Question

A bullet moving with velocity collides against a wall, consequently half of its kinetic energy is converted to heat. If the whole heat is acquired by the bullet then the rise in temperature will be:

Answer 1

Moving particles lose energy mostly through friction so that their kinetic energies eventually get transferred to heat energy.
First, one should know about the kinetic energy expression.
Then here is the question, it is mentioned that half of the kinetic energy will be converted to heat energy
Hence after writing the kinetic energy expression just do half of the energy that will be converted to heat.
After that use the formula for heat absorbed by a body in terms of specific heat and change in temperature and then finally equate that with the previously found heat energy absorbed by the bullet.

Formula used:
Kinetic energy = $0.5m{v^2}$
Heat absorbed = $ms({t_2} - {t_1})$

Complete step-by-step solution:
For a bullet with mass=$m$ and velocity=$v$ ;
the kinetic energy of bullet= = $0.5m{v^2}$
Half of the kinetic energy is converted to Heat energy
So, heat absorbed by bullet = $\dfrac{{0.5m{v^2}}}{2}$
Therefore, heat absorbed by bullet = $0.25m{v^2}$
Now, we will use the expression for heat absorbed in terms of specific heat and change in temperature
Heat absorbed = $ms({t_2} - {t_1})$; where s= specific heat and $({t_2} - {t_1})$= change in temperature
Accordingly now,
$0.25m{v^2} = ms({t_2} - {t_1})$
Therefore, from the above equation $({t_2} - {t_1}) = \dfrac{{{v^2}}}{{4s}}$ is the temperature rise.
Hence option A is correct.
Note: When the bullet hits the target then both the bullet and the target get deformed and hence the Kinetic energy of the bullet gets converted to heat energy. When we heat a substance past its melting point, whatever energy we add to it, goes into melting another portion of the mass hence the concept of latent heat is useful in solving such problems. We can define the Heat capacity of a substance as the heat absorbed by the substance for change in temperature by one degree. Specific heat capacity is defined as heat absorbed per unit gram of a substance to result in a change of temperature by one degree.