Two billiard balls undergo an elastic collision as shown in...

Two billiard balls undergo an elastic collision as shown in Figure P7.27. Ball 1 is initially traveling along $x$ with a speed of $10 \mathrm{m} / \mathrm{s},$ and ball 2 is at rest. After the collision, ball 1 moves away with a speed of $4.7 \mathrm{m} / \mathrm{s}$ at an angle $\theta=60^{\circ} .$ a. Find the speed of ball 2 after the collision. b. What angle does the final velocity of ball 2 make with the $x$ axis?

Two billiard balls undergo an elastic collision as shown in Figure P7.27. Ball 1 is initially traveling along $x$ with a speed of $10 \mathrm{m} / \mathrm{s},$ and ball 2 is at rest. After the collision, ball 1 moves away with a speed of $4.7 \mathrm{m} / \mathrm{s}$ at an angle $\theta=60^{\circ} .$
a. Find the speed of ball 2 after the collision.
b. What angle does the final velocity of ball 2 make with the $x$ axis?

Key Concepts

Vector Decomposition

Vector decomposition involves breaking a vector into its component parts, typically along mutually perpendicular axes such as x and y. This is essential in collision problems where the angles of motion change, as it allows one to analyze how the momentum and energy are distributed in each direction after the collision.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. In problems involving elastic collisions, the total kinetic energy before the collision equals the total kinetic energy after the collision, making it possible to solve for unknown speeds and directions of objects post-collision.

Conservation of Momentum

Conservation of momentum is the principle that the total momentum of a closed system remains constant if no external forces act on it. In two-dimensional collisions, momentum conservation is applied separately along each axis (commonly x and y), allowing the vector components of the velocities to be analyzed independently.

Conservation of Kinetic Energy

In elastic collisions, the conservation of kinetic energy is a key concept. This principle states that the sum of the kinetic energies of colliding objects remains unchanged before and after the collision, which can be used in conjunction with momentum conservation to solve for unknown quantities in the collision.

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