The function f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer function (also known as the floor function), gives the greatest integer less than or equal to x. Continuity of the floor function: The floor function is discontinuous at integer values of x. This is because, at any integer n, the function juRead more
The function f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer function (also known as the floor function), gives the greatest integer less than or equal to x.
Continuity of the floor function:
The floor function is discontinuous at integer values of x. This is because, at any integer n, the function jumps from n-1 to n. Hence, the function is not continuous at integer points.
Continuity at non-integer points:
At non-integer points, the function is continuous since it is a constant between integers.
Checking the given points:
– At x = 4, f(x) is not continuous since it jumps at an integer value.
– At x = -2, f(x) is also not continuous since it jumps at an integer value.
– At x = 1.5, f(x) = 1, which is continuous since it is not at an integer point.
– At x = 1, the function f(x) is discontinuous because it hops at the integer value.
To calculate the power generated by the turbine, we follow these steps: Step 1: Calculate the total power available due to falling water The power available is given by: Power = m × g × h Where: - m = mass flow rate of water = 15 kg/s - g = acceleration due to gravity = 10 m/s² - h = height = 60 m SRead more
To calculate the power generated by the turbine, we follow these steps:
Step 1: Calculate the total power available due to falling water
The power available is given by:
Power = m × g × h
Where:
– m = mass flow rate of water = 15 kg/s
– g = acceleration due to gravity = 10 m/s²
– h = height = 60 m
Substitute the values:
Power = 15 × 10 × 60
Power = 9000 W
Power = 9 kW
Step 2: Loss due to frictional forces
Since 10% of the energy is lost due to frictional forces, the effective power output would be
Effective Power = Total Power (1 − Loss Percentage)
Effective Power = 9 kW (1 − 0.1)
Effective Power = 9 kW × 0.9
Effective Power = 8.1 kW
The amount of power generated by the turbine is 8.1 kW.
To find the maximum speed of the child on the swing, we use the principle of conservation of energy. The total mechanical energy is conserved, so the loss in potential energy is converted to kinetic energy at the lowest point. Step 1: Calculate the change in potential energy The difference in heightRead more
To find the maximum speed of the child on the swing, we use the principle of conservation of energy. The total mechanical energy is conserved, so the loss in potential energy is converted to kinetic energy at the lowest point.
Step 1: Calculate the change in potential energy
The difference in height between the maximum and minimum positions is:
h = Maximum height − Minimum height
h = 2 m − 0.75 m
h = 1.25 m
Step 2: Apply conservation of energy law
At the highest point all the energy will be potential. At the lowest point, it will be the kinetic energy where the potential is converted into this kinetic energy.
K.E = P.E.
(1/2)mv² = mgh
Given:
– m = mass of child (gets eliminated on both the sides)
g = acceleration due to gravity = 10 m/s²
h = height = 1.25 m
– v = maximum speed
Rearrange the equation to find v:
v² = 2gh
v = √(2gh)
Substitute the values:
nv = √(2 × 10 × 1.25)
nv = √(25)
nv = 5 m/s
The maximum speed of the child on the swing is 5 m/s.
To calculate the work done overcoming the resistance of air, we will need to determine the change in kinetic energy. Step 1: Write down the formula for work done The work done by air resistance is equal to the loss in the kinetic energy of the bullet. The formula for kinetic energy is: K.E. = (1/2)Read more
To calculate the work done overcoming the resistance of air, we will need to determine the change in kinetic energy.
Step 1: Write down the formula for work done
The work done by air resistance is equal to the loss in the kinetic energy of the bullet. The formula for kinetic energy is:
K.E. = (1/2) m v²
Where:
– m = mass of the bullet = 10 g = 0.01 kg
– v = velocity of the bullet (initial and final)
Step 2: Calculate the initial and final kinetic energies
Initial velocity, u = 100 m/s
Final velocity, v = 500 m/s
To calculate the percentage change in momentum when the kinetic energy increases by 300%, let’s use the relationship between kinetic energy and momentum. Step 1: Relationship between kinetic energy and momentum Kinetic energy (K.E.) is given by: K.E. = (1/2) m v² Momentum (p) is given by: p = m v WeRead more
To calculate the percentage change in momentum when the kinetic energy increases by 300%, let’s use the relationship between kinetic energy and momentum.
Step 1: Relationship between kinetic energy and momentum
Kinetic energy (K.E.) is given by:
K.E. = (1/2) m v²
Momentum (p) is given by:
p = m v
We know:
K.E. = p² / (2m)
Step 2: Express the relationship between K.E. and p
Rearranging:
p² = 2m K.E.
Step 3: Analyze the change in K.E.
If kinetic energy increases by 300%, then the new K.E. is:
New K.E. = Initial K.E. × (1 + 300/100)
New K.E. = 4 × Initial K.E.
.
Step 4: Calculate the change in momentum
From the relationship (p² ∝ K.E.):
If K.E. increases by a factor of 4, then:
New p = √(4 × Initial p²)
New p = 2 × Initial p
Thus, momentum increases by 100%.
The function f(x) = [x], where [x] is the greatest integer function that is less than or equal to x, is continuous at
The function f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer function (also known as the floor function), gives the greatest integer less than or equal to x. Continuity of the floor function: The floor function is discontinuous at integer values of x. This is because, at any integer n, the function juRead more
The function f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer function (also known as the floor function), gives the greatest integer less than or equal to x.
Continuity of the floor function:
The floor function is discontinuous at integer values of x. This is because, at any integer n, the function jumps from n-1 to n. Hence, the function is not continuous at integer points.
Continuity at non-integer points:
At non-integer points, the function is continuous since it is a constant between integers.
Checking the given points:
– At x = 4, f(x) is not continuous since it jumps at an integer value.
– At x = -2, f(x) is also not continuous since it jumps at an integer value.
– At x = 1.5, f(x) = 1, which is continuous since it is not at an integer point.
– At x = 1, the function f(x) is discontinuous because it hops at the integer value.
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Water falls from a height of 60 m at the rate 15 kg/s to operate a turbine. The losses due to frictional forces are 10 % of energy. How much power is generated by the turbine? Take g = 10 m / s².
To calculate the power generated by the turbine, we follow these steps: Step 1: Calculate the total power available due to falling water The power available is given by: Power = m × g × h Where: - m = mass flow rate of water = 15 kg/s - g = acceleration due to gravity = 10 m/s² - h = height = 60 m SRead more
To calculate the power generated by the turbine, we follow these steps:
Step 1: Calculate the total power available due to falling water
The power available is given by:
Power = m × g × h
Where:
– m = mass flow rate of water = 15 kg/s
– g = acceleration due to gravity = 10 m/s²
– h = height = 60 m
Substitute the values:
Power = 15 × 10 × 60
Power = 9000 W
Power = 9 kW
Step 2: Loss due to frictional forces
Since 10% of the energy is lost due to frictional forces, the effective power output would be
Effective Power = Total Power (1 − Loss Percentage)
Effective Power = 9 kW (1 − 0.1)
Effective Power = 9 kW × 0.9
Effective Power = 8.1 kW
The amount of power generated by the turbine is 8.1 kW.
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A child is sitting on a swing. Its minimum and maximum heights from the ground 0.75 m and 2 m respectively, its maximum speed will be
To find the maximum speed of the child on the swing, we use the principle of conservation of energy. The total mechanical energy is conserved, so the loss in potential energy is converted to kinetic energy at the lowest point. Step 1: Calculate the change in potential energy The difference in heightRead more
To find the maximum speed of the child on the swing, we use the principle of conservation of energy. The total mechanical energy is conserved, so the loss in potential energy is converted to kinetic energy at the lowest point.
Step 1: Calculate the change in potential energy
The difference in height between the maximum and minimum positions is:
h = Maximum height − Minimum height
h = 2 m − 0.75 m
h = 1.25 m
Step 2: Apply conservation of energy law
At the highest point all the energy will be potential. At the lowest point, it will be the kinetic energy where the potential is converted into this kinetic energy.
K.E = P.E.
(1/2)mv² = mgh
Given:
– m = mass of child (gets eliminated on both the sides)
g = acceleration due to gravity = 10 m/s²
h = height = 1.25 m
– v = maximum speed
Rearrange the equation to find v:
v² = 2gh
v = √(2gh)
Substitute the values:
nv = √(2 × 10 × 1.25)
nv = √(25)
nv = 5 m/s
The maximum speed of the child on the swing is 5 m/s.
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A bullet of mass 10 g leaves a rifle at an initial velocity of 100 m/s and strikes the earth at the same level with a velocity of 500 m/s. The work done in joule overcoming the resistance of air will be
To calculate the work done overcoming the resistance of air, we will need to determine the change in kinetic energy. Step 1: Write down the formula for work done The work done by air resistance is equal to the loss in the kinetic energy of the bullet. The formula for kinetic energy is: K.E. = (1/2)Read more
To calculate the work done overcoming the resistance of air, we will need to determine the change in kinetic energy.
Step 1: Write down the formula for work done
The work done by air resistance is equal to the loss in the kinetic energy of the bullet. The formula for kinetic energy is:
K.E. = (1/2) m v²
Where:
– m = mass of the bullet = 10 g = 0.01 kg
– v = velocity of the bullet (initial and final)
Step 2: Calculate the initial and final kinetic energies
Initial velocity, u = 100 m/s
Final velocity, v = 500 m/s
Initial kinetic energy:
K.E.₁ = (1/2) m u²
K.E.₁ = (1/2) × 0.01 × (100)²
K.E.₁ = 0.005 × 10000
K.E.₁ = 50 J
Final kinetic energy:
K.E.₂ = (1/2) m v²
K.E.₂ = (1/2) × 0.01 × (500)²
K.E.₂ = 0.005 × 250000
K.E.₂ = 1250 J
Step 3: Find the work done
The work done in overcoming air resistance is the difference in kinetic energy:
Work done = K.E.₂ − K.E.₁
Work done = 1250 − 50
Work done = 1200 J
The work done in overcoming the resistance of air is 1200 J.
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If kinetic energy of a body is increased by 300% then percentage change in momentum will be
To calculate the percentage change in momentum when the kinetic energy increases by 300%, let’s use the relationship between kinetic energy and momentum. Step 1: Relationship between kinetic energy and momentum Kinetic energy (K.E.) is given by: K.E. = (1/2) m v² Momentum (p) is given by: p = m v WeRead more
To calculate the percentage change in momentum when the kinetic energy increases by 300%, let’s use the relationship between kinetic energy and momentum.
Step 1: Relationship between kinetic energy and momentum
Kinetic energy (K.E.) is given by:
K.E. = (1/2) m v²
Momentum (p) is given by:
p = m v
We know:
K.E. = p² / (2m)
Step 2: Express the relationship between K.E. and p
Rearranging:
p² = 2m K.E.
Step 3: Analyze the change in K.E.
If kinetic energy increases by 300%, then the new K.E. is:
New K.E. = Initial K.E. × (1 + 300/100)
New K.E. = 4 × Initial K.E.
.
Step 4: Calculate the change in momentum
From the relationship (p² ∝ K.E.):
If K.E. increases by a factor of 4, then:
New p = √(4 × Initial p²)
New p = 2 × Initial p
Thus, momentum increases by 100%.
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