Question

Calculate the velocity of the rod when its length will appear 90% of its proper length

a) 2c
b) 1c
c) 4c
d) 5c

URGENT​

Answer 1

Answer:

4 option is correct answer

Answer 2

Answer:

None of the given option is correct. The correct velocity is 0.4358 c.

Explanation:

The length of a moving object becomes shorter when observed in a rest frame, this phenomenon is called as Length contraction.

The length of the rod in a rest frame and in the rod's frame are related as

$l_o=\gamma l$.

where,

• $l$ = length of the rod observed in the rest frame.

• $l_o$ = length of the rod observed in its own moving frame also called as proper length of the rod.

• $\gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$.

• $v$ = speed with which the rod is moving.

• $c$ = speed of light in vacuum.

Given that, the length of the moving rod appears 90% of its proper length.

$l=90\%\ \text{of}\ l_o=\dfrac{90}{100} \times l_o=0.7l_o.$

Using the equation of length contraction,

$l_o=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}l\\l_o=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}0.9l_o\\1=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}0.9\\\sqrt{1-\dfrac{v^2}{c^2}}=0.9\\1-\dfrac{v^2}{c^2}=0.9^2=0.81\\\dfrac{v^2}{c^2} = 1-0.81=0.19\\\dfrac vc = \sqrt{0.19}=0.4358\\v=0.4358\ c.$

Thus, none of the given option is correct.