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Chapter 33: Problem 19

How fast would you have to move relative to a meter stick for it to be 99 cm long in your reference frame?

Short Answer

Expert verified
To make a meter stick appear 99 cm long to an observer, one would have to move at about 14.1% of the speed of light.

Step by step solution

01

Conversion of units

Convert all lengths to the same unit to maintain consistency in calculations. Here both \(L\) and \(L'\) are converted to meters. Therefore, \(L = 1\) meter and \(L' = 0.99\) meters.
02

Insert the values into the formula

Substitute the known values into the formula for length contraction. \(0.99 = 1 \sqrt{1 - v^2/c^2}\) When this equation is squared, it simplifies to \(0.98 = 1 - v^2/c^2\)
03

Solve for v

Rearrange the equation to solve for \(v\). The equation becomes \(v^2/c^2 = 1 - 0.9801\). Multiplying both sides by \(c^2\), we get \(v^2 = 0.0199c^2\). Taking the square root of both sides to solve for \(v\), we find that \(v = 0.141c\) or about 14.1% of the speed of light.

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