Work And Energy

Concept: Momentum ( ) of an object is given by , where is mass and is velocity. Kinetic Energy (KE) of an object is given by . We can also express kinetic energy in terms of momentum: Since , substitute this into the KE formula: . So, . Step 1: Define the given information for the two objects Let the first object have mass , momentum , and kinetic energy . Let the second object have mass , momentum , and kinetic energy . Given: The objects have equal momentum. So, . Let this common momentum be . Step 2: Write the kinetic energy for each object using the formula For the first object: For the second object: Step 3: Find the ratio of their kinetic energies ( ) We need to find . To divide these fractions, multiply the numerator by the reciprocal of the denominator: The terms cancel out: So, the ratio of their kinetic energies is . This matches option (1).

Concept: kWh (kilowatt-hour) is a unit of energy, commonly used for electrical energy consumption. The Joule (J) is the SI unit of energy. We need to convert 1 kWh to Joules. Energy = Power Time. The SI unit of power is Watt (W), where 1 Watt = 1 Joule per second (1 W = 1 J/s). Step 1: Break down 1 kWh into its components 1 kWh means:
Power = 1 kilowatt (kW)
Time = 1 hour (h) Step 2: Convert kilowatt to watts 1 kilowatt (kW) = 1000 watts (W). So, Power = 1000 W. Step 3: Convert hour to seconds 1 hour (h) = 60 minutes 1 minute = 60 seconds So, 1 hour = . Time = 3600 s. Step 4: Calculate the energy in Joules Energy = Power Time Substitute the values in SI units (Watts for power, seconds for time): Energy = Since 1 W = 1 J/s: Energy = The 's' (seconds) unit cancels out: Energy = Energy = Step 5: Express the energy in scientific notation To write 3,600,000 in scientific notation: Move the decimal point 6 places to the left to get 3.6. So, . Therefore, . This matches option (2).

Concept: Kinetic energy (KE) is the energy possessed by an object due to its motion. It is given by the formula: where:
is the mass of the object (in kilograms, kg).
is the velocity (or speed) of the object (in meters per second, m/s or ).
KE is measured in Joules (J). Step 1: Identify the given values
Mass of the object,
Uniform velocity of the object, (which is 4 m/s) Step 2: Substitute the values into the kinetic energy formula Step 3: Calculate Step 4: Calculate the kinetic energy We can simplify the calculation: . The unit of kinetic energy is Joules (J), since mass is in kg and velocity is in m/s. So, . The kinetic energy possessed by the object is 120 J. This matches option (2).

Concept: Work done by a constant force is a scalar quantity that measures the energy transferred to or from an object when it is moved by that force. It depends on the magnitude of the force, the magnitude of the displacement, and the angle between the force and displacement vectors. Step 1: Recall the Formula for Work Done The work done ( ) by a constant force ( ) that causes a displacement ( ) is given by: where:
is the magnitude of the constant force applied.
is the magnitude of the displacement of the object.
is the angle between the direction of the force vector and the direction of the displacement vector . Step 2: Identify the Condition for Maximum Work For the work done ( ) to be maximum, given that and are constant magnitudes, the value of must be maximum. Step 3: Determine the Maximum Value of The cosine function, , ranges from -1 to +1. The maximum value of is . Step 4: Find the Angle for which is Maximum The cosine function when the angle . Step 5: Calculate Maximum Work and Evaluate Options If (when ), the work done is: Let's check the values for the given options:
For : . So, .
For : . So, .
For : . So, .
For : . So, . Comparing these values, (when ) is the largest. Therefore, maximum work is done when the angle between force and displacement is .

Concept: When an object is lifted against gravity, the work done on the object is stored as gravitational potential energy. The work done in lifting an object is equal to the change in its potential energy. Step 1: Identify the given quantities Mass of the object, Height to which the object is lifted, Acceleration due to gravity, . The standard value for is approximately . We will use this value. Step 2: Recall the formula for work done against gravity (or change in potential energy) The work done ( ) in lifting an object of mass to a height against gravity is given by: This formula is derived from . Here, the force required to lift the object is equal to its weight ( ), the displacement is the height , and the angle between the upward lifting force and the upward displacement is ( ). Step 3: Substitute the values into the formula Step 4: Calculate the work done First, multiply the mass and height: Now, multiply this by : The unit is equivalent to Joules (J), which is the SI unit of work and energy. Looking at the options, matches our calculation. Therefore, the work done will be .