Answer
$v = 400~m/s$
Work Step by Step
The energy $Q$ required to raise the temperature of a substance is:
$Q = mc~\Delta T$
We can find the heat energy required to raise the temperature of the lead bullet and the wood.
$Q = m_l~c_l~\Delta T + m_w~c_w~\Delta T$
$Q = (0.015~kg)(130~J/kg~C^{\circ})(0.020~C^{\circ}) + (35~kg)(1700~J/kg~C^{\circ})(0.020~C^{\circ})$
$Q = 1190~J$
The kinetic energy of the bullet provides the energy to raise the temperature of the system. We can find the initial speed of the bullet.
$KE = Q$
$\frac{1}{2}mv^2 = Q$
$v^2 = \frac{2Q}{m}$
$v = \sqrt{\frac{2Q}{m}}$
$v = \sqrt{\frac{(2)(1190~J)}{0.015~kg}}$
$v = 400~m/s$
