Kinetic Energy

We know that the average translational kinetic energy of molecules in an ideal gas is related to the pressure, volume, and temperature by the ideal gas law: For a monatomic gas, the average kinetic energy of a molecule is given by: We can also use the equation for the pressure-volume relationship for gases: Given the total energy is and the volume , we can now calculate the pressure by substituting these values into the equation. The pressure comes out to be . Thus, the pressure in the vessel is .

Since the two cars have the same kinetic energy, we can use the formula for kinetic energy: From this, the velocity of each car can be calculated. The linear momentum is directly proportional to the mass of the car, meaning the heavier car will have more momentum.

Given that both bodies have the same kinetic energy, we know that: Since the momentum of the two bodies is and , and both have the same kinetic energy: The above equation implies that the body with the larger mass will have a smaller momentum because the kinetic energy is constant for both. Therefore, the light body must have less momentum than the heavy body. Thus, .

Given: Step 1: Formula for Kinetic Energy
The kinetic energy of a body is given by: Step 2: Substitute the Values
Substitute the given values into the formula: Thus, the kinetic energy is:

Rotational kinetic energy , where is the moment of inertia and is the angular velocity. For a solid sphere, , where is the mass and is the radius. Since the sphere rolls without slipping, , where is the linear velocity. . Translational kinetic energy . Total kinetic energy . The ratio of rotational to total kinetic energy is .