91Ó°ÊÓ

In any collision, understanding the conservation of momentum is essential. During an inelastic collision, like a football tackle where two players stick together, the total momentum of the system before and after the collision remains constant. Momentum, defined as mass multiplied by velocity (\(p = m \times v\), ensures that the overall momentum is preserved.
In this problem, a 110-kg linebacker and an 85-kg halfback collide, causing them to move as a single mass post-collision. Their individual momenta before collision are calculated based on their masses and velocities, with the linebacker moving north and the halfback moving east. The key to solving such problems is realizing that the sum of the momenta in each direction (north and east) will equal the total momentum after they collide and merge into one entity, moving with a common velocity.

To find out how the combined mass moves after the collision, it's crucial to separate the momentum into components. Here, we'll think of each movement direction—north and east—as distinct momentum components.
By multiplying the linebacker's mass by his velocity, you find the total northward momentum. Similarly, the total eastward momentum is the halfback’s mass times his velocity. This separation into momentum components helps because the players aren't initially moving in the same direction.
The northward component calculates to 968 kg m/s, and the eastward component calculates to 612 kg m/s. Understanding these components lays the groundwork for determining the actual velocity and direction post-collision by combining these influences at a right angle.

Once momentum components in both directions are calculated, the Pythagorean theorem becomes handy to find the resultant velocity's magnitude. The players form a right-angle movement since one is moving north and the other east.
By considering the north and east components as perpendicular, we can use the Pythagorean theorem to resolve these into a single combined velocity magnitude. So, the formula becomes \[v = \sqrt{v_{north}^2 + v_{east}^2}\]. This helps us find the total speed by plugging in the component velocities derived earlier (4.964 m/s north and 3.138 m/s east).
The verse of utilizing this theorem is turning complex motion into a single manageable solution, giving us the magnitude of the movement after the collision.

With the velocity's magnitude found, figuring out the direction requires applying trigonometry, specifically the inverse tangent function. Since the motion forms a right-angled triangle, the angle relative to the north can be found by comparing the east and north velocities.
The formula to use is \(\theta = \tan^{-1}\left(\frac{v_{east}}{v_{north}}\right)\). This formula focuses on the ratio of the eastward to northward velocity, thus yielding the angle with respect to due north.
In this problem, using the inverse tangent reveals an angle of approximately 32.33 degrees towards the east of north. This calculation provides a clear direction of the combined movement post-collision, which is essential for capturing both magnitude and direction of their net velocity.