Free neutrons have a characteristic half-life of 10.4 min . What fraction of a group of free neu

step 1 We are given that neutrons have half -life, the half of 10 .4 minutes. We want to find what is t

Free neutrons have a characteristic half-life of 10.4 min ....

Free neutrons have a characteristic half-life of 10.4 $\mathrm{min}$ . What fraction of a group of free neutrons with kincticncrgy 0.0400 $\mathrm{cV}$ decays before traveling a distance of 10.0 $\mathrm{km}$ ?

Key Concepts

Exponential Decay

Exponential decay describes the process in which a quantity decreases at a rate proportional to its current value. In radioactive decay, the number of undecayed nuclei decreases exponentially over time according to the relation N(t) = N(0) e^(-?t), where ? is the decay constant.

Half-life

The half-life is the time required for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by the formula t?/? = ln(2)/?. This concept is widely used to characterize the stability of unstable particles and isotopes, and it underpins calculations of decay fractions over time.

Decay Constant

The decay constant (?) is a measure of the probability per unit time that a nucleus (or particle) will decay. It is directly related to the half-life, providing the bridge from the discrete concept of half-life to the continuous process described by exponential decay.

Kinetic Energy and Velocity

The kinetic energy of a moving particle is related to its velocity by classical formulas such as KE = (1/2)mv² when non-relativistic approximations apply. Determining the speed of a particle from its kinetic energy is essential in time-of-flight calculations, where the time it takes to travel a given distance is needed.

Time-of-Flight Calculation

The time-of-flight calculation involves determining the duration a particle takes to travel a specific distance given its velocity (time = distance/velocity). This time is then used in the exponential decay formula to assess the fraction of particles that decay before covering that distance.

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