Key 2

Up a level : Time dilation
  1.  Spaceship:
    1. The one year on the spaceship is a proper time, and the time on the earth is the calculated time. So, we get \displaystyle t=\frac{1}{{\sqrt{{1-{{{0.3}}^{2}}}}}}=\text{1}\text{.15470 years} or one year and 5.65 days. The time on the spaceship will thus go slower than the time on the Earth, according to the people on the Earth.
    2. The one year on the Earth is now the proper time for the Earth,  so the situation is completely symmetrical to the one above, so the answer is 1.15470 years. The time on the spaceship runs slower than the time on the Earth according to the Earth, and the time on Earth runs slower than the time on the spaceship, according to the people on the spaceship.
    3. Now t=1 year, so t0=γt=(1/1.15470)t=0.866 years.
  2.  The Muon:
    1. The distance is s=ct=3·108 · 2.2·10–6=660 m.
    2. \displaystyle \gamma =\frac{1}{{\sqrt{{1-{{{\left( {\frac{v}{c}} \right)}}^{2}}}}}}=\frac{1}{{\sqrt{{1-{{{0.9992}}^{2}}}}}}=25
    3. \displaystyle t=\gamma \ {{t}_{0}}=25\cdot 2.2\cdot {{10}^{{-6}}}=5.5\cdot {{10}^{{-5}}}\text{ s}
    4.    25 times further or 16500 m.  The muons are created by cosmic rays interacting with atoms in the upper atmosphere. According to classical physics, we should not detect those at ground level, but we do. This is a strong indication that time dilation is a real phenomenon.
  3.  v=10000/3.6=2777.778 m/s.  This gives us

\displaystyle t=\frac{1}{{\sqrt{{1-{{{\left( {\frac{{(10000/3.6}}{{3\cdot {{{10}}^{8}}}}} \right)}}^{2}}}}}}=1.00000000004287\text{ years}

or a difference of 1.35 ms.

  1. v=100/3.6=2.778 m/s. If we try this, then we get 1 in Excel and on most calculators.   But, if you know enough calculus, then you could try with a series expansion of

\displaystyle y=\frac{1}{{\sqrt{{1-{{x}^{2}}}}}}

This gives us

\displaystyle y=1+\frac{{{{x}^{2}}}}{2}+...

So, to find out how much more than a year we are talking about, we only need to look at the last term for x=v/c. This gives us one year + 4.28669·10-15 years, or one year +4.28669·10-15·365.24·24·60·60·109=135 ns.

By the way, the approximation of the extra part is correct to 14 decimal digits.

Up a level : Time dilationLast modified: Jun 4, 2025 @ 15:47