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A point particle moving with velocity v strikes, in a perfectly elastic collision, a stationary sphere of the same mass and radius r. The line of motion of the particle passes at a perpendicular distance rho (with r > rho) from the centre of the sphere. Find the velocities of the two bodies after the collision.

1. Sphere moves with v*sqrt(1-(rho/r)²) along the line of centres; particle moves with v*(rho/r) perpendicular to it
2. Both move with v/2 in the original direction
3. Sphere acquires v and the particle stops, regardless of rho
4. Sphere moves with v*(rho/r); particle moves with v*sqrt(1-(rho/r)²)

Correct answer: Sphere moves with v*sqrt(1-(rho/r)²) along the line of centres; particle moves with v*(rho/r) perpendicular to it

Solution

For equal-mass elastic collision, the impact-line velocity component transfers completely to the sphere while the tangential component stays with the particle, and the two final velocities are perpendicular.

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