Question

A bomb of mass $M$ at rest explodes into two fragments of masses $m_{1}$ and $m_{2}$. The total energy released in the explosion is $E$. If $E_{1}$ and $E_{2}$ represent the energies carried by masses $m_{1}$ and $m_{2}$ respectively, then which of the following is correct?

• A. $E_{1}=\frac{m_{2}}{M} E$
• B. $E_{1}=\frac{m_{1}}{m_{2}} E$
• C. $E_{1}=\frac{m_{1}}{M} E$
• D. $E_{1}=\frac{m_{2}}{m_{1}} E$

Answer 1

Answer: (A)

Conserving momentum for the exploding bomb, $0=p_{1}+p_{2}$
$\Rightarrow p_{1} =-p_{2}$ or $p_{1}^{2}=p_{2}^{2}$
$E_{1}=\frac{p_{1}^{2}}{2 m_{1}}$ and $E_{2}=\frac{p_{2}^{2}}{2 m_{2}}$
$\Rightarrow 2 m_{1} E_{1}=2 m_{2} E_{2}$
$m_{1} E_{1}=m_{2} E_{2} ; \frac{E_{2}}{E_{1}}=\frac{m_{1}}{m_{2}} ; \frac{E_{2}}{E_{1}}+1=\frac{m_{1}}{m_{2}}+1$
$\frac{E_{2}+E_{1}}{E_{1}}=\frac{m_{1}+m_{2}}{m_{2}}$
$\Rightarrow \frac{E}{E_{1}}=\frac{M}{m_{2}}$ or $E_{1}=\frac{m_{2} E}{M}$