Question

When a ${\text{U}}^{238}$ nucleus, originally at rest, decays by emitting an $\alpha$-particle, say with speed of v m/sec, the recoil speed of the residual nucleus is :(in m/sec.)

• A. $-4v/234$
• B. $-4v/238$
• C. $4v/238$
• D. $-v/4$

Answer 1

The correct option is A $-4v/234$

Let mass of a nucleon be $m$.

So, mass of the $\alpha$ - particle $m_{\alpha }=4m$

Mass of residual nucleus $M=238m-4m=234m$

Total linear momentum of the system before decay $p_i=0$

Let the recoil speed of the residual nucleus be $V$.

Total linear momentum of the system after decay $P_f=m_{\alpha }v+MV$

$\therefore$ $p_f=2mv+234mV$

Using conservation of linear momentum : $P_f=p_i$

$\therefore$ $4mv+234mV=0$

$⟹$ $V=\frac{-4v}{234}$