Final answer:
The angle of the combined velocity vector of two cars after a collision with respect to north can be determined by using the inverse tangent function (atan) to calculate the angle relative to one of the initial momentum vectors. This is based on the principles of conservation of momentum and trigonometry.
Step-by-step explanation:
The student is asking about determining the angle of the combined velocity vector of two cars after a collision with respect to north, given a certain initial condition. In this scenario, we are dealing with the conservation of momentum principle, where momentum before collision equals momentum after collision. The problem involves solving for the resultant velocity vector of the two cars that stick together after they collide. To find the angle \( \theta \) made by the velocity vector with respect to the north, we would need to know the angle of the resultant velocity with respect to one of the initial velocity vectors, let's denote it by \( \phi \).
Assuming we have two perpendicular momentum vectors before the collision (due south and due west), and we define east as the positive x-direction and north as the positive y-direction, after the collision, the two cars will have a combined momentum vector pointing in a direction between these two axes. The angle \( \theta \) can be found using trigonometry, specifically the inverse tangent function (atan), which gives us the angle whose tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle.
For example, if the velocity vector due south is \( v_{1} \), for the first car with mass \( m_{1} \), and the velocity vector due west is \( v_{2} \), for the second car with mass \( m_{2} \), then the angle \( \theta \) can be calculated as:
\[ \theta = \text{atan}\left( \frac{m_{1}v_{1}}{m_{2}v_{2}} \right), \]
expressed with respect to north. The result would give us the angle relative to the westward axis, which we can then convert to be relative to north.