Question

The carbon isotope 14C is used for carbon dating of archeological artifacts. 14C(mass 2.34×10−26kg) decays by the process known as beta decay in which the nucleus emits an electron (the beta particle) and a subatomic particle called a neutrino. In one such decay, the electron and the neutrino are emitted at right angles to each other. The electron (mass 9.11×10−31kg) has a speed of 3.40×107 m/s and the neutrino has a momentum of 6.70×10−24 kg⋅m/s .
What is the recoil speed of the nucleus?
The answer is not 1900 or 1600 m/s...

Answer 1

Answer:

2240.92365 m/s

Explanation:

$m_1$ = Mass of electron = $9.11\times 10^{−31}\ kg$

$v_1$ = Speed of electron = $5.7\times 10^7\ m/s$

$p_2$ = Neutrino has a momentum = $7.3\times 10^{-24}\ kg m/s$

M = total mass = $2.34\times 10^{-26}\ kg$

In the x axis as the momentum is conserved

$Mv_x=m_1v_1\\\Rightarrow v_x=\dfrac{m_1v_1}{M}\\\Rightarrow v_x=\dfrac{9.11\times 10^{−31}\times 5.7\times 10^7}{2.34\times 10^{-26}}\\\Rightarrow v_x=2219.10256\ m/s$

In the y axis

$Mv_y=p_2\\\Rightarrow v_y=\dfrac{p_2}{M}\\\Rightarrow v_y=\dfrac{7.3\times 10^{-24}}{2.34\times 10^{-26}}\\\Rightarrow v_y=311.96581\ m/s$

The resultant velocity is

$R=\sqrt{v_x^2+v_y^2}\\\Rightarrow R=\sqrt{2219.10256^2+311.96581^2}\\\Rightarrow R=2240.92365\ m/s$

The recoil speed of the nucleus is 2240.92365 m/s