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On a frictionless horizontal line, a ball A of mass m moves with…

Question

On a frictionless horizontal line, a ball A of mass m moves with speed u and collides head-on elastically with a stationary ball B of mass 2m. Ball B then moves on and collides elastically with a fixed vertical wall, rebounding with its speed unchanged, after which it travels back and collides head-on elastically again with ball A. After this second collision between A and B, the final speed of A equals (p/q)u in lowest terms, where p and q are positive integers. Find p + q.

✓ Verified answer: 16checked by our engine — not a guess

Step-by-step solution

Take u = 1.
Elastic collision of A (mass m, speed u) with B (mass 2m at rest): vA = (m-2m)/(3m) u = -u/3, vB = (2m)/(3m) u = 2u/3.

B (moving +2u/3) hits the wall and rebounds to -2u/3.

Now A moves at -u/3 and B at -2u/3 (both leftward, B faster, so B catches A).

Elastic collision of A (m, -u/3) with B (2m, -2u/3): vA' = [(m-2m)(-u/3) + 2(2m)(-2u/3)]/(3m) = [(u/3) - 8u/3]/3 = (-7u/3)/3 = -7u/9.
Final speed of A = (7/9)u.
So p/q = 7/9, p + q = 16.

Final answer16

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