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

`v = 2.99* 10^(8) m/s`

v = 0.99 c

Step-by-step explanation:

Given data:

Distance travelled by neutron

d = 25000 light years

`=25000* 9.4* 10^(15) m`

`= 235* 10^(18) m`

Total time taken by neutron is
`t = (d)/(v)`

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

`t =\frac{t_0}{\sqrt{1 -(v^2)/(c^2)}}`

`(235* 10^(18))/(v) = \frac{610}{\sqrt{1 -(v^2)/((9* 19^(18))^2)}}`

solving for v we get

`v = 2.99* 10^8 m/s`

v = 0.99 c