Answer
The nail's temperature will rise $43~C^{\circ}$
Work Step by Step
We can find the kinetic energy of the hammer head as:
$KE = \frac{1}{2}mv^2$
$KE = \frac{1}{2}(1.20~kg)(7.5~m/s)^2$
$KE = 33.75~J$
We can find the total energy in 8 blows of the hammer head as:
$E = 8\times (33.75~J) = 270~J$
The energy $Q$ required to raise the temperature of a substance is
$Q = mc~\Delta T$
We can assume the all the kinetic energy from the hammer raises the temperature of the nail. We can find the temperature rise of the nail.
$Q = m_i~c_i~\Delta T$
$\Delta T = \frac{Q}{m_i~c_i}$
$\Delta T = \frac{270~J}{(0.014~kg)(450~J/kg~C^{\circ})}$
$\Delta T = 43~C^{\circ}$
The nail's temperature will rise $43~C^{\circ}$.
