Final answer:
The speed of the recoiling nucleus will be proportional to the speed of the alpha particle.
Step-by-step explanation:
When an atomic nucleus at rest decays radioactively into an alpha particle and a nucleus smaller than the original, the recoiling nucleus will have a speed equal to the difference in momentum between the initial and final state.
In this case, since the alpha particle is given a speed of 162,078 m/s, we can use conservation of momentum to find the speed of the recoiling nucleus.
The momentum of the alpha particle, p_alpha, is given by p_alpha = m_alpha * v_alpha, where m_alpha is the mass of the alpha particle and v_alpha is its speed.
The momentum of the recoiling nucleus, p_nucleus, is given by p_nucleus = m_nucleus * v_nucleus, where m_nucleus is the mass of the recoiling nucleus and v_nucleus is its speed.
By conserving momentum, we have p_alpha = p_nucleus, which implies m_alpha * v_alpha = m_nucleus * v_nucleus.
Since we know the speed of the alpha particle is 162,078 m/s, we can solve for the speed of the recoiling nucleus:
v_nucleus = (m_alpha * v_alpha) / m_nucleus = (4 * 162,078 m/s) / m_nucleus
Unfortunately, without additional information about the masses of the alpha particle and the recoiling nucleus, we cannot determine the exact speed of the recoiling nucleus. However, we can conclude that it will be proportional to the speed of the alpha particle.