What is this mathematical trickery? (removing a square root?)

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SUMMARY

The discussion revolves around a relativity problem involving time dilation, specifically calculating the time difference between an atomic clock moving at 1,000 km/hr and an identical clock on Earth. The solution involves substituting values into the time dilation equation, but the user encounters issues with calculator rounding due to the small velocity compared to the speed of light. A reference to a manipulation found on Cramster indicates the use of a Taylor expansion for the square root term in the time dilation formula, specifically for the condition where \(\frac{v^2}{c^2} << 1\).

PREREQUISITES
  • Understanding of special relativity concepts, particularly time dilation
  • Familiarity with the equation for time dilation: \( t' = t \sqrt{1 - \frac{v^2}{c^2}} \)
  • Basic knowledge of Taylor series expansions in mathematics
  • Proficiency in handling small values in calculations
NEXT STEPS
  • Study the derivation of the time dilation formula in special relativity
  • Learn about Taylor series and their applications in physics
  • Explore the implications of relativistic speeds on time measurement
  • Practice solving time dilation problems with varying velocities
USEFUL FOR

Students of physics, particularly those studying relativity, educators teaching these concepts, and anyone interested in the mathematical foundations of time dilation.

Raziel2701
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Homework Statement


So I was doing the following relativity problem: An atomic clock moves at 1 000 km/hr for 1 hr as measured by an identical clock on the Earth. At the end of the 1 hr interval, how many nanoseconds slow will the moving clock be compare with the Earth clock?


The problem itself is just based on substituting my given values into the equation for time dilation. However, because the velocity is rather small compared to the speed of light, my calculator rounds up values and I can't proceed, so I looked up the answer on cramster and they have this manipulation but I can't follow it because they do something I don't understand as depicted in this screen cap:


http://i.imgur.com/ZIT4q.png

So what exactly did they do with the square root?
 
Physics news on Phys.org
They taylor expanded the square root for [tex]\frac{v^2}{c^2}<<1[/tex]
 

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