Kinetic Energy

To solve this problem, we need to understand the relationship between momentum and kinetic energy. Let's break it down step-by-step:

1. **Formula for Momentum and Kinetic Energy**: The momentum ( ) of a body is given by the formula:

where is the mass of the body and is its velocity.

The kinetic energy ( ) of a body is given by the formula:

2. **Relating Momentum and Kinetic Energy**: We can express kinetic energy in terms of momentum. By substituting (from the momentum formula) into the kinetic energy equation, we have:

3. **Calculate the Increase in Kinetic Energy**: Given that the momentum of the body increases by 20%, the new momentum is:

The new kinetic energy based on the increased momentum is:

4. **Determine the Percentage Increase in Kinetic Energy**: The initial kinetic energy was . The percentage increase in kinetic energy can be calculated as:

Hence, when the momentum of a body is increased by 20%, its kinetic energy increases by 44%.

The correct answer is (D): 1 : 1

= ( At Lower temperature )

As temperature is same for both the gases.
⇒ Both gases will have same average kinetic energy.

⇒ =

The correct answer is (B):

KE_R = lω²

= × x ω² × (mR²)

KE_total = x x mR² x ω²

∴ =

To determine the wavelength required to double the kinetic energy of the emitted electron, we need to apply the concept of photoelectric effect. According to Einstein's photoelectric equation, the kinetic energy of the emitted electron when light of wavelength is incident on a metal surface is given by:

where:

• is Planck's constant
• is the speed of light
• is the work function of the metal

Given that the initial kinetic energy can be expressed as:

We want to double this kinetic energy, so we have:

where is the new wavelength. We can set up the equation for and equate it to the adjusted photoelectric equation:

Simplifying this equation:

Now solve for :

Thus, the new wavelength required to double the kinetic energy is:

This matches the provided option:

Therefore, the correct answer is indeed , which is consistent with the principles of the photoelectric effect.

=0.4
Therefore , the value of coefficient of friction is 0.4
So , the correct option is (B) : 0.4

The correct option is (A): 200 kg m/s

The correct option is (A)

Solution:
For a truck and a car moving with the same kinetic energy, the distance to stop under the same retarding force can be determined by using the equation: Since the initial kinetic energy is the same for both vehicles, the work done (force times distance) to bring them to rest will be equal. Thus, both vehicles come to rest in the same distance.

Statement II:

In Statement II, when a car changes its direction from east to north, its speed may remain constant, but its velocity is changing because velocity is a vector quantity. Since the direction of velocity is changing, the car has acceleration. Therefore, the acceleration is not zero.

As velocity is changing, acceleration .

Thus, Statement II is incorrect.

Thus, the correct answer is .

Solution:
For a rolling spherical shell, the angular velocity .
The rotational kinetic energy is given by: The total kinetic energy is: The ratio of rotational kinetic energy to total kinetic energy is: Thus, , so .

To solve this problem, we need to understand the concept of kinetic energy and the conservation of momentum in a system where a stationary particle breaks into two parts.

1. The kinetic energy ( ) of an object with mass moving at velocity is given by the formula: .
2. Let's denote the kinetic energies of the two parts as and , for masses and moving at velocities and respectively: and .
3. We need to find the ratio of the kinetic energies, . Thus:

1. Applying the principle of conservation of momentum, where the total initial momentum is zero (since the particle is initially stationary), we have: .
2. Using the above momentum relation, we can express: .
3. Substituting this into the kinetic energy ratio equation gives:

1. Therefore, the ratio of their kinetic energies is .

To determine the ratio of the magnitudes of linear momentum for two bodies with equal kinetic energies, we need to understand the relationship between kinetic energy, mass, and momentum.

Concepts Used:

• Kinetic energy of an object is given by the formula: , where is the mass and is the velocity of the object.
• Linear momentum of an object is given by: .

Given:

• Mass of body 1, g
• Mass of body 2, g
• Kinetic energies are equal:

Step-by-step Solution:

1. Write the expression for kinetic energy for each body:

• For body 1:
• For body 2:

2. Since the kinetic energies are equal, we equate the expressions:

Cancel the factor and rearrange the equation:

3. From this, express the velocities in terms of each other:

4. Solve for the ratio of velocities:

5. The ratio of the magnitudes of linear momentum can be expressed as:

Substitute the values and expression for :

The ratio of their linear momentum is .

Conclusion:

The correct answer is 2:5. This matches the given correct answer in the options.

To determine the total kinetic energy of 1 mole of oxygen at 27°C, we can use the formula for the kinetic energy of gases as derived from the kinetic theory of gases. The kinetic energy of one mole of a diatomic gas like oxygen can be given by:

where:

• is the number of moles, which is 1 mole in this case.
• is the universal gas constant, given as 8.31 J/mole K.
• is the absolute temperature in Kelvin.

First, convert the given temperature from Celsius to Kelvin:

T = 27 + 273.15 = 300.15 \text{ K}

For simplicity, we round off to 300 K. Substitute these values into the kinetic energy formula:

Calculating this gives:

Thus, the total kinetic energy of 1 mole of oxygen at 27°C is 6232.5 J. This confirms that the correct answer is

6232.5 J

.

The other options are incorrect as they do not match the calculated value based on the kinetic theory of gases.

The distance covered by an object is given by the area under the velocity-time graph.
Here, the graph shows a combination of trapezoidal and rectangular areas.
The total area under the graph between and represents the distance traveled.
By calculating the area from the graph:

We are asked to find the kinetic energy of a satellite of mass revolving around the Earth at a height above the surface.

Given Data:

Concept Used:

For a satellite in a circular orbit, the gravitational force provides the centripetal force:

where is the orbital radius.

The kinetic energy of the satellite is:

Step-by-Step Solution:

Step 1: Compute the orbital radius:

Step 2: Substitute in the kinetic energy formula:

Step 3: Substitute numerical values:

Step 4: Simplify step by step:

Final Computation & Result:

The kinetic energy of the satellite is:

Hence, the required answer is 3 × 10¹⁰ J.

At the highest point, the vertical component of the velocity of the ball becomes zero, while the horizontal component remains the same. The initial kinetic energy at the point of projection is: where and . The initial velocity has two components:
- Horizontal component: ,
- Vertical component: . The kinetic energy at the highest point is: Thus, the decrease in kinetic energy is: Thus, the decrease in kinetic energy is .