91Ó°ÊÓ

In an inelastic collision, such as when two hockey players stick together after colliding, the conservation of momentum plays a crucial role.
The law states that the total momentum of a closed system remains constant if no external forces are acting on it.
Here’s what that means in simple terms:

• Before the collision, each player has their own momentum, which is a product of mass and velocity.
• After they collide and stick together, their total momentum is the sum of their initial individual momentums.

Because the collision is inelastic, they move together as one mass after they collide. To find the final speed, we equate the total momentum before the collision to the total momentum after. This conservation helps us determine how fast they move collectively after colliding.

Trigonometry is a valuable mathematical tool to break down vector quantities, like momentum, into components.
Understanding momentum in two dimensions requires insight into how angles affect direction and magnitude.In our problem, the players hit each other at a 115-degree angle, so we use trigonometry to split their velocities into x (horizontal) and y (vertical) components.

• Player 1, moving purely horizontally, has an x-component of momentum.
• Player 2 moves at an angle, contributing both x and y components to their momentum.

Using trigonometric functions: - The cosine of the angle gives us the x-component: \( p_{x2} = mv \cos(115^{\circ}) \).- The sine gives the y-component: \( p_{y2} = mv \sin(115^{\circ}) \).These help visualize how each player’s movement contributes to the total momentum in the x and y directions, which is essential for calculating their speed post-collision.

Momentum components are essential for analyzing motion in multiple directions, especially in collisions.
They allow us to separate and understand how movement is distributed through different paths.In the scenario of our hockey players:

• The x-component represents movement along a straight horizontal path.
• The y-component deals with vertical or perpendicular movement driven by any angular force.

To solve the problem, we start by calculating the individual player's x and y momentum components: For player 1, all momentum is in the x-direction, while player 2, because of the angle, has portions in both x and y.This is expressed mathematically as:- Total initial x-momentum: \( mv + mv \cos(115^{\circ}) \).- Total initial y-momentum: \( mv \sin(115^{\circ}) \).By summing and balancing these components following the collision, which aligns with the conservation laws, we can find the final speed of the combined mass.