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In an inelastic collision, two or more objects collide and move together as a single mass post-collision. This type of collision is characterized by the loss of kinetic energy, though the momentum of the system is conserved. In our scenario with objects A and B, they stick together after the collision, meaning they move as a single unit. The key point here is that the momentum, which is the product of mass and velocity, remains the same before and after the collision. This principle of momentum conservation holds true regardless of the collision being elastic or inelastic. Understanding inelastic collisions is crucial in physics because it applies to real-world incidents, such as car crashes, where two entities might become deformed and entangled, yet their total momentum is unchanged.

When dealing with the momentum of objects moving in different directions, vector addition is a crucial concept. Momentum is a vector, which means it has both magnitude and direction. In the exercise, object A is moving due east, while object B is moving due north. To find the total momentum after they collide, we need to add these momentum vectors together. This is done by placing the vectors tail to tail and resolving them into their components. The overall momentum vector is found by calculating the resultant vector, which is the vector that connects the start of the first vector to the end of the last. This vector addition method allows for the conservation of momentum to be visually demonstrated and calculated mathematically.

The Pythagorean Theorem is a fundamental mathematical tool used to find the magnitude of a resultant vector formed by two perpendicular vectors. In the context of our exercise, after determining the momentum vectors for objects A and B, which are at right angles to each other, you can use the Pythagorean Theorem to find the magnitude of the total momentum vector. The formula you apply is \[|\vec{p}_{\text{total}}| = \sqrt{p_{A}^2 + p_{B}^2}\]where \(p_A\) is the momentum of object A and \(p_B\) is the momentum of object B. By substituting the given values into this formula, you obtain the total momentum magnitude in a straightforward manner. This theorem is convenient in physics whenever you're dealing with perpendicular components, simplifying the process of finding the combined effect of two such vectors.

Trigonometry is a powerful tool used in physics to resolve vector quantities and analyze angles between different forces or motions. In our problem, after determining the magnitudes of the momentum vectors through the Pythagorean Theorem, trigonometry is used to find the direction of the resultant momentum vector. This is done by calculating the angle \(\theta\) using the tangent function, which compares the opposite side of a right triangle to the adjacent side:\[\tan(\theta) = \frac{p_{B}}{p_{A}}\]where \(p_B\) and \(p_A\) are the momenta of objects B and A, respectively. Use the inverse tangent function, \(\tan^{-1}\), to find the angle. In our exercise, this angle tells us the direction of the total momentum in relation to the eastward path of object A, revealing that the vector moves away from it at an angle of approximately 46.7 degrees toward the north. Trigonometry thus bridges the gap between vector magnitudes and their orientation in space.