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Physics / Chapter 37: Relativity / Section 37-8

The Relativity of Velocities — Question 3

Section 37-8 · Physics pages 6176–6177

Question 3 Velocity measurements in two frames determine the boost A particle moves at u = 0.80c in S and u′ = 0.50c in a collinear frame S′. S′ u′ v Schematic; labels, not drawing lengths, specify the data. (a) Solve u′ = (u −v)/(1 −uv/c2) for v. (b) Find v and verify the original velocity transformation.
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Question 3 - Solution Velocity measurements in two frames determine the boost S′ u′ v (a) Method and calculation u′ −uu′v/c2 = u −v ⇒ v(1 −uu′/c2) = u −u′. v = u −u′ 1 −uu′/c2 . (b) Interpretation and further calculation v/c = .80 −.50 1 −.80(.50) = .50 . .80 −.50 1 −.80(.50) = .50 = u′/c. Physical check The denominator is positive for subluminal collinear speeds. The sign of v follows from which frame sees the larger forward speed.

Question 3

Velocity measurements in two frames determine the boost

A particle moves at u=0.80cu=0.80c in S and u′=0.50cu\prime=0.50c in a collinear frame S′S\prime.

Diagram shown in the original SVG worksheet.

  1. Solve u′=(u−v)/(1−uv/c2)u\prime=(u-v)/(1-uv/c^2) for v.

  2. Find v and verify the original velocity transformation.

Question 3 - Solution

Velocity measurements in two frames determine the boost

Diagram shown in the original SVG worksheet.

(a) Method and calculation

u′−uu′v/c2=u−v⇒v(1−uu′/c2)=u−u′.u\prime-uu\prime v/c^2=u-v\quad\Rightarrow\quad v(1-uu\prime/c^2)=u-u\prime.v=u−u′1−uu′/c2.\boxed{v=\frac{u-u\prime}{1-uu\prime/c^2}}.

(b) Interpretation and further calculation

v/c=.80−.501−.80(.50)=.50.\boxed{v/c=\frac{.80-.50}{1-.80(.50)}=.50}..80−.501−.80(.50)=.50=u′/c.\frac{.80-.50}{1-.80(.50)}=.50=u\prime/c.

Physical check

The denominator is positive for subluminal collinear speeds. The sign of v follows from which frame sees the larger forward speed.

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