From the \(y\) component of momentum conservation, we can find an expression for the speed of the nucleus: \[\begin{aligned} m_p v'_p\sin\theta &= m_N v'_N\sin\phi\\ \therefore v'_N &= \frac{m_p}{m_N}...From the \(y\) component of momentum conservation, we can find an expression for the speed of the nucleus: \[\begin{aligned} m_p v'_p\sin\theta &= m_N v'_N\sin\phi\\ \therefore v'_N &= \frac{m_p}{m_N}v'_p\sin\theta \frac{1}{\sin\phi}\end{aligned}\] which we can substitute into the \(x\) equation for momentum conservation to solve for the angle \(\phi\): \[\begin{aligned} m_p v_p &= m_p v'_p\cos\theta + m_N v'_N\cos\phi\\ m_p v_p &= m_p v'_p\cos\theta + m_N\frac{m_p}{m_N}v'_p\sin\theta \frac{\co…