Answer
The recoil speed of the thorium nucleus is $~2.57\times 10^5~m/s$
Work Step by Step
The initial momentum of the system is zero. By conservation of momentum, if the alpha particle gains a certain momentum $p_a$ in one direction, the remaining nucleus will gain the same magnitude of momentum $p_t$ in the opposite direction.
We can find the recoil speed $v_r$ of the thorium nucleus:
$p_t = p_a$
$m_t~v_r = m_a~v_a$
$v_r = \frac{m_a~v_a}{m_s}$
$v_r = \frac{(4.0~u)(0.050~c)}{234~u}$
$v_r = 8.55\times 10^{-4}~c$
$v_r = (8.55\times 10^{-4})(3.0\times 10^8~m/s)$
$v_r = 2.57\times 10^5~m/s$
The recoil speed of the thorium nucleus is $~2.57\times 10^5~m/s$.
