Question

Neutrons have a proper half-life of 610 seconds. It has been theorized that extremely energetic neutrons could be produced by collisions of ultra-energetic protons (accelerated by an unknown mechanism) with nuclei. For a collection of neutrons to reach earth with 50% surviving, how close to the speed of light must their velocity be (i.e., what is c – v in m/s) if they are created near Sagittarius A (black hole at the center of our galaxy), 25,000 light-years distant?

Hint: the γ factor is extremely large so use an approximation of β for large γ

Answer 1

Answer:

$v = 2.99\times 10^{8} m/s$

v = 0.99 c

Explanation:

Given data:

Distance travelled by neutron

d = 25000 light years

  $=25000\times 9.4\times 10^{15} m$

  $= 235\times 10^{18} m$

Total time taken by neutron is $t = \frac{d}{v}$

we know that relativistic formula of proper time can be computed as

$t =\frac{t_0}{\sqrt{1 -\frac{v^2}{c^2}}}$

$\frac{235\times 10^{18}}{v} = \frac{610}{\sqrt{1 -\frac{v^2}{(9\times 19^{18})^2}}}$

solving for v we get

$v = 2.99\times 10^8 m/s$

v = 0.99 c