Problem 10: Speed of Radioactive Sample
Concept:
A radiation counter records a count rate $C$ which is proportional to the activity of the source $A(t)$ and inversely proportional to the square of the distance $r(t)$ (due to solid angle considerations).
$$ C(t) \propto \frac{A(t)}{r(t)^2} $$We are given that the recorded count rate remains constant. Let initial distance be $r_0$ and decay constant be $\lambda$.
Activity at time $t$: $A(t) = A_0 e^{-\lambda t}$.
Setting the condition for constant count rate:
$$ \frac{A_0 e^{-\lambda t}}{r(t)^2} = \frac{A_0}{r_0^2} $$ $$ r(t)^2 = r_0^2 e^{-\lambda t} $$ $$ r(t) = r_0 e^{-\lambda t / 2} $$To find the speed, we differentiate the position $r(t)$ with respect to time:
$$ v(t) = \frac{dr}{dt} = r_0 \left( -\frac{\lambda}{2} \right) e^{-\lambda t / 2} $$The speed is the magnitude of velocity:
$$ |v| = \frac{r_0 \lambda}{2} \exp\left(-\frac{\lambda t}{2}\right) $$