Impacts of smooth spheres: identify the always-correct statement Which one of the following statements is always true for collisions of smooth spheres (head-on or oblique) under classical impact theory?

Difficulty: Medium

If two equal and perfectly elastic smooth spheres impinge directly, they interchange their velocities.
If a sphere impinges directly on an equal sphere at rest, a fraction (1 − e^2) of the original kinetic energy is lost.
If a smooth sphere impinges on another sphere at rest, the struck sphere moves along the line of centres.
If two equal perfectly elastic spheres impinge at right angles, their directions after impact are still at right angles.
All the above

Correct Answer: If two equal and perfectly elastic smooth spheres impinge directly, they interchange their velocities.

Explanation:

Introduction / Context:
Classical impact theory for smooth spheres separates velocities into components along and perpendicular to the line of centres. The coefficient of restitution e acts only along the line of impact; perpendicular components remain unchanged. Identifying universally valid statements requires careful use of these rules.

Given Data / Assumptions:

Spheres are smooth (no tangential impulse).
Masses can be equal or unequal; impacts can be direct (collinear) or oblique.
Coefficient of restitution 0 ≤ e ≤ 1 acts along the line of centres.

Concept / Approach:
For a head-on elastic impact (e = 1) of equal masses, momentum conservation and restitution give a neat result: the two bodies exchange velocities along the line of centres. For other statements, conditions or numerical factors matter, making them not always true.

Step-by-Step Solution:

Equal masses, e = 1, direct hit: u₁ → v₂ and u₂ → v₁ (interchange), hence option (a) is always true.For option (b): when one equal sphere is initially at rest, the fraction of kinetic energy lost is (1 − e^2)/2, not (1 − e^2). Hence (b) is false in general.Option (c) is true for the struck sphere’s component along the line of centres, but with oblique impacts its overall motion includes unchanged perpendicular components; the statement as written can be misread as exclusive line-of-centres motion, so it is not universally accurate.Option (d) is only true for special geometries; not guaranteed for all right-angle approaches.

Verification / Alternative check:
Resolve velocities into normal and tangential components; apply restitution only to the normal component and confirm the interchange in the special case of equal masses and e = 1.

Why Other Options Are Wrong:
See analysis above; each relies on extra conditions or has incorrect numeric factors.

Common Pitfalls:
Applying restitution to tangential components; assuming energy loss fraction without considering mass equality and initial states.

Final Answer:
If two equal and perfectly elastic smooth spheres impinge directly, they interchange their velocities.

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Impacts of smooth spheres: identify the always-correct statement Which one of the following statements is always true for collisions of smooth spheres (head-on or oblique) under classical impact theory?

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