Question

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

Answer 1

mass of hammer = M kg
mass of nail = m kg
initial velocity of hammer = um/s
distance =  a meters.
Resistance of wood =?
Kinetic energy of hammer is (Mxu²)/2
this energy is transferred to the nail.
∴ energy of nail is equal to that of hammer.
let initial velocity of nail be v.
Kinetic energy of nail= (mx v² )/2
mv²/2 = Mu²/2
∴v²=(Mxu²)/m
⇒v=u√M/m ------------(1)
let the retardation of nail in the block be a₁m/s²
v²=u²+2as
⇒0=v²-2x a₁xa
2xa₁=v²/a
∴a₁=v²/ax2
⇒a₁=(u²(√M/m)²)/2x a (from equation 1)
⇒(u²xM)/2xaxm
∴a₁=(u²xM)/2xaxm
so, force on wooden block is
F=mx a1
=[m x (u²XM)]/2Xaxm⇒u²M/2a
∴u²M/2a is the average  resistance of wood.