Animated Solution for Physics - Atoms and Nuclei: There is a stream of neutrons with a kinetic energy of $0.0327\text{ eV}$. If the half-life of neutrons is $700\text{ s}$, what fraction of neutrons will decay before they travel a distance of $10\text{ m}$?

Have you ever wondered what happens to a neutron when it's not bound inside an atomic nucleus? Free neutrons are actually unstable! They undergo beta decay, transforming into a proton, an electron, and an antineutrino. In this fascinating problem, we are looking at a stream of such free neutrons traveling through space. We need to find out what fraction of them will decay before they cover a distance of $10\text{ m}$.

We are given their kinetic energy: $K=0.0327\text{ eV}$.

Before we plug this into our classical kinetic energy formula, we must convert it into the standard SI unit, Joules.

Now, using the kinetic energy formula $K=\frac{1}{2}m{v}^2$, we can solve for the velocity $v$:

The rest mass of a neutron $m$ is approximately $1.675\times 10^{-27}\text{ kg}$. Substituting our values:

Now that we know they are zipping along at $2.5\times 10^3\text{ m/s}$, how long does it take them to travel the $10\text{ m}$ distance?

Time is simply distance divided by speed:

This is a mere 4 milliseconds! Given that the half-life of a free neutron is $700\text{ s}$, this time of flight is incredibly short. We should expect only a minuscule fraction of the neutrons to decay during this brief window.

The number of surviving neutrons $N$ after a time $t$ is given by:

We are interested in the fraction of neutrons that have decayed. The number of decayed neutrons is $\Delta N=N_0-N$. Therefore, the fraction decayed is:

It is related to the half-life $T_{1/2}$ by:

Now, we need to evaluate $1-e^{-\lambda t}$. Since $t=4.0\times 10^{-3}\text{ s}$ is so much smaller than $700\text{ s}$, the product $\lambda t$ is extremely close to zero. This allows us to use the brilliant first-order Taylor approximation for the exponential function:

Applying this to our fraction:

This beautifully simplifies our calculation! We just need to multiply the decay constant by the time of flight:

And there we have it! Only about 4 out of every million neutrons will decay during this 10-meter sprint. It's a beautiful demonstration of how classical kinematics and quantum decay laws intertwine.