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At the speed of 0.44 c as a fraction of c, the moving rod will have a length of 80% that of an identical rod at rest.
The speed of the light is c.
c = 3.0 × 10⁸ m/a
Let the length of the moving rod be L.
Let the length of the rod at rest be l.
Length contraction of the rod is,
[tex]L= \sqrt{ 1 - \frac{ v ^{2} }{c ^{2} } \times l} [/tex]
The length of the moving rod is,
Length of moving road = 80% of the length of the rod at rest.
L = 80 % of the l.
[tex] = \frac{80}{100} [/tex]
= 0.80 l
The moving rod will have a length 80% that of an identical rod at rest at the speed of,
[tex]L ^{2} = (1 - \frac{ v ^{2} }{c ^{2} } )\times l ^{2} [/tex]
[tex] \frac{L ^{2}}{l ^{2} }= (1 - \frac{ v ^{2} }{c ^{2} } )[/tex]
[tex]v ^{2} = (1 - \frac{L ^{2}}{l ^{2} }) \: {c^{2} } [/tex]
[tex]v ^{2} = ( \sqrt{1 - \frac{L ^{2}}{l ^{2} }}) \: {c^{2} } [/tex]
[tex]= ( \sqrt{ 1 - \ \frac{0.80 \: l^{2}}{l ^{2} }}) \: {c^{2} } [/tex]
[tex] = \sqrt{( 1 - 0.80) } c[/tex]
v = 0.44 c
Therefore, at the speed of 0.44 c as a fraction of c, the moving rod will have a length of 80% that of an identical rod at rest.
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