First slide

Second law

Question

A hammer of mass M strikes a nail of mass m with velocity of u m/s and drives it a metre into fixed block of wood. The average resistance of wood to the penetration of nail is

Moderate

A

$[\frac{M}{m + m}] \frac{u^{2}}{2 a}$
B

$[\frac{M^{2}}{(m + M)^{2}}] \frac{u^{2}}{2 a}$
C

$[\frac{M^{2}}{(m + M)}] \frac{u^{2}}{2 a}$
D

$[\frac{M + m}{m}] \frac{u^{2}}{2 a}$

Solution

Applying the conservation of momentum
$mu = (M + m) v_{0}$
$∴ v_{0} = (\frac{M}{M + m}) u$
The acceleration A can be obtained using the formula
$v^{2} = u^{2} + 2 as$
Here A is negative, y =0, u = v₀ and s = a
$∴ 0^{2} = v_{0}^{2} - 2 Aa or A = v_{0}^{2} / 2 a$
$or A = {(\frac{M}{m + M})}^{2} \frac{u^{2}}{2 a}$
$Resistance = (M + m) A = \frac{M^{2}}{(M + m)} \times \frac{u^{2}}{2 a}$ .

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