The drawing shows a collision between two pucks on an...

The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of $0.025 \mathrm{kg}$ and is moving along the $x$ axis with a velocity of $+5.5 \mathrm{m} / \mathrm{s} .$ It makes a collision with puck $\mathrm{B},$ which has a mass of $0.050 \mathrm{kg}$ and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing. Find the final speeds of (a) puck $A$ and $(b)$ puck $B$.

Key Concepts

Vector Decomposition in Collisions

When dealing with collisions that are not head-on, it is necessary to decompose the velocities into their components along mutually perpendicular axes (typically x and y axes). This approach simplifies the application of conservation laws by allowing the equations to be separated into independent directions, making it easier to solve for the unknown quantities.

Solving Systems of Equations

Collision problems often lead to systems of simultaneous equations derived from conservation laws (momentum in multiple directions and energy). Solving these systems using algebraic methods, such as substitution or elimination, is key to finding the final speeds and trajectories of the objects involved in the collision.

Conservation of Momentum

In collision problems, the total momentum of a closed system is conserved. This conservation applies separately in each independent direction (usually x and y). By decomposing the momenta into their vector components, one sets up equations that equate the sum of the initial momenta to the sum of the final momenta in each direction, which is essential for determining unknown velocities after a collision.

Conservation of Kinetic Energy

In an elastic collision, not only is momentum conserved, but kinetic energy is also conserved. This means that the sum of the kinetic energies before the collision equals the sum after the collision. This conservation provides an additional equation that, combined with the momentum conservation equations, allows one to solve for the final speeds of the colliding objects.

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