Answer:
2240.92365 m/s
Explanation:
[tex]m_1[/tex] = Mass of electron = [tex]9.11\times 10^{−31}\ kg[/tex]
[tex]v_1[/tex] = Speed of electron = [tex]5.7\times 10^7\ m/s[/tex]
[tex]p_2[/tex] = Neutrino has a momentum = [tex]7.3\times 10^{-24}\ kg m/s[/tex]
M = total mass = [tex]2.34\times 10^{-26}\ kg[/tex]
In the x axis as the momentum is conserved
[tex]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[/tex]
In the y axis
[tex]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[/tex]
The resultant velocity is
[tex]R=\sqrt{v_x^2+v_y^2}\\\Rightarrow R=\sqrt{2219.10256^2+311.96581^2}\\\Rightarrow R=2240.92365\ m/s[/tex]
The recoil speed of the nucleus is 2240.92365 m/s
The recoil speed of the nucleus in the beta decay process is gotten as; 1354.27 m/s
From principle of conservation of momentum, we will have;
Recoil momentum of nucleus = Recoil momentum of emitted particle
We are given;
Mass of electron; m_e = 9.11 × 10⁻³¹ kg
speed of electron; v_e = 3.4 × 10⁷ m/s
Momentum of electron; P = 6.7 × 10⁻²⁴ kg.m/s
Thus;
Momentum of electron ignoring relativistic effect is;
P_f = m_e * v_e
P_f = 9.11 × 10⁻³¹ × 3.4 × 10⁷
P_f = 3.0974 × 10⁻²³ kg.m/s
Resultant momentum is gotten from;
R = √[(3.0974 × 10⁻²³)² + (6.7 × 10⁻²⁴)²]
R = 3.169 × 10⁻²³ kg.m/s
The formula for the recoil speed of the nucleus is;
v = R/m_n
where m_n is mass of neutrino = 2.34 × 10⁻²⁶ kg
v = (3.169 × 10⁻²³)/(2.34 × 10⁻²⁶)
v = 1354.27 m/s
Read more about recoil speed at; https://brainly.com/question/26714844
