Consider again the \(0.10-\mathrm{kg}\) pendulum of Exercise 19.1. The pendulum starts out swinging at a maximum speed of \(0.80 \mathrm{~m} / \mathrm{s}\) and stops swinging in \(5.0 \mathrm{~min}\). It is in a box that contains \(1.0 \times 10^{23}\) nitrogen molecules, each having a mass of \(4.7 \times 10^{-26} \mathrm{~kg}\) and moving at \(500 \mathrm{~m} / \mathrm{s}\). The pendulum collides, on average, \(1.0 \times 10^{22}\) times per second with the nitrogen molecules.

(a) In one collision what amount of mechanical energy is transferred on average to a nitrogen molecule, and what is the ratio of this transferred energy to the average kinetic energy of the nitrogen molecule?

(b) By what fraction has the average kinctic energy of a nitrogen molecule increased when the pendulum stops swinging?

(c) If the final mechanical energy of the pendulum is equal to the average kinetic energy of a nitrogen molecule, what is the pendulum's average final speed?

Data from Exercise 19.1

Consider a \(0.10-\mathrm{kg}\) pendulum swinging at a maximum speed of \(0.80 \mathrm{~m} / \mathrm{s}\) inside a box that contains \(1.0 \times 10^{23}\) nitrogen molecules. The mass of a nitrogen molecule is \(4.7 \times 10^{-26} \mathrm{~kg}\), and at room temperature a typical nitrogen molecule moves at \(500 \mathrm{~m} / \mathrm{s}\). What are (a) the mechanical energy of the pendulum, (b) the average kinetic energy of one nitrogen molecule, and (c) the sum of the average kinetic energies of all the nitrogen molecules?