Three identical small spheres, each of weight 2 lb, can...

Three identical small spheres, each of weight 2 lb, can slide freely on a horizontal frictionless surface. Spheres $B$ and $C$ are connected by a light rod and are at rest in the position shown when sphere $B$ is struck squarely by sphere $A$ which is moving to the right with a velocity $v_{0}=(8 \mathrm{ft} / \mathrm{s})$ i. Knowing that $\theta=45^{\circ}$ and that the velocities of spheres $A$ and $B$ immediately after the impact are $v_{A}=0$ and $\dot{v}_{B}=(6 \mathrm{ft} / \mathrm{s}) \hat{\mathrm{i}}+\left(\mathrm{v}_{B}\right)_{y}$ j. determine $\left(v_{B}\right)_{y}$ and the velocity of $C$ immediately after impact.

Three identical small spheres, each of weight 2 lb, can slide freely
on a horizontal frictionless surface. Spheres $B$ and $C$ are connected by a light rod and are at rest in the position shown when sphere $B$ is struck squarely by sphere $A$ which is moving to the right with a velocity $v_{0}=(8 \mathrm{ft} / \mathrm{s})$ i. Knowing that $\theta=45^{\circ}$ and that the velocities of spheres $A$ and $B$ immediately after the impact are $v_{A}=0$ and
$\dot{v}_{B}=(6 \mathrm{ft} / \mathrm{s}) \hat{\mathrm{i}}+\left(\mathrm{v}_{B}\right)_{y}$ j. determine $\left(v_{B}\right)_{y}$ and the velocity of $C$ immediately after impact.

Key Concepts

Conservation of Linear Momentum

This principle states that in the absence of external forces, the total linear momentum of a system remains constant during collisions. For problems like this, where spheres are moving on a frictionless surface, it is essential to set up momentum conservation equations for both the horizontal and vertical directions to determine the unknown velocities after impact.

Rigid Body Constraint

When two objects are connected by a light rod, they must keep a fixed relative distance. This rigid connection imposes a constraint on their motion, meaning that their velocities are not independent but instead related through the geometry of the connection. Such constraints reduce the system's degrees of freedom and help relate the components of the velocities of the connected objects.

Component Analysis

Analyzing the motion by breaking the velocity vectors into their horizontal and vertical components is crucial in collision problems, especially when angles are involved. This method allows one to apply conservation laws independently in each direction and solve for unknown vector components effectively.

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