Question

A hammer of mass $M$ strikes a nail of mass $m$ with velocity of $u$ $m{s}^{-1}$ and drives it $a$ metres into a fixed block of wood. The average resistance of the wood to the penetration of the nail is:

• A. $\left(\frac{M}{M+m}\right)\frac{u^2}{2a}$
• B. $\left(\frac{M}{{(M+m)}^2}\right)\frac{u^2}{2a}$
• C. $\left(\frac{M+m}{M}\right)\frac{u^2}{2a}$
• D. $\left(\frac{M^2}{M+m}\right)\frac{u^2}{2a}$

Answer 1

The correct option is D $\left(\frac{M^2}{M+m}\right)\frac{u^2}{2a}$

Using conservation of momentum we have
$v=\frac{Mu}{M+m}$ ,
Thus we get
$FS=\frac{1}{2}(M+m)v^2$
$Fa=\frac{1}{2}(M+m)v^2$
$F=\frac{1}{2a}(M+m)(\frac{Mu}{M+m}{)}^2=(\frac{M^2}{M+m})\frac{u^2}{2a}$