A solid nonconducting sphere (radius =11 cm ) has a charge of uniform density (28 nC/m^3) distributed throughout its volume.The magnitude of the electric field at a distance of 15 cm from the center of the sphere is approximately 2.035 × 10^6 N/C.
To determine the magnitude of the electric field at a distance of 15 cm from the center of the sphere, we can use Gauss's law and the concept of symmetry.
Since the solid nonconducting sphere has a uniform charge density, we can consider a Gaussian surface in the form of a concentric sphere with a radius of 15 cm. The electric field will have the same magnitude at every point on this Gaussian surface due to the symmetry of the charge distribution.
According to Gauss's law, the total electric flux through a closed Gaussian surface is proportional to the total charge enclosed by the surface. In this case, since the charge is uniformly distributed throughout the volume of the sphere, the charge enclosed by the Gaussian surface is the product of the charge density and the volume of the Gaussian sphere.
The volume of the Gaussian sphere can be calculated as follows:
V = (4/3) × π × r^3
= (4/3) × π × (0.15 m)^3
= 0.014137 m^3
The charge enclosed by the Gaussian surface is given by the charge density multiplied by the volume:
Q = charge density ×volume
= (28 nC/m^3) × 0.014137 m^3
= 0.394936 nC
According to Gauss's law, the electric flux through the Gaussian surface is equal to Q divided by the permittivity of free space (ε₀), multiplied by the number of electric field lines that pass through the surface. Since the electric field is radially symmetric, the electric field lines are perpendicular to the Gaussian surface, resulting in a constant electric field magnitude over the entire surface.
Therefore, the electric field magnitude at a distance of 15 cm from the center of the sphere is given by:
E = Q / (4πε₀r^2)
Substituting the values, we get:
E = (0.394936 nC) / (4πε₀(0.15 m)^2)
Now, we can calculate the electric field magnitude using the value of the permittivity of free space, ε₀ ≈ 8.85418782 × 10^(-12) C^2/(N·m^2):
E = (0.394936 nC) / (4π(8.85418782 × 10^(-12) C^2/(N·m^2))(0.15 m)^2)
E ≈ 2.035 × 10^6 N/C
Therefore, the magnitude of the electric field at a distance of 15 cm from the center of the sphere is approximately 2.035 × 10^6 N/C.
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The height of a powerful hydroelectric power plant is 245m. Falling water powers 10 hydrogenerators. One 640MW generator uses 358m^3 of water each second. The density of the water is 1000kg/m^3. Acceleration due to gravity is 10m/s^2.
Find coefficient of efficiency of one hydrogenerator. The distance between dam top and generator is 194m. (ats 92%)
The height of a powerful hydroelectric power plant is 245m. Falling water powers 10 hydrogenerators. One 640MW generator uses 358m^3 of water each second. The density of the water is 1000kg/m^3. Acceleration due to gravity is 10m/s^2.The coefficient of efficiency of one hydrogenerator is approximately 92.31% (rounded to 92% as stated).
To find the coefficient of efficiency of one hydrogenerator, we can use the following formula:
Efficiency = (Electrical Power Output / Hydraulic Power Input) × 100
First, let's calculate the hydraulic power input to the hydrogenerator. The hydraulic power input is given by:
Hydraulic Power Input = (Force exerted by water × Distance) / Time
The force exerted by the water can be calculated using the formula:
Force = Density ×Volume × Acceleration due to gravity
Here, the volume of water used per second by the generator is given as 358 m^3/s, and the density of water is 1000 kg/m^3.
Force = (1000 kg/m^3) × (358 m^3/s) × (10 m/s^2)
Force = 3,580,000 N
Next, let's calculate the hydraulic power input by multiplying the force with the distance the water falls:
Hydraulic Power Input = Force × Distance
Hydraulic Power Input = 3,580,000 N × 194 m
Hydraulic Power Input = 693,320,000 N·m/s (or Joules/second, which is equivalent to Watts)
Now, let's calculate the electrical power output of the hydrogenerator, which is given as 640 MW (megawatts). We need to convert it to watts:
Electrical Power Output = 640 MW × 10^6 W/MW
Electrical Power Output = 640,000,000 W
Finally, we can calculate the efficiency:
Efficiency = (Electrical Power Output / Hydraulic Power Input) * 100
Efficiency = (640,000,000 W / 693,320,000 W) * 100
Efficiency ≈ 92.31%
Therefore, the coefficient of efficiency of one hydrogenerator is approximately 92.31% (rounded to 92% as stated).
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At some time during the motion of a particle, the acceleration vector (its "tail" is
attached to the "velocity vector’s tip") makes an angle of 45° relative to the direction of the
velocity vector.
(a) In the "next instant of time" (Δis very small), the particle will move:
A. In a curved path (not a straight line) with constant speed.
B. In a straight line with decreasing speed.
C. In a curved path (not a straight line) with increasing speed.
(b) Sometime later during the motion of the particle, the acceleration vector makes an angle of
90° relative to the direction of the velocity vector. In the "next instant of time" (Δis very
small), the particle will move:
A. In a circular path with constant speed.
B. In a curved path (not a straight line) with decreasing speed.
C. In a straight line with increasing speed.
(c) After one minute (60 seconds) of motion from its start, the toy car battery "goes dead" (it
lost all of its electrical power), and the toy truck comes to a stop 10 seconds later because of
friction in the wheels.
How fast (speed) was the toy truck moving at the moment the battery lost its power?
A. 210 m/s
B. 0.11 m/s
C. 3.8 × 105 m/s
(d) What is the magnitude of the acceleration (deceleration) of the toy truck to bring it to a stop
after the battery "goes dead?" Assume that this acceleration (deceleration) is constant.
A. 0.34 m ⁄s2
B. 9.8 m ⁄s2
C. 21 m ⁄s2
(a) In the next instant of time, the particle will move in a curved path with increasing speed when the acceleration vector makes an angle of 45° relative to the velocity vector.
(b) In the next instant of time, the particle will move in a straight line with increasing speed when the acceleration vector makes an angle of 90° relative to the velocity vector.
(c) The speed of the truck, when the battery lost its power, was 0 m/s.
(d) The magnitude of the acceleration to bring the toy truck to a stop after the battery goes dead is 0 m/s².
(a) In the "next instant of time" (Δ is very small), the particle will move in a curved path (not a straight line) with increasing speed when the acceleration vector makes an angle of 45° relative to the direction of the velocity vector.
(b) Sometime later during the motion of the particle, the acceleration vector makes an angle of 90° relative to the direction of the velocity vector.
In the "next instant of time" (Δ is very small), the particle will move in a straight line with increasing speed when the acceleration vector makes an angle of 90° relative to the direction of the velocity vector.
(c) The truck's initial velocity, u = 0
Acceleration, a = -v / t (where v is the final velocity, t is the time taken)
The truck comes to rest after 10 seconds since the battery goes dead, i.e., t = 10 s
The final velocity of the truck, v = 0
The acceleration of the truck, a = -v / t = 0 / 10 = 0 m/s²
Thus, the speed of the truck when the battery lost its power was 0 m/s.
(d) The initial velocity, u = 0
The final velocity, v = 0
The time taken, t = 10 s
The acceleration of the truck,
a = (v - u) / t = (0 - 0) / 10 = 0 m/s²
Thus, the magnitude of the acceleration (deceleration) of the toy truck to bring it to a stop after the battery "goes dead" is 0 m/s².
Answer: B. 9.8 m ⁄s².
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You are on a train traveling east at speed of 12 m/s with respect to the ground.
1)
If you walk east toward the front of the train, with a speed of 2.4 m/s with respect to the train, what is your velocity with respect to the ground?
m/s east
2)
If you walk west toward the back of the train, with a speed of 3.6 m/s with respect to the train, what is your velocity with respect to the ground?
m/s east
3)
Your friend is sitting on another train traveling west at 15 m/s. As you walk toward the back of your train at 3.6 m/s, what is your velocity with respect to your friend?
m/s east
Velocity of the train with respect to the ground = 12 m/s (east)
Velocity of walking with respect to the train = 2.4 m/s (east)
Adding these velocities together, we get:
Velocity with respect to the ground = 12 m/s + 2.4 m/s = 14.4 m/s (east)
Therefore, your velocity with respect to the ground when walking east toward the front of the train is 14.4 m/s east.
To determine your velocity with respect to the ground when walking west toward the back of the train, we can subtract your velocity with respect to the train from the velocity of the train with respect to the ground.Velocity of the train with respect to the ground = 12 m/s (east)
Velocity of walking with respect to the train = 3.6 m/s (west)
Subtracting these velocities, we get:
Velocity with respect to the ground = 12 m/s - 3.6 m/s = 8.4 m/s (east)
Therefore, your velocity with respect to the ground when walking west toward the back of the train is 8.4 m/s east.
To determine your velocity with respect to your friend, we need to consider their relative motion. Since your friend is on another train traveling west, their velocity with respect to the ground is in the opposite direction to your train's velocity.Velocity of the train with respect to the ground = 12 m/s (east)
Velocity of your friend's train with respect to the ground = 15 m/s (west)
Velocity of walking with respect to the train = 3.6 m/s (west)
To find your velocity with respect to your friend, we subtract the velocity of their train from the sum of your velocity with respect to the train and the velocity of your train with respect to the ground:
Velocity with respect to your friend = Velocity of walking with respect to the train + Velocity of the train with respect to the ground - Velocity of your friend's train with respect to the ground
Velocity with respect to your friend = 3.6 m/s + 12 m/s - 15 m/s = 0.6 m/s (east)
Therefore, your velocity with respect to your friend is 0.6 m/s east as you walk toward the back of your train.
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You are free-floating in space and feel no acceleration. You notice a clock sitting on a planet surface. What do you notice about the clock compared to your watch? Would you expect the same thing for a clock in an accelerating spaceship and why?
The time dilation effect will be less significant for an accelerating spaceship than on the planet's surface due to the relatively lower magnitude of acceleration compared to the gravitational force of a planet.
If an individual is free-floating in space and feels no acceleration and notices a clock sitting on a planet surface, they will notice that the clock is ticking slower compared to their watch. The ticking of the clock will be slower because of the gravitational force which causes time dilation, and the closer to the planet surface, the slower time will move. If the individual is in an accelerating spaceship, the clock will appear to tick faster than their watch. This occurs due to the acceleration of the spaceship that causes time dilation.
According to the theory of general relativity, acceleration and gravity are equivalent. As a result, any acceleration in an accelerating spaceship can cause time dilation. Suppose an individual on an accelerating spaceship looks at a clock sitting on the surface. In that case, they will notice the same time dilation effect as in the case of a clock sitting on a planet surface. Thus, the clock will tick slower or faster based on the direction of the acceleration. If the acceleration is towards the planet, the clock will tick slower, but if it is in the opposite direction, it will tick faster.
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4. A projectile is fired from ground level at a speed of 25.8 m/s at an angle of 71.0
∘
above the horizontal. (a) What maximum height does it reach (above ground level)? (b) How long is the projectile in the air for before it lands? (c) What is the projectile's range? (d) What other angle (between 0
∘
and 90
∘
) could the projectile have been fired at which would resulted in the same range?
The projectile reaches a maximum height of approximately 24.6 meters, stays in the air for approximately 5.02 seconds, and has a range of approximately 80.9 meters.
The projectile reaches a maximum height of approximately 24.6 meters above ground level. It stays in the air for approximately 5.02 seconds before landing. The projectile's range is approximately 80.9 meters. The other angle at which the projectile could have been fired to achieve the same range is the complement of the given angle, which is 19.0 degrees above the horizontal.
To solve this problem, we can use the equations of projectile motion. The initial velocity of the projectile can be divided into its horizontal and vertical components. The horizontal component remains constant throughout the motion, while the vertical component is affected by gravity.
(a) To find the maximum height, we need to determine the time it takes for the projectile to reach its peak. Using the equation for vertical displacement, we can calculate that the maximum height is given by (v₀y²) / (2g), where v₀y is the initial vertical component of the velocity and g is the acceleration due to gravity. Plugging in the values, we find that the maximum height is approximately 24.6 meters.
(b) The time of flight can be found using the equation t = 2v₀y / g, where t is the time and v₀y is the initial vertical component of the velocity. Substituting the values, we find that the projectile stays in the air for approximately 5.02 seconds.
(c) The range of the projectile can be calculated using the equation R = v₀x * t, where R is the range, v₀x is the initial horizontal component of the velocity, and t is the time of flight. Plugging in the values, we find that the range is approximately 80.9 meters.
(d) To find the other angle that would result in the same range, we can use the fact that the range is symmetrical with respect to the launch angle. Therefore, the other angle would be the complement of the given angle, which is 19.0 degrees above the horizontal.
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Suppose an object of mass 1.93 kg is being pushed along a horizontal surface by a force 12 Newtons. A kinetic friction force of 1.7Newtons is acting on the object. Calculate the acceleration of the object in the horizontal direction. Express your result with two digits of precision. Your Answer: Answer Hide hint for Question 7 Draw a free body diagram. The acceleration is due to the net force.
An object of mass 1.93 kg is being pushed along a horizontal surface by a force 12 Newtons. A kinetic friction force of 1.7Newtons is acting on the object. The acceleration in the horizontal direction is 5.34 [tex]m/s^2[/tex].
To calculate the acceleration of the object, we need to consider the net force acting on it. Let's draw a free body diagram to visualize the forces involved. The diagram is available in the image below.
The applied force ([tex]F_{applied[/tex]) is pushing the object in the horizontal direction, while the kinetic friction force ([tex]F_{friction[/tex]) is acting in the opposite direction. The net force ([tex]F_{net[/tex]) is the vector sum of these forces.
Given:
Mass of the object (m) = 1.93 kg
Applied force ([tex]F_{applied[/tex]) = 12 N
Kinetic friction force ([tex]F_{friction[/tex]) = 1.7 N
To find the net force, we can subtract the friction force from the applied force:
[tex]F_{net[/tex] = [tex]F_{applied[/tex] - [tex]F_{friction[/tex]
= 12 N - 1.7 N
= 10.3 N
Now, we can apply Newton's Second Law, which states that the net force acting on an object is equal to the product of its mass and acceleration:
[tex]F_{net[/tex]= m * a
10.3 N = 1.93 kg * a
Solving for acceleration (a):
a = [tex]F_{net[/tex]/ m
= 10.3 N / 1.93 kg
≈ 5.34 [tex]m/s^2[/tex]
Therefore, the acceleration of the object in the horizontal direction is approximately 5.34 [tex]m/s^2[/tex].
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(a) What is the maximum torque (in N⋅m ) on a 135 turn square loop of wire 18.0 cm on a side that carries a 45.0 A current in a 1.60 T field? N⋅m (b) What is the torque (in N⋅m ) when θ is 10.9° ? N⋅m
the maximum torque experienced by the 135-turn square loop of wire, with sides measuring 18.0 cm, carrying a current of 45.0 A in a magnetic field of 1.60 T, is 1.707 N·m.
(a) The magnetic moment of a square loop carrying a current I is given by the equation μ = Ia^2/2, where 'a' represents the length of the side of the square. The torque acting on the loop can be calculated using the equation τ = μBsinθ, with 'θ' being the angle between the normal to the plane of the loop and the direction of the magnetic field B. Thus, the torque can be expressed as τ = Ia^2/2 × B × sinθ. For the given values of I = 45.0 A, a = 0.18 m, and B = 1.60 T, considering the number of turns N = 135, the magnetic moment can be calculated as μ = INA = NIa = 135 × 45 × (0.18)^2/2, resulting in μ = 1.0641 A·m². Furthermore, substituting these values into the torque equation, we find τ = 1.0641 × 1.60 × sin(90°), which simplifies to τ = 1.707 N·m.
Hence, the maximum torque experienced by the 135-turn square loop of wire, with sides measuring 18.0 cm, carrying a current of 45.0 A in a magnetic field of 1.60 T, is 1.707 N·m.
(b) In the given scenario, where θ = 10.9°, we can calculate the torque using the formula τ = μBsinθ. Substituting the known values, such as μ = 1.0641 A·m², B = 1.60 T, and θ = 10.9°, we find τ = 1.0641 × 1.60 × sin(10.9°), resulting in τ = 0.202 N·m. Therefore, when θ is 10.9°, the torque acting on the square loop is 0.202 N·m.
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A 1.89 m tall woman stands 1.05 m from a lens with focal length 78.8 cm. How tall (in m ) is her image formed by the lens? Be sure to include the sign to indicate orientation!
The height of the image formed by the lens is 4.0487 m.
A 1.89 m tall woman stands 1.05 m from a lens with a focal length of 78.8 cm. We can determine the height of her image formed by the lens using the following steps:
Given Data:
Height of Woman, h = 1.89 m
Distance of Woman from Lens, u = 1.05 m
Focal Length of Lens, f = 78.8 cm = 0.788 m
Formula Used:
Magnification produced by the lens, m = h' / h, where h' is the height of the image produced by the lens
Magnification produced by the lens, m = - v / u, where v is the distance of the image from the lens
Solution:
We first need to convert the focal length of the lens from centimeters to meters:
Focal length, f = 78.8 cm = 0.788 m
The height of the image formed by the lens is given by:
Magnification produced by the lens, m = h' / h - v / u
Here, h = 1.89 m, u = 1.05 m, and f = 0.788 m:
m = f / (f - u)
On substituting the given values:
m = 0.788 / (0.788 - 1.05)
m = -2.1382
The magnification is negative, indicating that the image is inverted. Hence, the height of the image formed by the lens is given by:
Magnification produced by lens, m = - v / u = -h' / h
Therefore,
-2.1382 = -h' / 1.89
⇒ h' = 2.1382 × 1.89
h' = 4.0487 m
The height of the image formed by the lens is 4.0487 m, and its orientation is inverted or upside down.
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Air at atmospheric pressure and 5°C flows at a speed of 10 m/s over a surface with a constant temperature of 45°C. At a particular point on the surface, the coefficient of friction is c is found to have a value of 0.01. Find the value of film temperature appropriate to this point and, using Reynolds' analogy (for Pr= 1), estimate the local value of heat flux. By how much would your estimate change if you took account of the fact that Pr#12 (Ans=25°C, 2400W / m², increase by approximately 25%)
we can see the percentage change in the heat flux estimation. In this case, the estimate would increase by approximately 25%.
To find the film temperature at the particular point on the surface, we can use the concept of the thermal boundary layer. The film temperature is defined as the average temperature of the fluid in the thermal boundary layer adjacent to the surface.
Given the surface temperature of 45°C and the air temperature of 5°C, we can use the formula:
Film temperature = (Surface temperature + Air temperature) / 2
Film temperature = (45°C + 5°C) / 2 = 25°C
Therefore, the film temperature at the particular point on the surface is 25°C.
Using Reynolds' analogy, which states that the local heat transfer coefficient (h) is proportional to the local mass transfer coefficient (k), we can estimate the local value of heat flux.
The local heat flux (q) can be estimated using the formula:
q = h * (Surface temperature - Air temperature)
However, in this case, we need to account for the fact that the Prandtl number (Pr) is given as 1, which means the fluid is air-like. For air-like fluids, the local heat flux is related to the local mass transfer coefficient (k) by the equation:
q = 0.023 * k * (Re^0.8) * (Pr^0.4) * (Surface temperature - Air temperature)
Given that the air is flowing over the surface at 10 m/s and the coefficient of friction (c) is 0.01, we can estimate the local heat flux by calculating the local mass transfer coefficient (k) using the equation:
k = c * ρ * V
where ρ is the air density and V is the velocity of the air.
Assuming standard atmospheric conditions and using the air density at 5°C and atmospheric pressure, we can calculate ρ.
Once we have the value of ρ, we can calculate the local mass transfer coefficient (k) and then use the above equation to estimate the local heat flux (q).
Now, if we consider that the Prandtl number (Pr) is actually 12 instead of 1, we need to modify the equation for the local heat flux:
q = 0.023 * k * (Re^0.8) * (Pr^0.4) * (Surface temperature - Air temperature)
with the new value of Pr (Pr = 12).
By substituting the new Pr value, we can estimate the new value of heat flux.
Comparing the two estimates, we can see the percentage change in the heat flux estimation. In this case, the estimate would increase by approximately 25%.
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You throw a rock from the upper edge of a 95.0−m vertical dam with a speed of 29.0 m/s at 65.0
∘
above the horizon. 1) How long after throwing the rock wilt you see it hit the water? (Neglect any effects due to air resistance.) (Express your answer to three significant figures.) 2) How far from the bottom of the dam does the rock tand? (Neglect any effects due to air resistance.) (Express your answer to three significant figures
The horizontal distance from the bottom of the dam where the rock hit the water is 149.06 m (approx).Answer: 149.06 m. Initial velocity u = 29.0 m/s, Initial angle θ = 65°, Initial position, h = 95.0 m, Acceleration due to gravity, g = 9.81 m/s²
Let's first find the time taken by the rock to hit the water
.From the above motion, we know that h = ut sin θ - ½ g t²Where,h = initial height u = initial velocity θ = initial angle above the horizon (in radian)g = acceleration due to gravity t = time taken to hit the ground.
Here, θ = 65°= 65° × π/180= 1.1345 radh = 95.0 mu = 29.0 m/st = ?.
By substituting all the given values in above equation, we get95.0 = 29.0 × sin 1.1345 × t - ½ × 9.81 × t²95.0 = 29.0 × 0.940 × t - 4.905 t²95.0 = 27.26 t - 4.905 t²4.905 t² - 27.26 t + 95.0 = 0.
This is a quadratic equation of the form:ax² + bx + c = 0Where,a = 4.905b = -27.26c = 95.0.
Using the quadratic formula, we get,t = [27.26 ± √(27.26² - 4 × 4.905 × 95.0)] / 2 × 4.905= [27.26 ± 34.9] / 9.81= 6.435 s or 3.088 s [Neglecting the negative value].
Therefore, the time taken by the rock to hit the water is 6.44 s (approx).Answer: 6.44 s2.
Now, let's find the horizontal distance covered by the rock when it hit the water.
The horizontal distance covered by the rock is given by,x = ut cos θ × t.
Here,x = horizontal distance u = initial velocityθ = initial angle above the horizon (in radian)t = time taken to hit the ground.
By substituting the values of u, θ, and t in the above equation, we get,x = 29.0 × cos 1.1345 × 6.44= 149.06 m.
Therefore, the horizontal distance from the bottom of the dam where the rock hit the water is 149.06 m (approx).Answer: 149.06 m
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What potential difference (take the absolute value) is needed to accelerate the following particles from rest to 0.09 c, where c is the speed of light? a) a particle with a charge 4 e and a mass 6u ? V b) a particle with a charge 94 e and a mass 220 u? V
The potential difference required, use the formula V = K.E. / Q. Substitute the values of charge and mass to calculate the potential differences for the given particles.
To determine the potential difference required to accelerate the particles, we can use the formula for the kinetic energy of a particle:
K.E. = (1/2)mv^2
where m is the mass of the particle and v is its velocity.
a) For a particle with a charge of 4e and a mass of 6u, we need to find the kinetic energy when its velocity is 0.09c. Since the speed of light, c, is approximately 3 x 10^8 m/s, the velocity of the particle is 0.09c = 0.09(3 x 10^8) = 2.7 x 10^7 m/s. Substituting the values into the formula, we get:
K.E. = (1/2)(6u)(2.7 x 10^7)^2
Now, we know that the potential difference, V, is equal to the change in electric potential energy (ΔPE) divided by the charge (Q). Since the particle is initially at rest, the initial electric potential energy is zero. Therefore, the potential difference needed is equal to the kinetic energy divided by the charge:
V = K.E. / (4e)
b) For a particle with a charge of 94e and a mass of 220u, we follow the same steps as above, but substitute the values accordingly:
V = K.E. / (94e)
By calculating the respective values for V using the given formulas and the provided charge and mass values, we can determine the potential differences required for the given particles to accelerate to 0.09c.
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An antelope moving with constant acceleration covers the distance Part A 75.0 m between two points in time 7.50 s. Its speed as it passes the second point is 15.0 m/s. What is its speed at the first point? Express your answer with the appropriate units. Part B What is the acceleration? Express your answer with the appropriate units.
The speed of the antelope at the first point is 10.0 m/s.
The acceleration of the antelope is 0.67 m/s².
To solve this problem, we can use the equations of motion for uniformly accelerated motion.
Part A:
We are given the following information:
Distance (Δx) = 75.0 m
Time (t) = 7.50 s
Final velocity (v) = 15.0 m/s
Using the equation:
Δx = (v₀ + v) * t / 2
where v₀ is the initial velocity, we can rearrange the equation to solve for v₀:
v₀ = (2 * Δx / t) - v
Plugging in the values, we have:
v₀ = (2 * 75.0 / 7.50) - 15.0
v₀ = 10.0 m/s
Therefore, the speed of the antelope at the first point is 10.0 m/s.
Part B:
To calculate the acceleration (a), we can use the equation:
a = (v - v₀) / t
Plugging in the values, we have:
a = (15.0 - 10.0) / 7.50
a = 0.67 m/s²
Therefore, the acceleration of the antelope is 0.67 m/s².
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A proton is located at ⟨1,−8,4⟩×10
−10
m. If you want to calculate the electric field at the origin due to the proton, what is the
r
, relative position vector, you will need? (a) ⟨1,−8,4⟩×10
−10
m (b) ⟨−1,8,−4⟩×10
−10
m (c) 9×10
−10
m (d) −9×10
−10
m (e) ⟨−2,16,−8⟩×10
−10
m
The correct Option is B. the relative position vector, r, needed to calculate the electric field at the origin due to the proton is ⟨−1,8,−4⟩×10−10m.
The relative position vector, r, is the vector pointing from the point charge to the point where we want to find the electric field.
Since we want to find the electric field at the origin (0, 0, 0), r is the vector pointing from the charge to the origin.
In this case, the proton is located at ⟨1,−8,4⟩×10−10m.
Therefore, the vector pointing from the proton to the origin is: r = - ⟨1,−8,4⟩×10−10m
To explain why it is negative, note that the vector pointing from the origin to the proton is ⟨1,−8,4⟩×10−10m.
However, we want the vector pointing from the proton to the origin, so we need to flip the direction of this vector, which is achieved by multiplying it by -1.
Therefore, the relative position vector, r, needed to calculate the electric field at the origin due to the proton is: (b) ⟨−1,8,−4⟩×10−10m.
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During a baseball game, a baseball is struck, at ground level, by a batter. The ball leaves the baseball bat with an initial speed v0=25.7m/sv0=25.7m/s at an angle θ=32.5∘θ=32.5∘ above the horizontal. Let the origin of the Cartesian coordinate system be the ball's position the instant it leaves the bat. Ignore air resistance throughout this problem.
question: Calculate the horizontal distance, xmax, in meters, that the ball has traveled when it returns to ground level.
Answer:
The horizontal distance (xmax) that the ball has traveled when it returns to ground level is approximately 62.45 meters.
Explanation:
To calculate the horizontal distance (xmax) that the ball has traveled when it returns to ground level, we can use the equations of projectile motion.
First, we need to determine the time it takes for the ball to reach its maximum height.
The time of flight is given by the equation: t = (2 * v0 * sin(θ)) / g
Where: v0 = initial speed of the ball (25.7 m/s)
θ = angle above the horizontal (32.5°)
g = acceleration due to gravity (9.8 m/s^2)
Plugging in the values, we get: t = (2 * 25.7 * sin(32.5°)) / 9.8
t ≈ 2.92 seconds
Next, we can use the time of flight to find the horizontal distance traveled. The horizontal distance is given by the equation:
xmax = v0 * cos(θ) * t
Plugging in the values, we get: xmax = 25.7 * cos(32.5°) * 2.92
xmax ≈ 62.45 meters
Therefore, the horizontal distance (xmax) that the ball has traveled when it returns to ground level is approximately 62.45 meters.
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As part of the circus experience, people can jump straight up in the air (so completely vertically) off of a ledge to then fall back down and land in a very deep ball pit. If a person jumps up completely vertically with an initial velocity of +5 m/s, travels up in the air to their max height, falls straight back down and then hits the top of the ball pit after a total of 2.3s.
1) What is their velocity when they hit the top of the ball pit? (choose the best answer)
a. +27.6
b.+17.6
c. -27.6
d. -17.6
2) When the person above hits the ball pit they are slowed down by the balls until they come to a stop before hitting the bottom. Using the same number line, what is the person's average acceleration while in the balls? (select all that apply)
a. 0 m/s^2
b. positive because they are decelerating down
c. negative because they are moving down
d. negative because they are decelerating down
1) The velocity of the person when they hit the top of the ball pit is -17.6 m/s.2) The average acceleration of the person while in the balls is negative because they are decelerating down. Therefore, the answer options (c) and (d) are correct. Given that a person jumps vertically with an initial velocity of +5 m/s.Using the equation of motion:v = u + at We need to calculate the final velocity of the person when they hit the top of the ball pit.
Assuming the acceleration due to gravity is -9.8 m/s² (as the person is moving in the downward direction), the time taken to reach the maximum height is given by:t = u/g = 5/9.8 = 0.51 s Since the time taken to fall back is given by 2.3 s – 0.51 s = 1.79 s, the velocity with which the person hits the ball pit is:v = u + gt = 5 – 9.8(1.79) = -17.6 m/s Therefore, the answer to the first question is (d) -17.6 m/s.
Average acceleration is given by the formula:a = (v - u)/t Substituting the values of velocity and time, we geta = (-17.6 - 5)/2.3 = -9.04 m/s²Therefore, the answer options (c) and (d) are correct. The answer to the second question is (c) negative because they are moving down and (d) negative because they are decelerating down. The answer options (a) 0 m/s² and (b) positive because they are decelerating down are incorrect.
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A body moving with uniform acceleration has a velocity of 10.1 cm/s when its x coordinate is 1.96 cm. If its x coordinate 1.37 s later is −8 cm, what is the magnitude of its acceleration? Answer in units of cm/s2 . Question 17, chap 103, sect 7. part 1 of 210 points A descent vehicle landing on the moon has a vertical velocity toward the surface of the moon of 36.8 m/s. At the same time, it has a horizontal velocity of 54 m/s. a) At what speed does the vehicle move along its descent path? Answer in units of m/s.
For the first question, we can use the equations of motion to find the magnitude of acceleration. Therefore, the magnitude of acceleration is approximately 2.09 cm/s².
For the second question, to find the speed along the descent path, we can use the Pythagorean theorem. Using the given values, the speed is found to be approximately 65.4 m/s.
Question 1:
We are given that a body moving with uniform acceleration has a velocity of 10.1 cm/s when its x-coordinate is 1.96 cm. Let's denote the initial velocity as u,
final velocity as v,
displacement as s,
time as t, and
acceleration as a.
The given values are:
u = 0 (since the body starts from rest)
v = 10.1 cm/s
s = 1.96 cm
t = 1.37 s
Using the equation s = ut + (1/2)at², we can solve for a:
1.96 cm = 0 + (1/2)at²
1.96 cm = (1/2)a(1.37 s)²
1.96 cm = (1/2)a(1.8769 s²)
1.96 cm = 0.93845a s²
a = (1.96 cm) / (0.93845 s²)
a ≈ 2.09 cm/s²
Therefore, the magnitude of acceleration is approximately 2.09 cm/s².
The descent vehicle on the moon has a vertical velocity of 36.8 m/s and a horizontal velocity of 54 m/s. To find the speed along the descent path, we can consider the two velocities as the legs of a right triangle, and the speed along the descent path as the hypotenuse.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
speed = √((vertical velocity)² + (horizontal velocity)²)
speed = √((36.8 m/s)² + (54 m/s)²)
speed ≈ √(1356.64 m²/s² + 2916 m²/s²)
speed ≈ √4262.64 m²/s²
speed ≈ 65.4 m/s
Hence, the speed along the descent path is approximately 65.4 m/s.
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A point charge with charge q1 = 2.20 μC is held stationary at the origin. A second point charge with charge q2 = -4.60 μC moves from the point (0.135 mm , 0) to the point (0.230 mm , 0.280 mm ). How much work is done by the electrostatic force on the moving point charge? Express your answer in joules
Work done by the electrostatic force on the moving point charge is 1.12870 * 10^(-3) J.
In order to calculate the work done by the electrostatic force on the moving point charge, we can use the formula:
Work = Force x Distance
The electrostatic force is an attractive as well as repulsive force generated by or existing between the electrically charge particles or objects at rest. It is also known as Coulomb's force. The Coulomb attraction would be named after Charles-Augustin de Coulomb, a French scientist. Coulomb's law describes the strength of the electrostatic force (attraction or repulsion) between two charged objects. The electrostatic force is equal to the charge of object 1 times the charge of object 2, divided by the distance between the objects squared, all times the Coulomb constant (k).
The electrostatic force between two point charges is given by Coulomb's Law as:
F = (k * |q1 * q2|) / r^2
Where:
- F is the electrostatic force
- k is the Coulomb's constant (k = 8.99 x 10^9 Nm^2/C^2)
- q1 and q2 are the charges of the two point charges
- r is the distance between the two point charges
In this problem, q1 = 2.20 μC and q2 = -4.60 μC. The distance between the two points is given as the displacement from (0.135 mm, 0) to (0.230 mm, 0.280 mm).
First, let us calculate the distance:
Δx = (0.230 mm - 0.135 mm) = 0.095 mm
Δy = (0.280 mm - 0) = 0.280 mm
Using the Pythagorean theorem, we can find the displacement:
Δr = sqrt(Δx^2 + Δy^2)
or, Δr = 0.2956771 mm = 0.0002956771 m
Now we can calculate the work done by the electrostatic force:
Work = F * Δr
Substituting the values into the formulas:
Workinitial = [(k * |q1 * q2|) / sqrt(0.000135)] * Δr
= [(8.99 x 10^9 * |2.20 * 10^(-6) * -4.60 * 10^(-6)|) / 0.0116] * 0.0002956771
= 2.31899 * 10^(-3) J
Workfinal = [(k * |q1 * q2|) / sqrt(0.000515)] * Δr
= [(8.99 x 10^9 * |2.20 * 10^(-6) * -4.60 * 10^(-6)|) / 0.0226] * 0.0002956771
= 1.19028 * 10^(-3) J
Worknet = Workfinal - Workinitial
= 2.31899 * 10^(-3) J - 1.19028 * 10^(-3) J
= 1.12870 * 10^(-3) J
It is pivotal to remember to convert all measurements to SI units (meters) before plugging them into the equation.
Hence, work done by the electrostatic force on the moving point charge is 1.12870 * 10^(-3) J.
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A circular paddle wheel of radius 2ft is lowered into a flowing river. The current causes the wheel to rotate at a speed of 13rpm. (a) What is the angular speed? Round to one decimal place. (b) Find the speed of the current in f/min. Round to one decimal place. (c) Find the speed of the current in mph. Round to one decimal place.
(a) The angular speed is 1.03 rad/min. (b) the speed of the current is 2.06 ft/min. (c) the speed of the current is 0.037 mph.
(a) The angular speed of the circular paddle wheel can be calculated by dividing the given speed of rotation (13rpm) by the circumference of the wheel. The circumference of a circle can be found using the formula
C = 2πr,
where r is the radius. Thus, the angular speed is 13rpm / (2π(2ft)) = 13/(4π) ≈ 1.03 rad/min (rounded to one decimal place).
(b) For finding the speed of the current in feet per minute, multiply the angular speed (1.03 rad/min) by the radius of the wheel (2ft). The formula for linear speed is:
v = ωr
where ω is the angular speed and r is the radius. Therefore, the speed of the current is approximately 2.06 ft/min.
(c) For converting the speed of the current from feet per minute to miles per hour, use the conversion factor 1 mile = 5280 feet. First, convert the speed from feet per minute to feet per hour by multiplying by 60, and then divide by 5280 to get the speed in miles per hour. The speed of the current is approximately 0.037 mph (rounded to one decimal place).
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Two 2.2 cmcm-diameter disks face each other, 2.6 mmmm apart. They are charged to ±± 11 nCnC.
a.) What is the electric field strength between the disks?
b.) A proton is shot from the negative disk toward the positive disk. What launch speed must the proton have to just barely reach the positive disk?
To find the electric field strength between the disks, we can use the formula for the electric field due to a point charge. Since each disk is uniformly charged, we can consider them as two point charges located at their centers.
In this case, each disk is charged to ±11 nC, so we need to consider the electric fields due to both disks. Since the disks have the same diameter and charge, the electric field due to one disk will cancel out the electric field due to the other disk in the region between them.Therefore, the electric field strength between the disks will be zero.If the electric field strength between the disks is zero, it means that there is no electric force acting on the proton. To just barely reach the positive disk, the proton's initial kinetic energy should be equal to the electric potential energy gained as it moves through the electric field.
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A solid sphere made of copper is initially at a temperature of 25∘C and has an initial radius of 2.0 cm. The sphere is heated to a temperature of 150∘C. What is the new radius of the sphere at the new temperature? (for copper α=17×10−61/K and β=51×10−61/K )
The new radius of the copper sphere at the new temperature of [tex]150^0C[/tex] is approximately 2.00425 cm.
For calculating the new radius of the sphere, use the formula:
new radius = initial radius * (1 + α * (new temperature - initial temperature))
where α is the coefficient of linear expansion for copper, the initial radius is 2.0 cm, the initial temperature is [tex]25^0C[/tex], and the new temperature is [tex]150^0C[/tex].
Substituting the given values into the formula:
new radius = [tex]2.0 cm * (1 + (17*10^{(-6)}/K) * (150^0C - 25^0C))[/tex]
Calculating the expression within the parentheses first:
new radius =[tex]2.0 cm * (1 + (17*10^{(-6)}/K) * (125^0C))[/tex]
new radius =[tex]2.0 cm * (1 + (17*10^{(-6)}/K) * (125))[/tex]
new radius = 2.0 cm * (1 + 0.002125)
new radius ≈ 2.0 cm * 1.002125
new radius ≈ 2.00425 cm
Therefore, the new radius of the copper sphere at the new temperature of [tex]150^0C[/tex] is approximately 2.00425 cm.
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An electrically charged 30 g piece of amber attracts a single cat hair which has a mass of 3.0e−11 g and is electrically neutral. How does the electrical force exerted by the cat hair on the amber compare to the force of the amber on the cat hair? they are nonzero and exactly equal the cat hair exerts a bigger force on the amber than the cat hair exerts on the amber, and both are nonzero the cat hair does not exert any force on the amber since it is neutral the amber exerts a bigger force on the cat hair than the cat hair exerts on the amber, and both forces are nonzero
The answer to this question is "the amber exerts a bigger force on the cat hair than the cat hair exerts on the amber, and both forces are nonzero".It is a known fact that when two substances are rubbed together, they become electrically charged.
This occurs when the amber is rubbed against the cat hair, which makes the amber positively charged, while the cat hair becomes negatively charged. It is also known that objects with opposite charges are attracted to one another, while those with similar charges are repelled.The electrical force exerted by the cat hair on the amber is equal in magnitude and opposite in direction to the force exerted by the amber on the cat hair. This is according to Newton's third law of motion which states that for every action, there is an equal and opposite reaction.
Both forces are nonzero because they are being exerted on two different objects. However, the amber exerts a bigger force on the cat hair than the cat hair exerts on the amber. This is because the amber has a much greater mass than the cat hair, and the force between two objects is directly proportional to the mass of each object. Therefore, the bigger the mass of an object, the greater the force it exerts on other objects.
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b. Evaluate the field strength if \( L=10 \mathrm{~cm} \) and \( Q=30 \mathrm{nC} \). expression for the electric field \( \vec{E} \) at point \( P \). Give your answer in component form. Figure P23.4
if \( L=10 \mathrm{~cm} \) and \( Q=30 \mathrm{nC}, the magnitude of the electric field at point P is approximately 2.698 × 10⁵ N/C.
The electric field at point P, denoted as E, can be calculated using the formula:
E = (1 / (4πε₀)) * (Q / r²) * R
where E₀ is the permittivity of free space, Q is the charge, r is the distance from the charge to point P, and R is the unit vector pointing from the charge to point P.
Given:
Distance from the charge to point P, L = 10 cm = 0.1 m
Charge, Q = 30 nC = 30 × [tex]10^-^9[/tex] C
First, let's calculate the denominator:
r² = (0.1 m)² = 0.01 m²
Next, let's determine the value of E₀, which is the permittivity of free space. The value of E₀ is approximately [tex]8.854 * 10^-^1^2[/tex] C²/N·m².
Now, let's substitute the values into the formula:
[E] = (1 / (4π([tex]8.854 * 10^-^1^2[/tex])) * (30 × [tex]10^-^9[/tex] / 0.01)
Simplifying further:
[E] = (1 / (4π([tex]8.854 * 10^-^1^2[/tex]))) * (3 × [tex]10^-^7[/tex]/ 0.01)
[E] ≈ 2.698 × 10⁵ N/C
Therefore, the magnitude of the electric field at point P is approximately 2.698 × 10⁵ N/C.
To express the electric field in component form, we can multiply the magnitude by the unit vector in the direction from the charge to point P. The unit vector R only has a non-zero component in the x-direction, so the electric field in component form is:
E = (Eₓ, 0, 0)
where Eₓ is the magnitude of the electric field and indicating its magnitude in the x-direction and zero magnitudes in the y and z directions.
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. How far apart are two conducting plates that have an electric field strength of 4.50×10
3
V/m between them, if their potential difference is 15.0kV ?
The distance between the two conducting plates is approximately 3.33 meters.
To find the distance between the two conducting plates, we can use the formula:
Electric field strength (E) = Voltage (V) / Distance (d)
Given:
Electric field strength (E) = 4.50 × 10³ V/mPotential difference (V) = 15.0 kV = 15.0 × 10³ VWe can rearrange the formula to solve for the distance (d):
d = V / E
Substituting the given values:
d = (15.0 × 10³ V) / (4.50 × 10³ V/m)
Simplifying the expression:
d = 3.33 m
Therefore, the distance between the two conducting plates is approximately 3.33 meters.
The correct format of the question should be:
How far apart are two conducting plates that have an electric field strength of 4.50×10³V/m between them, if their potential difference is 15.0kV ?
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avid Blaine performed a stunt in which he fell from atop a 27 m pole onto a stack of cardboard boxes 3.7 m high. Assuming Blaine had a mass of 85 kg, estimate the minimum force that the boxes exerted on Blaine as he crashed through the stack.
The minimum force that the boxes exerted on Blaine as he crashed through the stack is 8.2 kN.
In the given problem, David Blaine performed a stunt in which he fell from atop a 27 m pole onto a stack of cardboard boxes 3.7 m high. Assuming Blaine had a mass of 85 kg, we have to estimate the minimum force that the boxes exerted on Blaine as he crashed through the stack.
Force is defined as the rate of change of momentum and its unit is Newton (N). It is the product of mass and acceleration.
The formula for calculating force is given as:
F = ma
Where,
F is force,
m is mass
a is acceleration
We can use the formula of conservation of energy to calculate the minimum force exerted on Blaine. The formula of conservation of energy is given as:
PE (potential energy) + KE (kinetic energy) = constant
Hence, the potential energy of Blaine on the top of the pole = mgh (mass * gravity * height)
The kinetic energy of Blaine when he crashes into the boxes = 1/2 mv² (0 velocity at the top of the pole)
Since the potential energy is converted to kinetic energy,
F_gravity * d = 1/2 mv² + mgh
Here,
F_gravity = the force of gravity on the object of mass m,
d = the distance from the top of the pole to the top of the cardboard box
We know that F_gravity is given as:
F_gravity = m * g
Here,
m = 85 kg,
g = 9.8 m/s²
F_gravity = 85 kg * 9.8 m/s² = 833 N
We know that the distance from the top of the pole to the top of the cardboard box = 27 m - 3.7 m = 23.3 m
Putting all the given values in the above equation:
F_gravity * d = 1/2 mv² + mgh
833 N * 23.3 m = 1/2 * 85 kg * v² + 85 kg * 9.8 m/s² * 3.7 m
Force (F) exerted on Blaine as he crashed through the stack = 8,158.05 N or 8.2 kN (rounded to 3 significant figures)
Therefore, the minimum force that the boxes exerted on Blaine as he crashed through the stack is 8.2 kN.
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Charge q
2
in (Figure 1) is in equilibrium. Assume Part A q=5.4nC. What is q
1
? Express your answer with the appropriate units.
Answer: q1 = 5.2734375 × 10⁻¹⁰ C
Given:
Charge q2 in (Figure 1) is in equilibrium.
Part A q=5.4nC.
We need to find the value of q1.
Since the system is in equilibrium, the net force acting on the charges is zero.
Hence, we can use Coulomb's Law to relate q1 and q2:
F1 = F2F1 = kq1q2/r1² ;
F2 = kq1q2/r2²
Now, since the charges are in equilibrium:
F1 + F2 = 0
(i.e.) kq1q2(1/r1² - 1/r2²) = 0r1
= 2.50 cm ;
r2 = 8.00 cm
=> r1² = 6.25 cm²; r2² = 64 cm²
Thus, we can calculate q1: 5.4 × 10⁻⁹ × q2/ (64 cm²/6.25 cm²)
= q2 × 0.09765625 q2 gets canceled on both sides,
hence q1 = 5.4 × 10⁻⁹ × 0.09765625
= 5.2734375 × 10⁻¹⁰ C
Answer: q1 = 5.2734375 × 10⁻¹⁰ C
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A football is kicked at 25.0 m/s at an angle of 33.0
∘
above the horizontal. Assume it starts 1.00 m above ground level. (Neglect any effects due to air resistance.) Describe the velocity vector of the ball in terms of its γ - and y-components, v
x
and v
y
, respectively, when it is 4.00 m above ground level and still moving upward. You attach a meter stick to an oak tree, such that the top of the meter stick is 1.27 meters above the ground. Later, an acorn falls from somewhere higher up in the tree. If the acorn takes 0.181 seconds to pass the length of the meter stick, how high h
0
above the ground was the acorn before it fell, assuming that the acorn did not run into any branches or leaves on the way down? While standing at the edge of the roof of a building, a man throws a stone upward with an initial speed of 6.95 m/s. The stone subsequently falls to the ground, which is 15.9 m below the point where the stone leaves his hand. At what speed does the stone impact the ground? Ignore air resistance and use g=9.81 m/s
2
for the acceleration due to gravity. impact speed: How much time is the stone in the air? elapsed time:
1.The impact speed of the stone is approximately 19.6 m/s.
At 4.00 m above ground level and still moving upward, the velocity vector of the ball in terms of its γ- and y-components, v x and v y , respectively is:
v x = 25.0 m/s cos (33.0°) ≈ 20.98 m/sv y = 25.0 m/s sin (33.0°) ≈ 13.51 m/s2.
For the acorn to pass the length of the meter stick (1.27 meters) in 0.181 seconds, the acorn was this high h0 above the ground before it fell: h0 = 0.5 gt2
= 0.5 (9.81 m/s2)(0.181 s)2
≈ 0.157 m3.
To find the time the stone is in the air, we can use the fact that the time taken for the stone to reach its maximum height is the same as the time it takes for the stone to fall from its maximum height to the ground.
Using the vertical motion of the stone, we have:
v = v0 + gt, where v0 = 6.95 m/s and g = 9.81 m/s2.
When the stone reaches its maximum height, its final velocity is zero,
so we have:
0 = v0 - gt, or t = v0 / g = 6.95 m/s / 9.81 m/s2 = 0.707 s.
The total time the stone is in the air is twice this time: t total = 2t = 1.41 s.4.
To find the impact speed of the stone, we can use the equation:
v2 = v02 + 2gh,
where h = 15.9 m is the height of the building and v0 = 6.95 m/s is the initial velocity of the stone.
Rearranging this equation to solve for v,
we have:
v = sqrt(v02 + 2gh) = sqrt((6.95 m/s)2 + 2(9.81 m/s2)(15.9 m)) ≈ 19.6 m/s.,
the impact speed of the stone is approximately 19.6 m/s.
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You have landed on an unknown planet, Newtonia, and want to know what objects will weigh there. You find that when a certain tool is pushed on a frictionless horizontal surface by a $12.4 \mathrm{~N}$ force, it moves $16.4 \mathrm{~m}$ in the first $2.00 \mathrm{~s}$, starting from rest. You next observe that if you release this tool from rest at $11.0 \mathrm{~m}$ above the ground, it takes $2.88 \mathrm{~s}$ to reach the ground.
A)What does the tool weigh on Newtonia?
B)What would it weigh on Earth?
The weight of the object on Newtonia is 12.4 N.
The tool weighs 12.4 N on Newtonia.
Given:
Force on horizontal surface, F = 12.4 N
Distance covered, s = 16.4 m
Time taken, t = 2.00 s
Height from which the object was released = 11.0 m
Time taken to reach the ground, t = 2.88 s
Here, we are given that the surface is frictionless, so there is no work done against friction. Hence, the kinetic energy of the object is converted into potential energy due to the gravity of the planet.
The weight of an object is defined as the force with which the planet pulls the object downwards. The formula to calculate the weight is given as:
Weight = mg
where, m = mass of the object
g = acceleration due to gravity
On Newtonia, the weight of an object will be given as:
Weight = mass of the object × acceleration due to gravity of Newtonia
Since we don't know the acceleration due to gravity of Newtonia, let's assume it to be g_N. Now, applying the first equation of motion, we get:
v = u + at
where, u = initial velocity = 0
a = acceleration
t = 2.00 s
v = ?
We have:
s = ut + (1/2)at²
⇒ a = (2s)/(t²) = (2 × 16.4)/(2.00²) = 16.4 m/s
The force applied on the object on Newtonia is the resultant force and is given as:
Resultant force = ma
where m is the mass of the object. Let's assume the mass of the object to be m_N on Newtonia, then, the force acting on it will be F. Hence, we get:
F = m_N × g_N --- (1)
F = m_N × a --- (2)
On dividing equation (1) by equation (2), we get:
(g_N/a) = (F/m_N)
⇒ g_N = (F/m_N)
Substituting the values of F and g_N in equation (1), we get:
12.4 = m_N × (F/m_N)
⇒ F = 12.4 N
So, the weight of the object on Newtonia is 12.4 N.
Mass of the object, m_N = (F/g_N) = (12.4/7.18)
Now, we need to calculate g_N. To calculate g_N, let's use the second observation of the tool falling freely from a height of 11 m. The time taken to fall from a height of 11 m is given as t = 2.88 s.
The formula to calculate the distance travelled by a freely falling object is given as:
s = (1/2)gt
Substituting the given values, we get:
11 = (1/2)g_N × (2.88)²
⇒ g_N = (2 × 11)/(2.88²)
⇒ g_N = 7.18 m/s²
Substituting this value of g_N in the expression of m_N, we get:
m_N = (12.4/7.18) = 1.73 kg
So, the weight of the object on Earth is given as:
Weight = mg
where, m = mass of the object
g = acceleration due to gravity on Earth
g = 9.8 m/s²
Weight = m × g = 1.73 × 9.8 = 16.97 N
Therefore, the tool weighs 12.4 N on Newtonia.
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A spring oscillating with a frequency of 40/p Hz has a constant K = 640 N/m.
The mass that is tied to the spring is:
a. 1.0
b. 10
c. 100
d. 1.0 X 10 -1
e. Other: which one? ___
The mass that is tied to the spring whose frequency is 40/p Hz and the spring constant is 640 N/m is 0.4 kg.
The frequency of the spring is expressed in terms of its physical properties, i.e., mass and spring constant. We can use the following equation to solve the problem:
f = 1 / 2π √(K / m), where m is the mass of the spring and K is the spring constant of the spring.
Therefore,
m = K / π2 f2
m = 640 / (π2 × (40/π)2)
m = 640 / (1600)kg
m = 0.4 kg
Therefore, The mass tied to the spring, whose frequency is 40/p Hz and spring constant is 640 N/m is 0.4 kg.
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Before working this problem, review Conceptual Example 14. A pellet gun is fired straight downward from the edge of a cliff that is 19.2 m above the ground, The pellet strikes the ground with a speed of 30.5 m/s. How far above the cliff edge would the pellet have gone had the gun been fired straight upward?
If the pellet gun is fired straight upward from the edge of the cliff that is 19.2 m above the ground, the pellet would reach a height of approximately 49.2 m above the cliff edge before falling back down.
When the pellet gun is fired straight downward, the initial velocity is 30.5 m/s in the downward direction. We can use the kinematic equation for vertical motion to find the time it takes for the pellet to hit the ground. Since the initial velocity is in the negative direction, we can use the equation:
h = v_i * t + (1/2) * g * t²
where h is the height above the ground, v_i is the initial velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s²). Plugging in the known values:
0 = 30.5 * t - (1/2) * 9.8 * t²
Solving this quadratic equation, we find two solutions: t = 0 s (which represents the time at the top of the trajectory) and t ≈ 2.20 s (which represents the time it takes for the pellet to hit the ground).
Now, if the gun is fired straight upward, the initial velocity would be positive 30.5 m/s. Using the same kinematic equation, we can calculate the maximum height reached by the pellet. The final velocity at the maximum height is 0 m/s since the pellet momentarily stops before falling back down. Thus, the equation becomes:
0 = 30.5 * t - (1/2) * 9.8 * t²
Solving for t, we find t ≈ 3.13 s. Plugging this value back into the equation to find the maximum height (h) reached by the pellet, we have:
h = 30.5 * 3.13 - (1/2) * 9.8 * (3.13)²
h ≈ 49.2 m
Therefore, if the pellet gun is fired straight upward, the pellet would reach a height of approximately 49.2 m above the cliff edge before falling back down.
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a car moves at 18 m/s and coasts up the hill with uniform acceleration of -1.8m/s^2. what is displacement after 9 seconds?
The displacement of the car after 9 seconds is 89.1 meter
To find the displacement of the car after 9 seconds, we can use the equation of motion:
s = ut + (1/2)at^2
where:
s is the displacement
u is the initial velocity
a is the acceleration
t is the time
Given:
u = 18 m/s (initial velocity)
a = -1.8 m/s^2 (acceleration)
t = 9 s (time)
Plugging in the values, we get:
s = (18)(9) + (1/2)(-1.8)(9)^2
s = 162 - 72.9
s = 89.1 meters
Therefore, the displacement of the car after 9 seconds is 89.1 meter.
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