Assuming that 50% of the initial kinetic energy of the bullet is converted into thermal energy in the bullet and the specific heat of lead is c, write an expression that would allow you to determine the bullet's temperature increase.

the expression is ΔT = v² / (4*c)

Explanation:

the kinetic energy K of the bullet is given by

K=1/2*m*v²

where m = mass, v = velocity

the heat Q absorbed by the bullet is given by

Q = m * c * (T final - T initial), where c = specific heat of lead, T initial = initial temperature, T final = final temperature

then

thermal energy = kinetic energy converted into thermal energy

Q = K - 1/2 * K = 1/2*K

replacing values

m * c * (T final - T initial) = 1/2 * 1/2*m*v² = 1/4*m*v²

the increase in temperature ΔT = (T final - T initial), then

ΔT = (T final - T initial) = 1/4*m*v² / (m * c) = v² / (4*c)

ΔT = v² / (4*c)