A sphere P of mass m and velocity v⃗ undergoes an oblique and perfectly elastic collision with a

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A sphere P of mass m and velocity v⃗ undergoes an oblique...

A sphere $P$ of mass $\mathrm{m}$ and velocity $\vec{v}$ undergoes an oblique and perfectly elastic collision with an identical sphere $Q$ initially at rest. The angle $\theta$ between the velocities of the spheres after the collision shall be (a) 0 (b) $45^{\circ}$ (c) $90^{\circ}$ (d) $180^{\circ}$

A sphere $P$ of mass $\mathrm{m}$ and velocity $\vec{v}$ undergoes an oblique and perfectly elastic collision with an identical sphere $Q$ initially at rest. The angle $\theta$ between the velocities of the spheres after the collision shall be
(a) 0
(b) $45^{\circ}$
(c) $90^{\circ}$
(d) $180^{\circ}$

Key Concepts

Conservation of Kinetic Energy

In a perfectly elastic collision, not only is momentum conserved, but kinetic energy is conserved as well. This principle is used alongside momentum conservation to solve for unknown velocities. The combination of both conservation laws is essential for analyzing the energy distribution in collisions and determining outcomes such as scattering angles.

Conservation of Momentum

This principle states that in a closed system with no external forces, the total momentum remains constant before and after a collision. It is crucial in analyzing collision problems, especially in multiple dimensions, because it allows for the determination of the final velocities of objects when momentum is decomposed into components along chosen axes.

Elastic Collision

An elastic collision is one in which there is no loss of kinetic energy in the system. This characteristic distinguishes elastic collisions from inelastic ones and provides additional constraints that lead to specific results, such as the occurrence of perpendicular (90°) directions in the case of collisions between identical masses.

Oblique Collision

An oblique collision occurs when two objects collide at an angle other than head-on, leading to deflections in their paths. Analyzing oblique collisions involves resolving vector quantities into components and applying conservation laws in each direction, which in certain symmetric cases (like collisions between identical objects) can result in specific angular relationships between the final trajectories.

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