Neutron Time of Flight and Kinetic Energy

In this experiment, we determine the velocity of a non-relativistic neutron by measuring the distance it travels and the time taken. The velocity (v) is given by the distance (L) divided by the time of flight (t):

The kinetic energy (T) of the neutron is given by the formula:

where m is the mass of the neutron. Substituting the expression for velocity into the kinetic energy formula, we get:

We are given that the error in the measurement of L is negligibly small. The mass m of the neutron is also a constant. Therefore, the kinetic energy T depends only on the time of flight t.

The relationship is $T \propto t^{-2}$. To find the relationship between the relative error in kinetic energy ($|\delta T/T|$) and the relative error in time of flight ($|\delta t/t|$), we can use error propagation principles. For a function $f(x) = Cx^n$, where C is a constant, the relationship between relative errors is approximately $|\delta f/f| = |n| |\delta x/x|$.

In our case, $T = \frac{1}{2}mL^2 t^{-2}$. Here, $x$ is t, $n$ is -2, and $C = \frac{1}{2}mL^2$ (which is treated as a constant since errors in m and L are negligible). Thus, the relationship between the relative errors is:

Calculating Permissible Time Error

We are given that we want to estimate the kinetic energy T to within 5% accuracy. This means the relative error in kinetic energy should be less than or equal to 0.05:

To find the maximum permissible error in measuring the time of flight ($|\delta t|$), we use the equality:

Substituting this into our error relationship:

We are given that the approximate time of flight is $t \approx 50$ ns. We can now solve for $|\delta t|$:

The maximum permissible error $|\delta t|$ in measuring the time of flight is 1.25 ns to achieve the desired 5% accuracy in the kinetic energy estimate.

Comparing this value to the given options, 1.25 ns is the nearest value.