Answer to Question #88429 in Mechanics | Relativity for Sridhar

Question

Free neutrons of mass, 1.67 × 10–27 kg and de Broglie wavelength 1 nm have a mean life
time of about 900 s. The distance from the source where the number of neutrons have
decayed to one-half its initial value is (h = 6.62 × 10–34 Js)
(1) 2.4 × 105 m (2) 2.4 × 10–4 m
(3) 4.8 × 105 m (4) 4.8 × 10–4 m

Answer 1

In order to define the distance from the source that is

$$l = v T,$$

one has to obtain both velocity and half-life period of neutrons.

1) according to the de Broglie theory, there exists the following relation for the momentum of a particle:

$$p = \frac{h}{\lambda}, \quad p = mv$$

$$\Rightarrow \, v = \frac{h}{m \lambda}$$

2) there exists the following relation between the half-life and average lifetime of a decaying object:

$$T_{1/2}\equiv T = \tau \ln{2}$$

Finally, we obtain:

$$l=\frac{h \tau \ln{2}}{m \lambda} = \frac{6.62 \cdot 10^{-34} \cdot 9 \cdot 10^{2} \ln{2}}{1.67 \cdot 10^{-27} \cdot 10^{-9}} \approx 2.4 \cdot 10^{5} \,m$$

Answer: (1) 2.4 * 10⁵ m.