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114 學年度臺灣大學轉學生招生考試｜第 65 題

普通物理學(A)

A point explosion with thermal energy E occurs at time t = 0, driving a spherical shock wave into a uniform medium of density ρ. From dimensional analysis, the radius of the shock R should have a time dependence of
  1. R ∝ t1/4
  2. R ∝ t2/5
  3. R ∝ t2/7
  4. R ∝ t
  5. R ∝ t6/5
提示
Build R from E, ρ, and t by matching dimensions (the Sedov–Taylor blast-wave problem). Solve for the exponent of t.
正確答案

正解：B

詳解
A. (A) R ∝ t^(1/4) — incorrect. Dimensional matching of E, ρ, t does not yield a 1/4 power.
B. (B) R ∝ t^(2/5) — correct. Seeking R = E^a ρ^b t^c and matching the dimensions of length, mass, and time gives a = 1/5, b = −1/5, c = 2/5. Hence R ∝ (E/ρ)^(1/5) t^(2/5) — the classic Sedov–Taylor result.
C. (C) R ∝ t^(2/7) — incorrect. The 2/7 exponent arises in a radiative (energy-losing) snowplow phase, not the energy-conserving Sedov phase set up here.
D. (D) R ∝ t — incorrect. Linear growth would mean constant expansion speed; a decelerating blast wave grows more slowly than that.
E. (E) R ∝ t^(6/5) — incorrect. This accelerating expansion is unphysical for a blast wave plowing into ambient gas.

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