A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of 1.4 × 107 m/s. Calculate the recoil speed of the residual nucleus thorium-234. Assume that the mass of a nucleus is proportional to the mass number.
Mass of the uranium-238 nucleus, m = 238
Initial speed of the nucleus, u = 0 m/s
Final speed of the nucleus = v
Final speed of the alpha particle, v2 = 1.4 × 107 m/s
Mass of the nucleus after emission, m1 = 234
Mass of the alpha particle, m2 = 4
Applying conservation of momentum, mu = m1v1 + m2v2
238×0 = 234×v + 4×1.4×107
234v = –5.6×107
v = –2.39×105 m/s ≈ –2.4×105 m/s
∴ Recoil speed of nucleus is 2.4×107 m/s opposite to the direction of emission of alpha particle.