Answer:
0.97c
Explanation:
From the relativistic equation for length contraction, we have
[tex]l[/tex] = [tex]l_{0}\sqrt{1 - \beta }[/tex]
where
[tex]l[/tex] is the final length of the object
[tex]l_{0}[/tex] is the original length of the object before contraction
β = [tex]v^{2} /c^2[/tex]
where v is the speed of the object
c is the speed of light in free space = 3 x 10^8 m/s
The equation can be re-written as
[tex]l[/tex]/[tex]l_{0}[/tex] = [tex]\sqrt{1 - \beta }[/tex]
For the length to contract to one-fourth of the proper length, then
[tex]l[/tex]/[tex]l_{0}[/tex] = 1/4
substituting into the equation, we'll have
1/4 = [tex]\sqrt{1 - \beta }[/tex]
substituting for β, we'll have
1/4 = [tex]\sqrt{1 - v^2/c^2 }[/tex]
squaring both side of the equation, we'll have
1/16 = 1 - [tex]v^2/c^2[/tex]
[tex]v^2/c^2[/tex] = 1 - 1/16
[tex]v^2/c^2[/tex] = 15/16
square root both sides of the equation, we have
v/c = 0.968
v = 0.97c
The object must travel at a speed of [tex]\frac{\sqrt{15}}{4}[/tex] times the lightspeed ([tex]\frac{\sqrt{15}}{4}\cdot c[/tex]) to be contracted to one-fourth its proper length.
The length contraction experimented by the object as it approaches lightspeed ([tex]c[/tex]), in meters per second, is described by Lorentz contraction formula:
[tex]\frac{L}{L_{o}} = \sqrt{1-\left(\frac{v}{c} \right)^{2}}[/tex] (1)
Where:
If we know that [tex]\frac{L}{L_{o}} = \frac{1}{4}[/tex] and [tex]\frac{v}{c} = r[/tex], then the value of [tex]r[/tex] is:
[tex]\frac{1}{4} = \sqrt{1-r^{2}}[/tex]
[tex]1-r^{2} = \frac{1}{16}[/tex]
[tex]r^{2}=\frac{15}{16}[/tex]
[tex]r= \frac{\sqrt{15}}{4}[/tex]
The object must travel at a speed of [tex]\frac{\sqrt{15}}{4}[/tex] times the lightspeed ([tex]\frac{\sqrt{15}}{4}\cdot c[/tex]) to be contracted to one-fourth its proper length.
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