91Ó°ÊÓ

In the world of physics, collisions are events where two or more bodies exert forces on each other for a relatively short time. In this exercise, we're dealing with a perfectly inelastic collision, where a linebacker tackles a halfback on a football field.

In such cases, the two players stick together after colliding. The main principle ruling this event is the conservation of momentum. Momentum is a vector quantity, meaning it has both magnitude and direction. In collision physics, we often assume no external forces are acting during the short time span of the collision. Therefore, the total momentum of the system before the collision is equal to the total momentum after the collision.

Specifically, the key steps involve calculating the momentum of each player before they collide and combining these to find the resulting motion of the system. This example highlights the importance of understanding vector nature of momentum in collision scenarios.

Adding vectors is a crucial step in solving the problem of two football players colliding. Each player has an initial momentum vector, which reflects both their speed and direction.

In our exercise, the linebacker moves north, and the halfback moves east. Thus, their momentum vectors are perpendicular. When vectors are perpendicular, we use the Pythagorean theorem to find their resultant. This gives us a new vector that represents the total momentum.

• The magnitude of the total momentum is calculated as \( \sqrt{(P_{L}^2 + P_{H}^2)} \).
• This resultant vector shows the direction and magnitude of the combined movement after collision.

Vector addition helps us merge these two separate momenta into a single vector, crucial for further calculations concerning velocity and direction after a collision.

Calculating momentum involves using the fundamental formula: \( \text{momentum} = \text{mass} \times \text{velocity} \). Each player in the exercise has their momentum calculated prior to the collision.

For the linebacker, the momentum is determined by multiplying his mass (110 kg) by his velocity (8.8 m/s), resulting in a northward momentum vector. Similarly, the halfback's momentum is found by multiplying his mass (85 kg) with his velocity (7.2 m/s), resulting in an eastward vector.

• Linebacker: \( P_{L} = 968 \, \text{kg}\cdot\text{m/s (north)} \)
• Halfback: \( P_{H} = 612 \, \text{kg}\cdot\text{m/s (east)} \)

These vector quantities are essential to understanding the complete picture of the collision. They represent how the different parts of the system are moving and allow us to apply conservation laws to find post-collision behaviors.

Determining the angle of the resultant velocity vector helps us understand the direction the players move together after the collision.

Using the tangent function, we can find the angle \( \theta \) between the resultant vector and the northward axis (the line of the linebacker's initial movement). This involves calculating the arctangent of the ratio of the eastward and northward momentum components. Specifically:

• The formula used is \( \theta = \tan^{-1} \left( \frac{P_{H}}{P_{L}} \right) \).
• In this exercise, it's calculated as \( \theta = \tan^{-1}(\frac{612}{968}) \).
• The resulting angle provides a measure of how much the combined motion veers from the north direction into the east.

This angle determination is critical in giving us a complete understanding not just of the speed, but also of the new direction the players collectively follow after the collision has occurred.