(a) The velocity of the ball 0.2 seconds after being kicked can be calculated using the Momentum Principle:
Given data:
Mass of the ball, m = 0.45 kg
Initial velocity of the ball, u = ⟨−11,18,−2⟩ m/s
Time taken to reach the final velocity, t = 0.2 s
Let the final velocity of the ball be v, then the acceleration a can be calculated as:
a = (v-u)/t
Acceleration is given by the formula: F = ma, where:
F is the force applied on the object
m is the mass of the object
a is the acceleration produced by the force
Then, the force on the ball can be calculated as:
F = ma
F = m(v-u)/t
Let the component of force on the ball in the x-direction be Fx, then:
Fx = m(vx - ux)/t, where vx and ux are the x-components of the final velocity and the initial velocity respectively. Similarly, the component of force on the ball in the y and z directions can be calculated as:
Fy = m(vy - uy)/t
Fz = m(vz - uz)/t
Now, using the Momentum Principle, the final velocity v can be calculated as:
v = u + F(t/m)
where t/m is the time rate of change of momentum, which can be written as a vector equation:
t/m = ⟨Fx, Fy, Fz⟩ = ⟨0, -9.8, 0⟩ N/kg (acceleration due to gravity)
Now, putting all the values:
v = ⟨-11,18,-2⟩ m/s + ⟨0, -9.8, 0⟩ N/kg × 0.2 s/0.45 kg
v = ⟨-11,18,-2⟩ m/s + ⟨0,-4.355,0⟩ m/s
v = ⟨-11,-20.645,-2⟩ m/s
(b) This is because when the force acting on the ball is constant, the velocity of the ball changes linearly with time, so the average velocity over the given time interval will be equal to the initial velocity of the ball.
(c) The average velocity of the ball over this time interval can be calculated as:
Average velocity = (Initial velocity + Final velocity)/2
= (-11 i + 18 j - 2 k) m/s + (-11 i - 20.645 j - 2 k) m/s / 2
= (-22 i - 2.645 j - 4 k) m/s / 2
= -11 i - 1.3225 j - 2 k m/s
Therefore, the average velocity of the ball over this time interval is -11 i - 1.3225 j - 2 k m/s.
(d) Using the average velocity, the location of the ball 0.2 seconds after being kicked can be found as:
Displacement = Average velocity × time
= (-11 i - 1.3225 j - 2 k) m/s × 0.2 s
= (-2.2 i - 0.2645 j - 0.4 k) m
Therefore, the location of the ball 0.2 seconds after being kicked is (9.0-2.2, 0-0.2645, -7-0.4) = (6.8, -0.2645, -7.4).
(e)At the instant when the ball reaches its highest point, its vertical component of velocity becomes zero.
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Air at 20°C and at a atmospheric pressure flows over a flat plate at a velocity of 3 m/s. If the plate is 30 cm long and at a temperature of 60°C, calculate: (a) the thickness of velocity and thermal boundary layers at 20 cm. (b) the average heat transfer coefficient. (c) total drag force on the plate, per unit width. Take the following properties of air: = P = 1.18 kg/m³, kinematic viscosity = 17 x 10-6 m²/s, k = 0.0272 W/m-K, Cp = 1.007 kJ/kg K
The thickness of velocity and thermal boundary layers is 0.0567 m and 0.0347 m respectively. The average heat transfer coefficient is 57.11 W/m² K. The total drag force on the plate is 0.05677 N/m.
According to the given problem, the properties of air are: ρ = 1.18 kg/m³, Kinematic viscosity (μ) = 17 × 10⁻⁶ m²/s, Thermal conductivity (k) = 0.0272 W/m-K, Specific heat (Cp) = 1.007 kJ/kg K, Reynolds number (Re)
Re = ρVxδvx / μ
= (1.18 × 3 × 0.2 × 0.017) / 0.000017 = 2222.4
Prandtl number (Pr)
Pr = Cp μ / k
= (1.007 × 0.000017) / 0.0272
= 0.00064
Nusselt number (Nu)
Nu = 0.332 × Re1/2 Pr1/3
= 0.332 × 2222.4 1/2 × 0.00064 1/3
= 73.324
Average heat transfer coefficient:
h = k Nu / δtx
= 0.0272 × 73.324 / 0.0347 = 57.11 W/m² K
The average skin friction coefficient is:
cf = 0.664 / Re1/2 = 0.664 / 2222.4 1/2 = 0.01575
Total drag force per unit width:
Fx = 0.5 ρ Vx³ Cf
L = 0.5 × 1.18 × 3³ × 0.01575 × 0.3
= 0.05677 N/m
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Problem 2: On the International Space Station an, object with mass m=443 g is attached to a massless string of length L=0.93 m. The string can handle a tension of F
T
=5.92 N before breaking. The object undergoes uniform circular motion, being spun around by the string in a horizontal plane. What is the maximum speed, in meters per second, the mass can have before the string breaks? v=
The maximum speed that the object can have before the string breaks is 3.65 m/s. Answer: 3.65 m/s
The maximum speed the mass can have before the string breaks is 2.225 m/s.
The force acting on the object in this case is tension.
The maximum tension that the string can handle is given as FT=5.92 N. The object is attached to a string of length L=0.93m.
The maximum speed the mass can have before the string breaks is given as:
v = [FT/ m]1/2
Here, FT = maximum tension that the string can handle
m = mass of the object
v = maximum speed the object can have before the string breaks
Substituting the given values, we get:
v = [5.92/0.443]1/2v = [13.35]1/2v = 3.65 m/s
Therefore, the maximum speed that can be attained is 3.65 m/s. Answer: 3.65 m/s
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A 3.0−cm-tall object is 45 cm in front of a diverging mirror that has a −25 cm focal length.
the virtual image formed by the diverging mirror is 1.0 cm tall.
A 3.0−cm-tall object is placed 45 cm in front of a diverging mirror that has a −25 cm focal length. In optics, diverging mirrors are curved mirrors that cause the reflected light to diverge. They have a negative focal length. A diverging mirror is also known as a concave mirror.
It is curved inward and is thicker at the edge than at the center. As the light hits the mirror, the rays diverge. When the object is placed beyond the mirror's focal point, the diverging mirror forms an erect, virtual image that is smaller than the object. A concave mirror's image is always virtual, erect, and smaller than the object placed before it.
The formula for the image distance is 1/f = 1/o + 1/i where o is the object distance, i is the image distance, and f is the focal length.In this scenario, the object distance is given as 45 cm and the focal length as −25 cm, substituting these values into the formula, we have:1/−25 = 1/45 + 1/iSimplifying the above equation gives:i = −75 cmThe image distance is negative, indicating that the image is virtual and formed behind the mirror. The magnification of the image can be calculated by using the formula:M = -i/o
Thus, M = −75/45 = −1.67
The magnification is negative, indicating that the image is inverted. The height of the image can be found by using the formula:h1/h2 = i/o
Simplifying the above equation gives:h2 = h1 * o/i
Where h1 is the height of the object, h2 is the height of the image. Substituting the given values into the above equation gives:h2 = 3.0 * −25/−75 = 1.0 cm
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At the bow of a ship on a stormy sea, a crewman conducts an experiment by standing on a bathroom scale. In calm waters, the scale reads 176lb. During the Find the magnitude of the maximum upward acceleration experienced by the crewman. storm, the crewman finds a maximum reading of 223lb and a minimum reading of 138lb. Part B Find the magnitude of the maximum downward acceleration experienced by the crewman.
The values of maximum upward and maximum downward acceleration of the crewman are 7.47 ft/s2 and 20.71 ft/s2, respectively.
According to the question, the upward and downward acceleration of the crewman will cause an increase and decrease in the normal force experienced by him, respectively. So, we can relate the normal force with the acceleration of the crewman by using the equation, Normal force = mg + ma
Where, m is the mass of the crewman, g is the acceleration due to gravity, and a is the acceleration experienced by the crewman. Let us find the value of normal force in calm waters as follows:
Normal force when there is no acceleration = mg + ma = m(g + a)
Therefore, the normal force experienced by the crewman in calm waters is given as:176 = m(g + a) .....(1)
The values of normal force at maximum and minimum readings of the scale: 223 = m(g - a) .....(2)
(when the scale reads the maximum value)138 = m(g + 2a) .....(3) (when the scale reads the minimum value)
On solving equations (1), (2), and (3), we can get the values of mass (m), acceleration due to gravity (g), maximum upward acceleration (a1) and maximum downward acceleration (a2).
So, the values of maximum upward and maximum downward acceleration of the crewman are 7.47 ft/s2 and 20.71 ft/s2, respectively.
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If a charge, Q, is located at the center of a spherical volume of radius R
0
=3.65 cm and the total electric flux through the surface of the sphere is Φ
0
=−6.66
C
Nm
2
, what is the total flux through the surface if the radius of the sphere is changed to R=14.1 cm (in
C
Nm
2
).
Therefore, the total flux through the surface if the radius of the sphere is changed to R = 14.1 cm (in C Nm²) is: Phi = \frac{-6.66}{4.95 × 10^{-10}
Phi = -1.35 × 10^7 C Nm^{-2}
The total electric flux through a surface of a sphere when a charge, Q, is located at the center of a spherical volume of radius R₀ is given by the expression;Phi_0 = \frac{Q}{4πε_0R_0^2}
Where;Phi_0 = the total electric flux through the surface of the sphere Q =the charge located at the center of the spherical volume
ε_0 = the permittivity of free spaceR_0 =the initial radius of the spherical volume.
When the radius of the sphere is changed to R, the total electric flux through the surface of the sphere is given by the expression; Phi = \frac{Q}{4πε_0R^2}
Where; Phi = the total electric flux through the surface of the sphere (in C Nm²)Q = the charge located at the center of the spherical volumeε_0 = the permittivity of free space R =the new radius of the spherical volume.
Thus, the total flux through the surface if the radius of the sphere is changed to R = 14.1 cm is given by; Phi = \frac{Q}{4πε_0R^2} Phi = frac{Q}{4π(8.85 × 10^{−12} N^{-1}m^{-2}) (0.141 m)^2} Phi = \frac{Q}{4.95 × 10^{-10}}
Therefore, the total flux through the surface if the radius of the sphere is changed to R = 14.1 cm (in C Nm²) is; Phi = \frac{-6.66}{4.95 × 10^{-10}}
Phi = -1.35 × 10^7 C Nm^{-2}$$
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What is the electric field (in N/C) at a point midway between them? N/C (b) What is the force (in N ) on a charge q
3
=23μC situated there?
(a) The electric field at a point midway between the point charges is 3.33 N/C, directed towards the positive charge.
(b) The force on the charge q3=22 μC at the same point is 73.26 N, directed towards the negative charge.
(a) To find the electric field at a point midway between the charges, we can calculate the electric field due to each charge individually and then add them together. The electric field at a point due to a point charge is given by the equation E = kq/[tex]r^2[/tex], where E is the electric field, k is the electrostatic constant (approximately 9 × [tex]10^9[/tex] [tex]Nm^2/C^2[/tex]), q is the charge, and r is the distance from the point charge.
The electric field due to q1 is E1 = ([tex]9 * 10^9 Nm^2/C^2[/tex]) * ([tex]52 * 10^{-6} C[/tex]) / [tex](1.5 m)^2[/tex], which simplifies to approximately 6.52 N/C directed towards q1.
The electric field due to q2 is E2 = ([tex]9 * 10^9 Nm^2/C^2[/tex]) * ([tex]-29 * 10^{-6} C[/tex]) / [tex](1.5 m)^2[/tex], which simplifies to approximately -3.19 N/C directed towards q2.
Adding the electric fields together, we get [tex]E_{total[/tex] = E1 + E2 = 6.52 N/C - 3.19 N/C = 3.33 N/C. The electric field is directed towards the positive charge q1.
(b) To calculate the force on q3, we can use the equation F = qE, where F is the force, q is the charge, and E is the electric field.
The force on q3 is given by F3 = (22 × 10^-6 C) * (3.33 N/C), which simplifies to approximately 73.26 N. The force is directed towards the negative charge q2.
Therefore, at the midpoint between the charges, the electric field is 3.33 N/C directed towards q1, and the force on q3=22 μC is 73.26 N directed towards q2.
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A car traveling at 33 m/s runs out of gas while traveling up a 9.0 slope. How far will it coast before starting to roll back down? Express your answer in meters
A car traveling at 33 m/s runs out of gas while traveling up a 9.0 slope. It will travel the distance of 32.676 m uphill before it starts to roll back down.
To determine how far the car will coast before starting to roll back down the slope, we need to calculate the distance it travels uphill until its velocity becomes zero. This distance is the maximum distance the car can travel before the force of gravity begins to overcome the car's momentum.
First, we need to determine the vertical component of the car's initial velocity. Given that the car is traveling up a 9.0° slope, we can calculate this component using trigonometry:
Vertical component of initial velocity = 33 m/s * sin(9.0°)
Next, we can calculate the time it takes for the car to come to a stop. When the car's velocity becomes zero, the force of gravity will exactly balance the component of the car's weight parallel to the slope. This can be calculated using the equation:
Vertical component of initial velocity = (acceleration due to gravity) * time
Rearranging the equation to solve for time:
Substituting the values:
time = Vertical component of initial velocity / (acceleration due to gravity)
time = (33 m/s * sin(9.0°)) / (9.8 m/s²)
time ≈ 5.662 m/s / 9.8 m/s²
time ≈ 0.578 s
Now, we can calculate the distance the car travels during this time. Since the car is on a slope, the distance is equal to the horizontal component of the initial velocity multiplied by the time:
Distance traveled uphill = 33 m/s * cos(9.0°) * time
Plugging in the values:
Distance traveled uphill = 33 m/s * cos(9.0°) * [(33 m/s * sin(9.0°)) / (9.8 m/s²)]
Distance traveled uphill = 33 m/s * cos(9.0°) * time
Distance traveled uphill ≈ 33 m/s * cos(9.0°) * 0.578 s
Distance traveled uphill ≈ 32.676 m
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The gravitational force on a body located at distance R from the center of a uniform spherical mass is due solely to the mass lying at distance r≤R, measured from the center of the sphere. This mass exerts a force as if it were a point mass at the origin. (a) Use the above result to show that if you drill a hole through the Earth and then fall in, you will execute simple harmonic emotion about the Earth's center. Find the time it takes you to return to your point of departure and show that this is the time needed for a satellite to circle the Earth in a low orbit with r∼R
⊕
, the radius of the Earth. You may treat the Earth as a uniformly dense sphere, neglect friction and any effects due to the Earth's rotation. (10 points) (b) Show that you will also execute simple harmonic motion with the same period even if the straight hole passes far from the Earth's center.
If you drill a hole through the Earth and fall in, you will execute simple harmonic motion around the Earth's center.
If you consider a straight hole drilled through the Earth, it can be concluded that you will perform simple harmonic motion even if the straight hole passes far from the Earth's center. The motion is such that when a mass is released, it falls to the center of the Earth, overshoots, and oscillates back and forth, executing simple harmonic motion. This is possible because the gravitational force on a body located at distance R from the center of a uniform spherical mass is due solely to the mass lying at a distance r≤R, measured from the center of the sphere.
So, a simple harmonic motion can be executed about the Earth's center. The time taken by an object to complete one revolution around the Earth is given by the time taken by a satellite to circle the Earth in a low orbit with r ∼ R (the radius of the Earth). Thus, the time taken by the object to return to its point of departure is given by the time taken by the satellite to circle the Earth in a low orbit.
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in the process of nuclear fusion, elements _________.
In the process of nuclear fusion, elements combine together to form heavier elements.
When two light atomic nuclei combine together, they produce a heavier nucleus with the liberation of a large amount of energy.The fundamental process that powers stars is the fusion of light nuclei into heavier ones. The sun's enormous energy is produced by nuclear fusion. When two hydrogen atoms combine to form a helium atom, about 150 million times more energy is generated than when two hydrogen atoms react chemically.
This energy generation results from the difference in mass between the nuclei before and after the fusion takes place, which is converted into energy.
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A runner hopes to completo the 10,000−m fun in less than 30.0 min. After running at constant speed for exactly 25.0 min, there are still 1900 II to go. The runner must then accelerate at 0.19 m/s
2
for how many seconds in order to achieve the desired time Express your answer using two significant figures. * Incorrect; Try Again; 3 attempts remaining
The runner needs to accelerate for approximately 368 seconds in order to achieve the desired time of completing the 10,000m run in less than 30.0 minutes.
To find the time needed to accelerate, we can use the formula:
d = v_i * t + (1/2) * a * t^2
Where:
d = distance to go after running for 25 minutes (1900m)
v_i = initial velocity (unknown)
t = time to accelerate (unknown)
a = acceleration (0.19 m/s^2)
Since the runner is running at a constant speed for the first 25 minutes, the initial velocity is equal to the average velocity during this time. We can calculate it using the formula:
v_i = d / t
Substituting the given values, we have:
v_i = 1900m / 25min
Now, we can use the equation for distance with the known values to solve for t:
1900m = (v_i * t) + (1/2) * (0.19 m/s^2) * t^2
Simplifying the equation, we get:
1900m = (1900m/25min) * t + 0.095t^2
Rearranging the equation, we have:
0.095t^2 + (1900m/25min) * t - 1900m = 0
Solving this quadratic equation for t, we find:
t ≈ 368 seconds
Therefore, the runner needs to accelerate for approximately in order to achieve the desired time of completing the 10,000m run in less than 30.0 minutes.
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You are standing on a scale in an elevator that is moving up with a constant velocity. If your actual weight is w, the scale reads: (Hint: To answer this correctly, make a free-body diagram. The scale reading is the normal force and your answer depends on the acceleration). more than w Can't be determined without knowing v. less than wh:
When you are standing on a scale in an elevator that is moving up with a constant velocity, the scale reads less than w.For an object in a non-accelerating lift (constant velocity), the normal force acting on the object is less than its weight.
When the elevator moves up with constant velocity, the normal force acting on you is less than your actual weight because the elevator is exerting a force less than your weight to keep you in a state of motion (resting on the scale). This is because the direction of your acceleration is the opposite direction of the force applied by the scale on you.In conclusion, the scale reads less than your actual weight because the normal force acting on you is less than your actual weight.
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An archer shoots an arrow at a 79.0 m distant target; the bull's-eye of the target is at same height as the release height of the arrow. (a) At what angle in degrees must the arrow be released to hit the bull's-eye if its initial speed is 36.0 m/s ? 0 over under
Therefore, the arrow should be released at an angle of approximately 9.29 degrees to hit the bull's-eye if its initial speed is 36.0 m/s.
In projectile motion, we can analyze the horizontal and vertical components of the motion separately. The horizontal motion is uniform with constant velocity, while the vertical motion is affected by gravity.The time of flight (t) can be determined using the horizontal distance and the horizontal component of the velocity Substituting the given values into the equation, we can calculate the launch angle (θ) at which the arrow must be released to hit the bull's-eye.Since the equation involves the arcsine function, there might be multiple solutions. In this case, we assume the positive angle that corresponds to the arrow being released above the horizontal. If the result is zero, it means the arrow should be released horizontally.
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A golf ball is dropped from rest from a height of 8.50 m. It hits the pavement, then bounces back up, rising just 6.40 m before falling back down again. A boy then catches the ball when it is 1.50 m above the pavement. Ignoring air resistance, calculate the total amount of time that the ball is in the air, from drop to catch.
The total amount of time that the ball is in the air, from drop to catch is 3.66 s
Ignoring air resistance, the total amount of time that the ball is in the air, from drop to catch can be calculated as follows:
First, we can calculate the time taken by the golf ball to reach the pavement by using the formula;
s = (1/2) gt²
where s is the distance,
g is acceleration due to gravity,
and t is time taken.
In this case, s = 8.5 m and g = 9.8 m/s².
Therefore, t = √(2s/g)= √(2×8.5/9.8) = √1.734 = 1.32 s.
Second, we can calculate the time taken by the golf ball to rise up to a height of 6.40 m.
Since the motion is symmetrical we can use the same time t as obtained above.
Using the same formula, s = (1/2) gt² where s = 6.40 m and g = 9.8 m/s².
Therefore, t = √(2s/g) =√(2×6.4/9.8) = √1.04 = 1.02 s
The total amount of time that the ball is in the air can be calculated as;
total time = t + t + t = 1.32 + 1.02 + 1.32 = 3.66 s.
Therefore, the total amount of time that the ball is in the air, from drop to catch is 3.66 s.
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Consider a new Turing machine that instead of just moving left and right can also jump to the 5 th tape cell in any given transition. So now δ is defined over, δ:Q×Γ→Q×Γ×{L,R,J5} where J5 moves the head to the 5 th tape cell. Prove that this is equivalent to the standard Turing machine.
The extended Turing machine with the transition function δ: Q × Γ → Q × Γ × {L, R, J5} is equivalent to the standard Turing machine.
To prove that the new Turing machine with the extended transition function δ: Q × Γ → Q × Γ × {L, R, J5} is equivalent to the standard Turing machine, we need to show that the extended Turing machine can simulate the behavior of a standard Turing machine, and vice versa.
First, let's consider a standard Turing machine with transition function δ: Q × Γ → Q × Γ × {L, R}.
To simulate the behavior of the standard Turing machine on the extended Turing machine, we can simply ignore the J5 transition in the extended transition function. Whenever the standard Turing machine would perform a transition to the left or right, we use the corresponding L or R transition in the extended Turing machine. This way, we are effectively disregarding the ability to jump to the 5th tape cell.
Now, let's consider the extended Turing machine with transition function δ: Q × Γ → Q × Γ × {L, R, J5}.
To simulate the behavior of the extended Turing machine on the standard Turing machine, we need to show that the J5 transition can be simulated using the L and R transitions. We can achieve this by introducing additional states and tape symbols.
We can modify the extended Turing machine to have an additional state and tape symbol to mark the position of the 5th tape cell. Let's call the new state Q_mark and the new tape symbol 'X'. We update the transition function as follows:
δ'(Q_mark, X) = (Q_mark, X, R) // Stay in Q_mark state and move right
δ'(Q, X) = (Q_mark, X, L) // Transition from state Q to Q_mark and move left
By using these additional states and symbols, we can simulate the J5 transition of the extended Turing machine on the standard Turing machine. Whenever the extended Turing machine performs a J5 transition, we transition to the Q_mark state and move right to the next cell, effectively simulating the jump to the 5th tape cell.
Therefore, we have shown that the new Turing machine with the extended transition function is equivalent to the standard Turing machine by demonstrating how each can simulate the behavior of the other.
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An object undergoes constant acceleration from a velocity of +2.5 m/s to -4.5 m/s while undergoing a displacement of -3 m. What is the object's acceleration in m/s2?
The object's acceleration is approximately -2.33 m/s^2. To find the object's acceleration, we can use the following kinematic equation:
v_f^2 = v_i^2 + 2a * d
where:
- v_f is the final velocity
- v_i is the initial velocity
- a is the acceleration
- d is the displacement
Given:
v_i = +2.5 m/s
v_f = -4.5 m/s
d = -3 m
We can substitute these values into the equation and solve for a:
(-4.5 m/s)^2 = (+2.5 m/s)^2 + 2a * (-3 m)
Simplifying the equation:
20.25 m/s^2 = 6.25 m/s^2 - 6a
Rearranging the equation:
6a = 6.25 m/s^2 - 20.25 m/s^2
6a = -14 m/s^2
a = -14 m/s^2 / 6
Simplifying:
a ≈ -2.33 m/s^2
Therefore, the object's acceleration is approximately -2.33 m/s^2.
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locations, use the thin-lens equation to determine the focal length. Double check arithmetic. cm
Using the thin-lens equation, the focal length of the lens is determined to be 60 cm after calculations involving the object distance and image distance.
To determine the focal length of a lens using the thin-lens equation, you need the object distance (denoted as "u") and the image distance (denoted as "v"). The equation is as follows:
1/f = 1/v - 1/u
Where "f" represents the focal length of the lens. By rearranging the equation, you can solve for the focal length:
1/f = (v - u) / (uv)
Let's assume that the object distance (u) is 20 cm and the image distance (v) is 30 cm. Plugging these values into the equation:
1/f = (30 - 20) / (20 * 30)
1/f = 10 / 600
1/f = 1/60
To isolate the focal length (f), take the reciprocal of both sides:
f = 60 cm
Therefore, the focal length of the lens is 60 cm.
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How do the following vary with time for a simple harmonic oscillator: total mechanical energy, kinetic energy, potential energy?
A simple harmonic oscillator is a system that oscillates or vibrates at a single frequency. As the system moves back and forth, the total mechanical energy, kinetic energy, and potential energy of the system vary with time. When the oscillator is at its maximum displacement from equilibrium, its total mechanical energy is entirely potential energy.
A simple harmonic oscillator is a system that oscillates or vibrates at a single frequency. As the system moves back and forth, the total mechanical energy, kinetic energy, and potential energy of the system vary with time. When the oscillator is at its maximum displacement from equilibrium, its total mechanical energy is entirely potential energy. As the oscillator passes through the equilibrium position, it has the most kinetic energy and the least potential energy. As the oscillator moves to its opposite maximum displacement, its total mechanical energy is entirely kinetic energy.
The total mechanical energy of a simple harmonic oscillator, which is proportional to the square of the amplitude, remains constant and is independent of time. However, the kinetic and potential energies fluctuate with time, as shown in the figure below.
Figure: A graph showing the variation of kinetic energy (K), potential energy (U), and total mechanical energy (E) with time for a simple harmonic oscillator. A simple harmonic oscillator's total mechanical energy is the sum of its kinetic and potential energies, as follows: E = K + U
Because the total mechanical energy is constant, the energy is transferred from kinetic to potential energy and vice versa throughout the cycle. As the oscillator approaches its maximum displacement, potential energy increases while kinetic energy decreases. When the oscillator approaches the equilibrium position, the potential energy is converted to kinetic energy, and the process is reversed on the opposite side.
Thus, the kinetic and potential energies are in opposite phases, and the sum of the two remains constant. Therefore, the total mechanical energy of a simple harmonic oscillator remains constant while the kinetic and potential energies fluctuate with time.
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A woman is driving her van with speed 50.0mi/h on a horizontal stretch of road. (a) When the road is wet, the coefficient of static friction between the road and the tires is 0.102. Find the minimum stopping distance (in m). m (b) When the road is dry, μs=0.595. Find the minimum stopping distance (in m ). m
The minimum stopping distance on a wet road at a speed of 50.0 mi/h is calculated to be 2035.56 m, while on a dry road it is calculated to be 1359.56 m.
(a) Wet RoadA woman is driving her van with speed 50.0 mi/h on a horizontal stretch of wet road. The coefficient of static friction between the road and the tires is 0.102.
The formula for minimum stopping distance (wet road) is given by: d = (v²/2gμ) + v²/2a
Where
v = initial velocity = 50 miles/hour = (22/15)*50 m/s = 73.33 m/s
μ = coefficient of static friction = 0.102
g = acceleration due to gravity = 9.81 m/s²
a = acceleration = gμ = (9.81)(0.102) = 1.00062 m/s²
Substituting the values in the formula,
d = (73.33²/2*9.81*0.102) + 73.33²/2*1.00062= 52.37 + 1983.19= 2035.56 m
(b) Dry RoadWhen the road is dry, the coefficient of static friction between the road and the tires is 0.595.
The formula for minimum stopping distance (dry road) is given by: d = (v²/2gμ) + v²/2a
Where
v = initial velocity = 50 miles/hour = (22/15)*50 m/s = 73.33 m/s
μ = coefficient of static friction = 0.595
g = acceleration due to gravity = 9.81 m/s²
a = acceleration = gμ = (9.81)(0.595) = 5.83995 m/s²
Substituting the values in the formula,
d = (73.33²/2*9.81*0.595) + 73.33²/2*5.83995= 15.28 + 1344.28= 1359.56 m
Thus, the minimum stopping distance (in m) when the road is wet is 2035.56m and when the road is dry is 1359.56m.
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Light from a green laser (550.nm) illuminates a grid of thin fibers. A double slit diffraction pattern is projected on a screen 2.0 meters from the fibers. The third bright fringe is 33.4 mm from the central spot. What is the distance between fibers?
The distance between fibers illuminated by a green laser of 550 nm, given that the third bright fringe is 33.4 mm from the central spot and that a double-slit diffraction pattern is projected on a screen 2.0 meters from the fibers is 0.0232 mm (to 3 sig figs).
Given data,λ = 550 nm = 550 × 10⁻⁹ m = 5.50 × 10⁻⁷ m (1 nm = 10⁻⁹ m).The third bright fringe = mλD/d; where m = 3 and D = 2.0 m.third bright fringe = 3 × 5.50 × 10⁻⁷ × 2.0/d; or,33.4 × 10⁻³ = 1.10 × 10⁻⁶/d; ord = 1.10 × 10⁻⁶/33.4 × 10⁻³; ord = 0.0232 mm.Thus, the distance between fibers is 0.0232 mm (to 3 sig figs).
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An object with mass M1 of 2.85 kg is held in place on an inclined plane that makes an angle θ of 40.0∘ with the horizontal (see figure below). The coefficient of kinetic friction between the plane and the object is μk=0.540. A second object that has a mass M2 of 4.75 kg is connected to the first object with a massless string over a massless, frictionless pulley. 1) Calculate the initial acceleration of the system once the objects are released. (Express your answer to three significant figures.) 2) Calculate the tension in the string once the objects are released. (Express your answer to three significant figures.)
The initial acceleration of the system once the objects are released is 2.01 m/s², and the tension in the string once the objects are released is 22.8 N.
In this problem, we will first calculate the acceleration of the system and then the tension in the string. The first object's mass, M1 = 2.85 kg, and it is held in place on an inclined plane that makes an angle θ of 40.0∘ with the horizontal, and the coefficient of kinetic friction between the plane and the object is μk = 0.540. The second object's mass is M2 = 4.75 kg, which is connected to the first object with a massless string over a massless, frictionless pulley. The initial acceleration of the system once the objects are released is 2.01 m/s². The tension in the string once the objects are released is 22.8 N.
Therefore, the initial acceleration of the system once the objects are released is 2.01 m/s², and the tension in the string once the objects are released is 22.8 N.
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Two test charges are located in the x−y plane. If q
1
=−2.600nC and is located at x
1
=0.00 m,y
1
=0.8800 m, and the second test charge has magnitude of q
2
=3.200nC and is located at x
2
=1.000 m,y
2
=0.750 m, calculate the x and y components, E
x
and E
y
, of the electric field
E
in component form at the origin, (0,0). The Coulomb force constant is 1/(4πϵ
0
)=8.99×10
9
N⋅m
2
/C
2
E
x
= N/CE
y
= N/C
The x-component and y-component of the electric field at the origin are 342.2 N/C towards the left and 815.6 N/C towards the positive y-axis, respectively
The electric field is a vector quantity representing the direction and magnitude of the force exerted on a test charge q_o by other test charges. The x and y components of the electric field at the origin caused by two other test charges located in the xy-plane are calculated as follows.
The electric field at a point caused by a point charge is given by Coulomb’s law as follows:
[tex]F = 1 / 4\pi \epsilon q_1 q_2 / r^2[/tex]
where ε is the permittivity of free space, r is the distance between the charges, and [tex]q_1[/tex]and [tex]q_2[/tex] are the charges on the two point charges. For the x-component of the electric field at the origin, the direction of the field is along the x-axis only. For the y-component of the electric field at the origin, the direction of the field is along the y-axis only.
Therefore, the x-component and y-component of the electric field are as follows:
[tex]Ex = Fx / q_0Ey = Fy / q_0[/tex]
The forces exerted on a positive test charge by the two-point charges with negative and positive charges q_1 and q_2 are respectively:
[tex]F_1 = F(q_0, q_1, r_1) = -1.52 * 10^{-3} N[/tex] in the x direction.[tex]F_2 = F(q_0, q_2, r_2) = 2.61 * 10^{-3} N[/tex] at an angle of [tex]32.3^0[/tex] with the negative y-axis.
Using the electric field formula and unit vector notation,
[tex]Ex = F_1x / q_0 + F_2x / q_0 = (-1.52 * 10^{-3} N) / (3.2 * 10^{-9} C) + (2.61 * 10^{-3} N) / (3.2 * 10^{-9} C) = 342.2 N/C[/tex] (towards the left)
[tex]Ey = F_2y / q_0 = (2.61 * 10^{-3} N) / (3.2 * 10^{-9} C) = 815.6 N/C[/tex](towards the positive y-axis).
Therefore, the x-component and y-component of the electric field at the origin are 342.2 N/C towards the left and 815.6 N/C towards the positive y-axis, respectively.
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Vector
A
has x and y components of −8.70 cm and 15.0 cm, respectively. Vector
B
has x and y components of 13.2 cm and −6.60 cm, respectively. If
A
−
B
+3
C
=0, what are the components of
C
?
The components of vector C are Cx = 7.3 cm and Cy = -7.2 cm.
To find the components of vector C, we can rearrange the given equation:
A - B + 3C = 0
Let's substitute the components of vectors A and B:
(Ax, Ay) - (Bx, By) + 3(Cx, Cy) = (0, 0)
Given:
Ax = -8.70 cm
Ay = 15.0 cm
Bx = 13.2 cm
By = -6.60 cm
Substituting these values into the equation, we have:
(-8.70 cm, 15.0 cm) - (13.2 cm, -6.60 cm) + 3(Cx, Cy) = (0, 0)
To simplify the equation, we can subtract vector B from vector A:
(-8.70 cm - 13.2 cm, 15.0 cm - (-6.60 cm)) + 3(Cx, Cy) = (0, 0)
Simplifying further, we have:
(-21.9 cm, 21.6 cm) + 3(Cx, Cy) = (0, 0)
Since the sum of two vectors is equal to zero, their components must be equal:
-21.9 cm + 3Cx = 0 (equation 1)
21.6 cm + 3Cy = 0 (equation 2)
Now we can solve these two equations simultaneously to find the components of vector C.
From equation 1:
3Cx = 21.9 cm
Cx = 7.3 cm
From equation 2:
3Cy = -21.6 cm
Cy = -7.2 cm
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The actual question is:
Vector A has x and y components of −8.70 cm and 15.0 cm, respectively. Vector B has x and y components of 13.2 cm and −6.60 cm, respectively.
If A−B+3C =0,
What are the components of C?
A copper wire has a diameter of 1.422 mm. What magnitude current flows when the drift velocity is 1.54 mm/s ? Take the density of copper to be 8.92×10
3
kg/m
3
.
The magnitude current flows, when the drift velocity is 1.54 mm/s, is 20.3 A.
When the drift velocity is 1.54 mm/s.
We will make use of the formula that relates drift velocity with the current.
We have:
vd = (I / n * A * q )
Where vd is the drift velocity
I is the current
n is the density of free electrons
A is the cross-sectional area of the wire
q is the charge carried by an electron
For copper, the value of n is 8.5 × 1028 electrons/meter cube.
Given that the wire is circular, its cross-sectional area is A = πr2 = πd2/4
where d is the diameter of the wire.
Hence we have:
d = 1.422 mm = 1.422 × 10-3 mA = π/4 x (1.422 × 10-3 m)2= 1.59 x 10-6 m^2
Now we can calculate the current as follows:
I = n * A * q * vd/I = (8.5 x 10^28)(π/4)(1.422 x 10^-3)^2(1.602 x 10^-19)(1.54 x 10^-3)=20.3A
Approximately, the magnitude of the current flowing in the copper wire is 20.3A.
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Suppose a 52-turn coil lies in the plane of the page in a uniform magnetic field that is directed out of the page. The coil originally has an area of 0.275 m2. It is stretched to have no area in 0.100 s. What is the magnitude (in V ) and direction (as seen from above) of the average induced emf if the uniform magnetic field has a strength of 1.60 T?
Faraday's law of electromagnetic induction is used to compute induced EMF, abbreviated as e. It expresses the relationship between the EMF generated and the magnetic flux's rate of change, abbreviated as φ.
The induced EMF in the coil with 52 turns that lies in the plane of the page in a uniform magnetic field of 1.60 T that is directed out of the page is calculated as follows:
Given values are as follows:52-turn coilInitial area, A1 = 0.275 m²Final area, A2 = 0m² (as it is stretched to have no area in 0.100s)Time, t = 0.100 s
Strength of the uniform magnetic field, B = 1.60 T
We need to calculate the magnitude (in V) and direction (as seen from above) of the average induced EMF, e.We know that the flux is defined as φ = B.A.
Therefore, we can write:[tex]B * A1 = B * A2 + [(ΔB / Δt) * A2][/tex]
By substituting the given values in the above formula,
we get: (1.60 T)(0.275 m²)
= [tex](1.60 T)(0 m²) + [(ΔB / Δt) * 0 m²]ΔB / Δt[/tex]
= [tex][(1.60 T)(0.275 m²)] / (0 m²)(0.100 s)ΔB / Δt[/tex]
= [tex]4.40 T/s[/tex]
Now, by using Faraday's law of electromagnetic induction, we can calculate the induced EMF.
[tex]e = -N (Δφ / Δt).[/tex]
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Tom was frightened by Jerry dropping a pot and he jumped 6ft in the air with a speed of 4 fus. Tom weighs 12lb. Piease give answers for the following. Pay attantian fo the units that it asking for the answer to be in and do not put the units
Tom's kinetic energy is 96 J, his potential energy is 2304 J, his total mechanical energy is 2400 J, his acceleration is 1.33 ft/s^2, and his force is 16 N.
1. What is Tom's kinetic energy in Joules?
KE = 1/2mv^2 = 1/2 * 12 * 4^2
KE = 96 J
2. What is Tom's potential energy at the top of his jump in Joules?
PE = mgh = 12 * 32 * 6
PE = 2304 J
3. What is Tom's total mechanical energy in Joules?
TE = KE + PE = 96 + 2304
TE = 2400 J
4. What is Tom's acceleration during his jump in feet per second squared?
a = v^2 / 2h = 4^2 / 2 * 6
a = 1.33 ft/s^2
5. What is Tom's force in pounds?
F = ma = 12 * 1.33
F = 16 N
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(a) How fast would a motorist have to be traveling for a yellow (λ=590.00 nm ) traffic light to appear green ( λ=550.00 nm ) because of the Doppler shift? (nm is nanometer and is 10
−9
meters) (b) Should the motorist be traveling toward or away from the traffic light to see this effect? (c) How fast would a motorist have to be traveling for a yellow (λ=590.00 nm) traffic light to appear red (λ=700.00 nm) because of the Doppler shift? Attach File
(a) A motorist has to be traveling with a speed of 1.26×10^7 m/s towards a yellow traffic light of wavelength 590.00 nm for it to appear green (wavelength 550.00 nm) because of the Doppler shift.For yellow light to appear red (700.00 nm), a motorist would have to be moving away from the traffic light at a high speed of 2.36×10^7 m/s, which is about 8.87% of the speed of light.
When an object, in this case, a motorist, is moving towards a traffic light, the apparent wavelength of the light received by the object is shorter than its original wavelength (known as blue-shift). As we know that green light has a shorter wavelength than yellow light, hence the yellow light will appear green to the motorist when he is moving towards it with enough speed. For yellow light to appear green (550.00 nm), a motorist would have to be moving towards the traffic light at a high speed of 1.26×10^7 m/s, which is about 4.74% of the speed of light.
(b) The motorist should be traveling towards the traffic light to observe this effect. (c) A motorist has to be traveling with a speed of 2.36×10^7 m/s away from a yellow traffic light of wavelength 590.00 nm for it to appear red (wavelength 700.00 nm) because of the Doppler shift. When an object, in this case, a motorist, is moving away from a traffic light, the apparent wavelength of the light received by the object is longer than its original wavelength (known as red-shift). As we know that red light has a longer wavelength than yellow light, hence the yellow light will appear red to the motorist when he is moving away from it with enough speed.
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At its peak, a tornado is 70.0 m in diameter and carries 350 km/h winds. What is its angular velocity in revolutions per second? rev/s Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 370rpm, and the plece of dust is 3.9 cm from the center, what is the total distance (in m ) traveled by the dust in 2 minutes? (Ignore accelerations due to getting the CD rotating.)
The angular velocity of the tornado at its peak, with a diameter of 70.0 m and wind speed of 350 km/h, is approximately 0.0379 revolutions per second (rev/s).
The angular velocity of the tornado at its peak, with a diameter of 70.0 m and wind speed of 350 km/h, is approximately 0.0379 rev/s. The angular velocity is calculated by converting the wind speed to m/s, finding the distance traveled in one revolution, and dividing the linear speed by the distance.
For the spinning CD, with a spin rate of 370 rpm and a piece of dust located 3.9 cm from the center, the total distance traveled by the dust in 2 minutes is approximately 73.8 meters. This is calculated by converting the spin rate to rev/s, finding the distance traveled in one revolution using the circumference formula, and multiplying it by the spin rate and time.
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Find the electric potential at point P in the figure (Figure 1). Part B Suppose the three charges shown in the figure are held in place. A fourth charge, with a charge of +6.82μC and a mass of 4.23 g, is released from rest at point P. What is the speed of the fourth charge when it has moved infinitely far away from the other three charges?
The speed of the fourth charge when it has moved infinitely far away from the other three charges is 46.3 m/s.
The electric potential at point P is the sum of the electric potentials due to each of the three charges. The electric potential due to a point charge is given by:
V = kQ/r
where k is the Coulomb constant, Q is the charge of the point charge, and r is the distance from the point charge to the point where the potential is being measured.
In this case, the three charges are located at (0, 0), (3, 0), and (0, 4). The distances from these points to point P are 3, 4, and 5, respectively.
The total electric potential at point P is then:
V = k(6.82 μC) / 3 + k(6.82 μC) / 4 + k(6.82 μC) / 5
= 20.4 V
When the fourth charge is released from rest at point P, it will accelerate away from the other three charges. The potential energy of the fourth charge is given by:
U = kQq/r
where k is the Coulomb constant, Q is the charge of the fourth charge, q is the charge of one of the other three charges, and r is the distance between the fourth charge and that other charge.
The total potential energy of the fourth charge is then:
U = k(6.82 μC)(-6.82 μC) / 3 + k(6.82 μC)(-6.82 μC) / 4 + k(6.82 μC)(-6.82 μC) / 5 = -85.9 V
When the fourth charge moves infinitely far away from the other three charges, its potential energy will be zero. The change in potential energy of the fourth charge is then:
ΔU = 0 - (-85.9 V) = 85.9 V
This change in potential energy is equal to the kinetic energy of the fourth charge when it is infinitely far away from the other three charges. The kinetic energy of the fourth charge is given by:
K = 1/2 mv^2
where m is the mass of the fourth charge and v is its speed.
Solving for v, we get:
v = sqrt(2K/m) = sqrt(2(85.9 V)(1/2)) = 46.3 m/s
Therefore, the speed of the fourth charge when it has moved infinitely far away from the other three charges is 46.3 m/s.
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Projectile motion describes objects projected outward near the surface of the earth objects orbiting the earth objects leaving the earth's gravitational field resistance due to friction The resultant displacement is the same as distance travelled (discussed in Ch 2) is always equal to the length of the path along which an object travels is the shortest distance from the starting point directly to the ending point is measured in units of distance divided by time Question 3 (0.5 points) When adding vectors, which of the following is NOT true the order in which the vectors are added is important drawing a diagram may be useful for solving the problem the tip-to-tail method of adding vectors is useful the arrow drawn from the tail of the first vector to the tip of the last vector represents the sum of the vectors The parallelogram method is used for adding vectors used for determining Kindergarden aptitude the same as the Pi method used for resolving a vector into its components The trigonometric function sin( theta) is equal to length of the side opposite the angle theta divided by the length of the hypotenuse (o/h) length of the side adjacent the angle theta divided by the length of the hypotenuse (a/h) length of the side opposite the angle theta divided by the length of the side adjacent the angle theta length of the side adjacent the angle theta divided by the length of the side opposite the angle theta The following are all examples of objects that experience projectile motion near the surface of the earth EXCEPT a thrown baseball a basket ball thrown toward the basket a speeding bullet a clock pendulum Projectile motion near the surface of the earth makes a path in the shape of a parabola triangle straight line trapezoid One yellow tennis ball is projected horizontally while at the same time a red tennis ball is dropped vertically from the same point near the earth's surface. The yellow tennis ball will reach the ground before the red tennis ball reach the ground after the red tennis ball reach the ground at the same time as the red tennis ball enter a lunar orbit You are hanging in a tree 3 meters off the ground. A friend on a nearby 3 meter high hill aims a tennis ball launcher horizontally directly at you. What will happen if you let go of the tree and fall directly to the ground at the same time the gun is fired? (neglecting air resistance) You will hit the ground and be in position to catch the ball as it arrives at the same spot at the same time you do. The tennis ball will pass through the spot where you were hanging. while you will have dropped down below. You will reach the ground before the tennis ball, which will arrive at the same spot after you reach the ground The tennis ball will pass over your head, but below the spot where you were hanging Question 10 (0.5 points) Two vectors can be added accurately by adding their components along chosen axes with the aid of trigonometric functions maximum and minumum magnitudes along each axis in a chosen coordinate system areas formed within right-angle triangles where the vectors form the hypotenuse of each triangle magnitudes, without taking into consideration their directions Projectile motion describes objects projected outward near the surface of the earth objects orbiting the earth objects leaving the earth's gravitational field resistance due to friction Question 2 (0.5 points) The resultant displacement is the same as distance travelled (discussed in Ch 2) is always equal to the length of the path along which an object travels is the shortest distance from the starting point directly to the ending point is measured in units of distance divided by time Question 3 (0.5 points) When adding vectors, which of the following is NOT true the order in which the vectors are added is important drawing a diagram may be useful for solving the problem the tip-to-tail method of adding vectors is useful the arrow drawn from the tail of the first vector to the tip of the last vector represents the sum of the vectors The parallelogram method is used for adding vectors used for determining Kindergarden aptitude the same as the Pi method used for resolving a vector into its components
For Question 3, the statement that is NOT true when adding vectors is the same as the Pi method used for resolving a vector into its components.
When adding vectors, the Pi method is not used for resolving a vector into its components. The Pi method, also known as the method of trigonometric components, is used to break down a single vector into its horizontal and vertical components. It involves using trigonometric functions (such as sine and cosine) to determine the magnitudes of the components.
On the other hand, when adding vectors, the tip-to-tail method is commonly used. It involves placing the vectors head-to-tail and drawing an arrow from the tail of the first vector to the tip of the last vector. The resulting arrow represents the sum or resultant of the vectors. The parallelogram method can also be used, which involves constructing a parallelogram using the vectors and drawing the resultant vector from the common point of the parallelogram.
Therefore, the statement that is NOT true when adding vectors is that the Pi method is used for resolving a vector into its components.
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. A student throws a ball vertically upwards from the top of the 7 m high CPUT roof. (a) If, after 2 seconds, he catches the ball on its ways down again, with what speed was thrown? (b) What was the velocity of the ball when its was caught? [9,8 m/s] (c) If the student fails to catch the ball with what speed will it hit the ground?
The answers to the given questions are as follows:
(a) The ball was thrown upwards from the roof with an initial velocity of -6.3 m/s.
(b) When the ball was caught on its way down after 2 seconds, its velocity was 13.3 m/s in the downward direction.
(c) If the student fails to catch the ball, it will hit the ground with a speed of approximately 13.31 m/s.
To solve the problem, we can use the equations of motion for vertical motion under constant acceleration. In this case, the acceleration is due to gravity and is equal to 9.8 m/s² (assuming no air resistance).
Given:
Initial height (h) = 7 m
Acceleration due to gravity (g) = 9.8 m/s²
Time (t) = 2 s
(a) To find the initial velocity at which the ball was thrown upwards:
Using the equation of motion:
h = ut + (1/2)gt², where u is the initial velocity.
Plugging in the known values, we have:
7 = u(2) + (1/2)(9.8)(2)²
7 = 2u + 19.6
2u = 7 - 19.6
2u = -12.6
u = -6.3 m/s
Therefore, the ball was thrown upwards with an initial velocity of -6.3 m/s (negative sign indicates the upward direction).
(b) To find the velocity of the ball when it was caught:
Since the ball is caught on its way down after 2 seconds, we can use the equation of motion:
v = u + gt, where v is the final velocity (when caught).
Plugging in the values, we have:
v = -6.3 + (9.8)(2)
v = -6.3 + 19.6
v = 13.3 m/s
Therefore, the velocity of the ball when it was caught is 13.3 m/s (positive sign indicates the downward direction).
(c) If the student fails to catch the ball, it will continue to fall freely under gravity until it hits the ground. To find the speed at which it will hit the ground, we can use the equation:
v² = u² + 2gh,
where
v is the final velocity,
u is the initial velocity,
g is the acceleration due to gravity, and
h is the initial height.
Plugging in the values, we have:
v² = (-6.3)² + 2(9.8)(7)
v² = 39.69 + 137.2
v² = 176.89
v = √176.89
v ≈ 13.31 m/s
Therefore, if the student fails to catch the ball, it will hit the ground with a speed of approximately 13.31 m/s.
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