Calculate the energy and velocity of the a-particle -

Question

Recently, a new element has been isolated, temporarily named Ununseptium, with atomic number 117. The longest-lived isotope obtained so far is ²⁹⁴Uus with an atomic mass of 294.2105 ± 0.0007u.

²⁹⁴Uus is unstable, and decays rapidly into other atomic species. One possible decay route is via emission of an a-particle (a Helium nucleus, mass m_α = 4.0001u) to form element 115 (Uup), with atomic mass m_p = 290.1959u. Assume that the initial kinetic energy of the ^29SUus atom can be neglected, so that the atom is approximately stationary in the lab frame.

(h) Conservation of total energy before and after the collision implies E_s = E_α + E_p

where E_α is the energy of the a particle, and E_p is the energy of the Uup atom. Similarly, conservation of total momentum before and after the collision implies

0 = p_α + Pp

Using these formulas, prove that the final energy of the a-particle is given by

E_α = -m_s² m_α² m_p²/ 2m_s c²

Hint: Solve for E_p, and use the mass-energy relation E² = c²p² +m²c⁴

(i) Hence calculate the energy and velocity of the a-particle.