A 32 -u oxygen molecule \(\left(\mathrm{O}_{2}\right)\) moving in the \(+x\) -direction at \(580 \mathrm{m} / \mathrm{s} \text { collides with an oxygen atom (mass } 16 \mathrm{u})\) moving at \(870 \mathrm{m} / \mathrm{s}\) at \(27^{\circ}\) to the \(x\) -axis. The particles stick together to form an ozone molecule. Find the ozone's velocity.

Momentum conservation is a fundamental principle in physics that states the total momentum of a closed system remains constant if no external forces act upon it. In the case of collisions or interactions between objects within the system, individual momenta might change, but the overall momentum remains the same. This principle can be mathematically represented by the equation:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
When working with momentum, ensure to consider the vector nature of the quantity, which includes both magnitude and direction. In the context of the provided exercise, we would calculate the momenta of both the \( \mathrm{O}_{2} \) molecule and \( \mathrm{O} \) atom before and after the collision and ensure they are equal to demonstrate the conservation of momentum in both the x and y directions.

To determine the velocity of an ozone molecule following a collision, you apply the laws of momentum conservation. Specifically, you split the problem into calculations along the x-axis and y-axis due to the vector nature of velocity and momentum. For the x-axis, the initial velocities are used to calculate their respective momenta, considering any angles relative to the axis. Similarly, for the y-axis, it's important to include the sine component of the angles when calculating momentum. These steps follow a systematic approach to eventually find the resulting direction and magnitude of the ozone's velocity after the interaction. By applying Pythagorean theorem and trigonometric functions, as outlined in the provided solution, you can find the resultant velocity's size and the angle of movement.

Atomic mass units (u) are a convenient way to express masses on the atomic scale, but physical equations typically require mass in kilograms (kg). To convert between these units, use the conversion factor 1 u = 1.66 x 10^-27 kg. So, when a mass is given in atomic mass units, as it was in the textbook problem with the \( \mathrm{O}_{2} \) molecule and \( \mathrm{O} \) atom, it's essential to convert it to kilograms before applying equations of momentum or other physics equations. This accuracy ensures that the calculated values are consistent and compatible with units used in standard principles like conservation of momentum.

During a collision, objects exchange momentum according to their velocities and masses. There are two types of collisions: elastic and inelastic. An elastic collision is one where both momentum and kinetic energy are conserved. However, the exercise deals with an inelastic collision, where the objects stick together after the collision, and kinetic energy is not conserved, though momentum is still conserved.

By breaking down the velocities into their x and y components, the conservation principles can be applied separately along each axis to discover the resultant velocity of the combined mass post-collision. This approach simplifies the handling of collisions in any dimension and aligns with the overall principle of momentum conservation in any physical interaction.