When you open the door to an air-conditioned room, you mix hot gas with cool gas. Saying that a gas is hot or cold actually refers to its average energy; that is, the hot gas molecules have a higher kinetic energy than the cold gas molecules. The difference in kinetic energy in the mixed gases decreases over time as a result of elastic collisions between the gas molecules, which redistribute the energy. Consider a two-dimensional collision between two nitrogen molecules \(\left(\mathrm{N}_{2},\right.\) molecular weight \(=28.0 \mathrm{~g} / \mathrm{mol}\) ). One molecule moves at \(30.0^{\circ}\) with respect to the horizontal with a velocity of \(672 \mathrm{~m} / \mathrm{s} .\) This molecule collides with a second molecule moving in the negative horizontal direction at \(246 \mathrm{~m} / \mathrm{s}\). What are the molecules' final velocities if the one that is initially more energetic moves in the vertical direction after the collision?

Kinetic energy fundamentally refers to the energy possessed by an object due to its motion. All moving objects have kinetic energy, which is directly proportional to the mass of the object, and the square of its velocity. It is given by the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \(m\) is the object's mass and \(v\) is its velocity. In the context of gases, such as in our exercise, molecules are constantly on the move, and their kinetic energy reflects their temperature. The 'hotter' a molecule is, the faster it moves, and thus, the higher its kinetic energy. In the given exercise, the collision between two nitrogen molecules redistributes their kinetic energy. It's worth noting how the collision leads to a dispersion of energy, eventually leading to a thermal equilibrium where both molecules would end up with similar kinetic energies, assuming a perfectly elastic collision in which no kinetic energy is lost.

The conservation of momentum is a fundamental concept in physics which states that if no external forces act on a closed system of objects, the total momentum of that system remains constant. Momentum is a vector quantity, meaning it has both magnitude and direction, and is a product of an object's mass and its velocity:
\[ \vec{p} = m\vec{v} \]
For collisions, momentum conservation holds true for each direction separately. This is crucial in analyzing the outcome of collisions, as seen in our exercise. With two molecules, A and B, moving in two dimensions, we set up separate equations for the horizontal (x) and vertical (y) momentum conservation. This principle allows us to predict the final velocities of each molecule after the collision, provided that the collision is elastic. By ensuring momentum is conserved, we can isolate the final velocities and find a solution that respects both the physical law and the conditions of the problem.

Collisions in two-dimensional space, like the one outlined in our exercise involving nitrogen molecules, are slightly more complex than one-dimensional collisions because they involve both the x (horizontal) and y (vertical) components of motion. The principles governing the collision—specifically, the conservation of momentum and kinetic energy—are the same, but must be applied to each direction independently.

For an elastic collision in two dimensions, both the total kinetic energy and the momentum of the system are conserved. As we solved in the exercise, breaking down the initial velocities into their component forms allows us to apply these conservation laws separately on the x and y axes. The final velocities are then determined by the requirement that the sum of the momenta and kinetic energies before the collision must equal the sum after the collision. The complexity of two-dimensional collisions often requires a systematic approach to problem-solving, such as clearly labeling the initial and final velocities, and carefully applying vector addition and resolution.