C_{eq}V^{2}[/tex]

[tex]E = \frac{1}{2}\times 6.0\times 10^{- 6}\times 300^{2} = 0.27 J[/tex]

Answer:

The speed of the block when it reaches the bottom is 17 m/s.

(b) is correct option.

Explanation:

Given that,

Height = 15 m

Mass = 2 kg

We need to calculate the speed of the block when it reaches the bottom

Using conservation of energy

[tex]mgh=\dfrac{1}{2}mv^2[/tex]

[tex]v^2=2gh[/tex]

[tex]v=\sqrt{2gh}[/tex]

Where, g = acceleration due to gravity

h = height

Put the value into the formula

[tex]v=\sqrt{2\times9.8\times15}[/tex]

[tex]v=17\ m/s[/tex]

Hence, The speed of the block when it reaches the bottom is 17 m/s.

Answer:

1. 6.672 kPa

2. 49.05 mm of mercury

Explanation:

h = 6400 m

Absolute pressure, p = 46 kPa = 46000 Pa

density of air, d = 0.823 kg/m^3

density of mercury, D = 13600 kg/m^3

(a) Absolute pressure = Atmospheric pressure + pressure due to height

46000 = Atmospheric pressure + h x d x g

Atmospheric pressure = 46000 - 6400 x 0.823 x 10 = 6672 Pa = 6.672 kPa

(b) To convert the pressure into mercury pressure

Atmospheric pressure = H x D x g

Where, H is the height of mercury, D be the density of mercury, g be the acceleration due to gravity

6672 = H x 13600 x 10

H = 0.04905 m

H = 49.05 mm of mercury

To calculate the local atmospheric pressure in the city, use the formula P = P0 * e^(-Mgh/RT), where P is the pressure at the current location, P0 is the pressure at sea level, M is the molar mass of air, g is the acceleration due to gravity, h is the altitude, R is the ideal gas constant, and T is the temperature. Plug in the given values into the formula and solve for P0 to find the local atmospheric pressure in kPa. To convert this pressure to mmHg, use the conversion factor that 101.325 kPa is equal to 760 mmHg.

To calculate the local atmospheric pressure in the city, we can use the relationship between altitude, pressure, and density of air. Altitude affects atmospheric pressure because as we go higher in the atmosphere, the air becomes less dense and therefore exerts less pressure. The formula to calculate pressure is given by:

P = P0 * e^(-Mgh/RT)

Where P is the local atmospheric pressure, P0 is the pressure at sea level, M is the molar mass of air, g is the acceleration due to gravity, h is the altitude, R is the ideal gas constant, and T is the temperature. Since we know the altitude and pressure at the current location, we can rearrange the formula to solve for P0:

P0 = P / e^(-Mgh/RT)

Using the given values, we can substitute them into the formula. The density of air is 0.828 kg/m³, the altitude is 6400 m, the absolute pressure is 46 kPa, the molar mass of air is the average molar mass of air (28.97 g/mol), the acceleration due to gravity is 9.8 m/s², the ideal gas constant is 8.314 J/(mol·K), and the temperature can be assumed to be the average temperature at that altitude (around 0°C or 273 K).

Plugging in these values into the formula, we get:

Calculating this expression will give us the local atmospheric pressure in kPa.

To convert this pressure to mmHg, we can use the conversion factor that 101.325 kPa is equal to 760 mmHg. So, to find the pressure in mmHg, we can multiply the pressure in kPa by the conversion factor.

Answer:

(a) 10 s

(b) 236.5 m

(c) Kathy's speed = 47.3 m/s

    Stan's speed = 42.9 m/s

Explanation:

Given:

Assumptions:

Part (a):

 Kathy will catch Stan only if the distances traveled by each of them are equal at the same instant.

[tex]\therefore s_s=s_k\\\Rightarrow u_st_s+\dfrac{1}{2}a_st_s^2=u_kt_k+\dfrac{1}{2}a_kt_k^2\\ \Rightarrow (0)(t+1)+\dfrac{1}{2}(3.9)(t+1)^2=(0)(t)+\dfrac{1}{2}(4.73)t^2\\ \Rightarrow \dfrac{1}{2}(3.9)(t+1)^2=\dfrac{1}{2}(4.73)t^2\\\Rightarrow (3.9)(t+1)^2=(4.73)t^2\\\Rightarrow \dfrac{(t+1)^2}{t^2}=\dfrac{4.73}{3.9}\\\textrm{Taking square root in both sides}\\\dfrac{t+1}{t}= 1.1\\\Rightarrow t+1=1.1t\\\Rightarrow 0.1t = 1\\\Rightarrow t = 10\\[/tex]

Hence, Kathy catches Stan after 11 s from the Stan's starting times.

Part (b):

Distance traveled by Kathy to catch Stan will be distance the distance traveled by her in 10 s.

[tex]s_s = u_kt_k+\dfrac{1}{2}a_kt_k^2\\\Rightarrow s_s= (0)(t)+\dfrac{1}{2}(4.73)t^2\\\Rightarrow s_s= \dfrac{1}{2}(4.73)(10)^2\\\Rightarrow s_s= 236.5[/tex]

Hence, Kathy traveled a distance of 236.5 m to overtake Stan.

Part (c):

[tex]v_k = u_k+a_kt_k\\\Rightarrow v_k = 0+(4.73)(t)\\\Rightarrow v_k = (4.73)(10)\\\Rightarrow v_k =47.3[/tex]

The speed of Kathy at the instant she catches Stan is 47.3 m/s.

[tex]v_s = u_s+a_st_s\\\Rightarrow v_s = 0+(3.9)(t+1)\\\Rightarrow v_s = (3.9)(10+1)\\\Rightarrow v_s =42.9[/tex]

The speed of Stan at the instant Kathy catches him is 42.9 m/s.

Answer:

The acceleration of proton will be [tex]431.137\times 10^8m/sec^2[/tex]

Explanation:

We have given electric field E = 450 N/C

Charge on proton [tex]=1.6\times 10^{-19}C[/tex]

Force on electron due to electric field is given by [tex]F=qE=1.6\times 10^{-19}\times 450=720\times 10^{-19}N[/tex]

Mass of electron [tex]m=1.67\times 10^{-27}kg[/tex]

Now according to second law of motion [tex]F=ma[/tex]

So [tex]720\times10^{-19}=1.67\times 10^{-27}a[/tex]

[tex]a=431.137\times 10^8m/sec^2[/tex]

So the acceleration of proton will be [tex]431.137\times 10^8m/sec^2[/tex]

Answer:

T₁ = 19.39N   with an angle of 52° with the horizontal.

T₂ = 15.64N  with an angle of 40° with the horizontal.

Explanation:

1)We calculate the weight of holiday decoration

W = m×g

W: holiday decoration weight in Newtons (N)

m: holiday decoration weight = 3 kg

g: acceleration due to gravity = 9.8 m/s²

W = m×g = 3 kg × 9.8 m/s² = 29.4 N

2)We apply Newton's first law to the system in equilibrium:

ΣFx = 0

T₁Cos52° - T₂Cos40°=0

T₁Cos52° = T₂Cos40°

T₁ = T₂(Cos40° / T₁Cos52°)

T₁ = 1,24×T₂  Equation (1)

ΣFy = 0

T₁Sin52° + T₂Sin40° - 29.4 = 0 Equation (2)

We replace T₁ of  Equation (1) in the Equation (2)

1.24×T₂×Sin52° + T₂Sin40° - 29.4 = 0

1.88*T₂ = 29.4

T₂ = 29.4/1.88

T₂ = 15.64N

We replace T₂ = 15.64N in the Equation (1)  to calculate T₁:

T₁ = 1.24×15.64N

T₁ = 19.39N

Answer: 20 m/s

Explanation: To solve this problem we have to consider the expression of the kinetic energy given by:

Ekinetic=(1/2)*(m*v^2)

then E=0.5*30Kg*(20 m/s)^2=15*400=6000J

Answer:

9.934 m/s²

Explanation:

Given:

Initial speed of the Bugatti Veyron Super Sport = 0 mi/h

Final speed of the Bugatti Veyron Super Sport = 60 mi/h

Now,

1 mi/h = 0.44704 m / s

thus,

60 mi/h = 0.44704 × 60 = 26.8224 m/s

Time = 2.70 m/s

Now,

The acceleration (a) is given as:

[tex]a=\frac{\textup{Change in speed}}{\textup{Time}}[/tex]

thus,

[tex]a=\frac{26.8224 - 0 }{2.70}[/tex]

or

a = 9.934 m/s²

Answer:

Velocity of the truck, v = 4 m/s                                                            

Explanation:

It is given that,

Distance covered by the truck, d = 100 m

Time taken by the truck, t = 25 s

Let v is the velocity of the truck. We know that the velocity is a vector quantity. Mathematically, it is given by :

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

[tex]v=\dfrac{100\ m}{25\ s}[/tex]

v = 4 m/s

So, the velocity of the truck is 4 m/s. Hence, this is the required solution.

Answer:

Light travels faster in a less dense medium

Explanation:

In a less dense medium, light will travel faster than it did in the more dense medium (an example of this transition can be light going from water to air).

This is because in a less dense medium there ares less particles, so light finds less resistence in its path.

The opposite happens in a more dense medium, there are more particles so light will travel slower because more particles will get in the way.

In summary, the more dense the medium, the more particles it has, and the slower that light will travel in it.

If[tex]\( q_1 = 0 \), then \( q_2 = 11 \)[/tex] which is positive.

Thus, [tex]\( q_1 = 0 \, \text{C} \) and \( q_2 = 11 \, \text{C} \)[/tex] are the valid solutions.

To find the charges [tex]\( q_1 \) and \( q_2 \),[/tex] we can use Coulomb's law, which states that the magnitude of the force between two point charges is given by:

[tex]\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \][/tex]

Given that the force exerted by one charge on the other has a magnitude of 8 mN (milli-Newtons) when they are separated by 4.5 m, we can set up the equation as follows:

[tex]\[ 8 \times 10^{-3} \, \text{N} = \frac{(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \cdot |q_1 \cdot q_2|}{(4.5 \, \text{m})^2} \][/tex]

Solving for[tex]\( |q_1 \cdot q_2| \[/tex]), we get:

[tex]\[ |q_1 \cdot q_2| = \frac{8 \times 10^{-3} \, \text{N} \cdot (4.5 \, \text{m})^2}{8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2} \][/tex]

[tex]\[ |q_1 \cdot q_2| = \frac{8 \times 10^{-3} \times 20.25}{8.99 \times 10^9} \, \text{C}^2 \][/tex]

[tex]\[ |q_1 \cdot q_2| = \frac{0.162}{8.99 \times 10^9} \, \text{C}^2 \][/tex]

[tex]\[ |q_1 \cdot q_2| = 1.802 \times 10^{-11} \, \text{C}^2 \][/tex]

Now, since both charges are positive and repel each other, we know that [tex]\( q_1 \)[/tex]is smaller than [tex]\( q_2 \)[/tex]. So, let's assume [tex]\( q_1 = x \) and \( q_2 = 11 - x \)[/tex]. Substituting these values into the equation for [tex]\( |q_1 \cdot q_2| \)[/tex], we get:

[tex]\[ x \cdot (11 - x) = 1.802 \times 10^{-11} \, \text{C}^2 \][/tex]

[tex]\[ 11x - x^2 = 1.802 \times 10^{-11} \, \text{C}^2 \][/tex]

Now, solve this quadratic equation to find  x , which represents [tex]\( q_1 \).[/tex] Once we find x , we can find [tex]\( q_2 = 11 - x \).[/tex] Then, we verify that both charges are positive and satisfy the given conditions.

To solve the quadratic equation [tex]\( 11x - x^2 = 1.802 \times 10^{-11} \, \text{C}^2 \)[/tex], we first rearrange it to the standard form:

[tex]\[ x^2 - 11x + 1.802 \times 10^{-11} = 0 \][/tex]

We can use the quadratic formula to find[tex]\( x \):[/tex]

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

where [tex]\( a = 1 \), \( b = -11 \), and \( c = 1.802 \times 10^{-11} \).[/tex]

Plugging in these values:

[tex]\[ x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(1.802 \times 10^{-11})}}{2(1)} \][/tex]

[tex]\[ x = \frac{11 \pm \sqrt{121 - 7.208 \times 10^{-11}}}{2} \][/tex]

[tex]\[ x = \frac{11 \pm \sqrt{121 - 7.208 \times 10^{-11}}}{2} \][/tex]

[tex]\[ x = \frac{11 \pm \sqrt{121 - 7.208 \times 10^{-11}}}{2} \][/tex]

[tex]\[ x = \frac{11 \pm \sqrt{121 - 7.208 \times 10^{-11}}}{2} \][/tex]

[tex]\[ x = \frac{11 \pm \sqrt{121 - 7.208 \times 10^{-11}}}{2} \][/tex]

[tex]\[ x \approx \frac{11 \pm \sqrt{121}}{2} \][/tex]

[tex]\[ x \approx \frac{11 \pm 11}{2} \][/tex]

[tex]\[ x_1 \approx \frac{11 + 11}{2} \][/tex]

[tex]\[ x_1 \approx \frac{22}{2} \][/tex]

[tex]\[ x_1 \approx 11 \][/tex]

[tex]\[ x_2 \approx \frac{11 - 11}{2} \][/tex]

[tex]\[ x_2 \approx \frac{0}{2} \][/tex]

[tex]\[ x_2 \approx 0 \][/tex]

So, we have [tex]\( x_1 = 11 \) and \( x_2 = 0 \).[/tex]

Since [tex]\( q_1 \)[/tex]  must be smaller than [tex]\( q_2 \),[/tex] we take [tex]\( x_1 = 11 \) as \( q_1 \) and \( x_2 = 0 \) as \( q_2 \).[/tex]

Therefore,[tex]\( q_1 = 11 \, \text{C} \) and \( q_2 = 0 \, \text{C} \)[/tex]. But we need both charges to be positive, so this solution is not valid.

Let's check the other root [tex]\( x_2 = 0 \):[/tex]

If[tex]\( q_1 = 0 \), then \( q_2 = 11 \)[/tex] which is positive.

Thus, [tex]\( q_1 = 0 \, \text{C} \) and \( q_2 = 11 \, \text{C} \)[/tex] are the valid solutions.

Answer:

- 196.2 J

Explanation:

Given:

Mass of the ball = 2.00 kg

Height through which ball drops = 10.0 m

Since it is mentioned that the ball comes to a stop in the mud, therefore the velocity of the ball will be zero i.e v = 0 m/s

Thus.

the kinetic energy = [tex]\frac{1}{2}mv^2[/tex]  

= [tex]\frac{1}{2}m\times0^2[/tex]  = 0 J

Now,

The potential energy = - mgh

here, g is the acceleration due to gravity

and, the negative sign depicts that the ball is falling down

thus,

The potential energy = - 2 × 9.81 × 10 =  - 196.2 J

Hence, the total energy = - 196.2 + 0 = - 196.2 J

Final answer:

The sum of the gravitational potential and kinetic energy of the 2.00-kg ball after it comes to a stop in the mud at the bottom of a 10.0-m deep well is 0 Joules, since both potential and kinetic energy are zero at rest.

Explanation:

When the 2.00-kg ball is dropped into a 10.0-m deep well, it initially has a certain amount of gravitational potential energy which is converted entirely to kinetic energy just before it hits the bottom, assuming there is no air resistance. The sum of its potential and kinetic energy at the top is equal to the gravitational potential energy it has due to its position. This can be calculated using the formula for gravitational potential energy (GPE), which is GPE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and h is the height above the reference point. In this case, when the ball comes to a stop in the mud at the bottom of the well, all its potential energy has been converted to kinetic energy during the fall, and then to other forms of energy (like heat or sound) upon impact. The ball's potential energy at the bottom of the well is zero and the kinetic energy is also zero since the ball is at rest, so the sum of its potential and kinetic energy after coming to a stop is:

mgh = 2*9.81*10 = -196.2 J (as the ball is falling down)

Sum = -196.2J

According to the conservation of energy principle, energy is neither created nor destroyed, only transformed from one form to another. Therefore, initially, the ball had potential energy which got converted to kinetic energy during the fall and finally to other forms of energy when it came to rest.

Answer:595.32 N

Explanation:

Given

Mechanic shop has a car of weight([tex]F_1[/tex]) 16,500 N

Area of lift([tex]A_1[/tex])=[tex]125 cm^2[/tex]

Area of small piston([tex]A_2[/tex])=[tex]4.51 cm^2[/tex]

According to pascal's law pressure transmit will be same in all direction in a closed container

[tex]P_1=P_2[/tex]

[tex]\frac{F_1}{A_1}=\frac{F_2}{A_2}[/tex]

[tex]F_2=F_1\frac{A_2}{A_1}[/tex]

[tex]F_2=16,500\times \frac{4.51}{125}=595.32 N[/tex]

Answer:

Explanation:

Relative velocity = 100 - 15 = 85 m /s

Distance to be covered up with this relative speed = 1.3km

= 1300m

If t be the time taken

85 x t = 1300

t = 1300/ 85

= 15.29

Time of fall of the bomb t₁ by 1 km

1000 = 1/2 g t₁²

t₁² = 2000/9.8

t₁ = 14.28 s

Since t₁≠ t₂

It will not score a direct hit.

Answer:

Answer

Explanation:

If the satellite is in perfect orbit around the earth then gravitational pull of the earth on the satellite and moving forward satellite is in perfect balance i.e gravitational pull does not effect the orbit as the velocity of the satellite is good enough to counter the gravitational pull of the earth and inertia makes the satellite move forward with greater velocity, but when its velocity is suddenly reduced, the gravitational pull will counter on the orbiting satellite and it will crash onto the earth.

This is same as in well of death when motorist moves in circular motion inside the well with centripetal and centrifugal forces balanced, suddenly crashes on the ground when velocity of the motorist is not enough to balance forces.

the satellite's mass has no affect on its orbit around the earth rather the mass of the earth is only affecting factor .

Answer:

Explanation:

Given

Velocity of women relative to the car is u

Let v be the velocity of car after women jump off

Therefore women velocity relative to the ground is v-u

Conserving momentum

Mv+Nm(v-u)=0

Mv+Nmv=Nmu

[tex]v=\frac{Nm}{M+Nm}[/tex]

(b)Let [tex]v_r[/tex] be the velocity of car after r women has jumped and [tex]v_N[/tex] be the final velocity of the car after N women have jumped.

After r women jumped and N-r on cart momentum is given by

[tex]P_r=\left [ M+\left ( N-r\right )m\right ]v_r[/tex]

After next woman jumps car has velocity of [tex]v_{r+1}[/tex] while the woman has velocity is [tex]v_{r+1}-u[/tex](relative to the ground)

Total momentum

[tex]P_{r+1}=\left [ M+\left ( N-r-1\right )m\right ]v_{r+1}+m\left ( v_{r+1}-u\right )[/tex]

Since total momentum in horizontal direction is conserved then

[tex]P_{r+1}=P_r[/tex]

[tex]v_{r+1}-v_r=\frac{mu}{\left ( M+\left ( N-r\right )m\right )}[/tex]

Summing the above expression from r=0 to r=N-1 we get

[tex]\sum_{j=1}^{j=N}\frac{mu}{\left ( M+jm\right )}[/tex]

(c)Answer in part b is greater because j\leq N[/tex]

Thus Velocity in part b is greater.

Intuitively if you jump after an another person you will impart extra momentum to the cart compared to when all persons jumped off simultaneously.

When all women jump off the flatcar at the same time, the final velocity of the flatcar is zero. However, when the women jump off one at a time, the flatcar gains momentum and moves in the opposite direction, resulting in a non-zero final velocity.

To answer the given question, we can apply the principle of conservation of momentum. When all the women jump off the flatcar at the same time, the total momentum of the system must remain constant. Since the women are jumping off the flatcar with equal and opposite velocities, the total momentum of the system before and after their jumps will be zero. Therefore, the final velocity of the flatcar will also be zero.

On the other hand, when the women jump off one at a time, the total momentum of the system does not remain constant. Initially, the flatcar and the first woman have a total momentum of zero. But as the first woman jumps off with a velocity u, the flatcar gains an equal and opposite momentum. When the second woman jumps off, the flatcar gains momentum again, and this process continues until all women have jumped off. As a result, the flatcar gains momentum with each jump and moves in the opposite direction of the women. Therefore, the final velocity of the flatcar when the women jump off one at a time will be greater than zero.

From a physical and intuitive perspective, the answer to (b) is greater because as each woman jumps off, the flatcar gains momentum in the opposite direction. This momentum accumulates with each jump, resulting in a higher final velocity for the flatcar compared to when all the women jump off simultaneously.

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Final answer:

The motorist's total displacement is 179.58 km north, and his average velocity is approximately 63.38 km/h north for the entire trip.

Explanation:

Displacement and Average Velocity Calculation

For the given problem, we can find the motorist's total displacement by adding the distances traveled in each segment of the journey, all in the same direction (north). First, the motorist drives north for 35.0 minutes at 85.0 km/h. The distance for this part of the trip is:

Distance = Speed × Time = 85.0 km/h × (35.0 min / 60 min/h) = 49.58 km (rounded to 2 decimal places)

The next segment involves a stop, so the displacement remains unchanged.

Finally, the motorist drives an additional 130 km north.

Therefore, the total displacement is the sum of these distances: Total Displacement = 49.58 km + 130 km = 179.58 km north.

To find the average velocity, we need to consider the total displacement and the total time, including the stop:

Total time = Driving time + Stopping time = (35.0 min + 2.00 h + 15.0 min) = 2.833 hours (2 hours and 50 minutes).

Average Velocity = Total Displacement / Total Time = 179.58 km / 2.833 h ≈ 63.38 km/h north.

The motorist's total displacement is 179.58 km north, and his average velocity over the entire trip is approximately 63.38 km/h north.