Assuming that 50% of a bullet's initial kinetic energy is converted to thermal energy and given the specific heat capacity of lead as \( c \), the formula to calculate the temperature increase, \( \Delta T \), of the bullet is: \[ \Delta T = \frac{0.5 \cdot KE}{m \cdot c} \] where \( KE \) is the initial kinetic energy of the bullet, and \( m \) is the mass of the bullet.

The formula provided to calculate the temperature increase (ΔT) of a bullet assumes that 50% of the bullet's initial kinetic energy (KE) is converted into thermal energy when the bullet is stopped (for example, when hitting a target). Here’s how you can understand this equation step-by-step:

1. Initial Kinetic Energy (KE) is the energy that the bullet has due to its motion. It can be calculated using the formula: \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass of the bullet and \( v \) is its velocity.

2. When the bullet stops, some of this kinetic energy is converted into other forms of energy, mainly heat, due to the forces of impact and friction.

3. According to the problem statement, 50% of this kinetic energy becomes thermal energy. Hence, the thermal energy (Q) that we use to calculate the temperature change is half of the initial kinetic energy: \[ Q = 0.5 \cdot KE \]

4. The specific heat capacity (c) is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin).

5. The increase in the temperature of the bullet (ΔT) can be calculated using the formula that relates the thermal energy, mass, and specific heat capacity: \[ Q = m \cdot c \cdot \Delta T \]

Combining this with the previous step, we have: \[ 0.5 \cdot KE = m \cdot c \cdot \Delta T \]

6. When you solve for ΔT to find the temperature increase, you rearrange the formula to: \[ \Delta T = \frac{0.5 \cdot KE}{m \cdot c} \]

So using this formula, you can calculate the amount the bullet's temperature will increase upon stopping, assuming that 50% of its initial kinetic energy is converted into thermal energy.

Extra: Understanding the concepts of energy conversion and specific heat capacity is crucial in physics.

- **Energy Conversion**: The Law of Conservation of Energy tells us that energy cannot be created or destroyed, only transformed from one form to another. In the case of a bullet being fired and stopped, the kinetic energy it possesses due to its movement is transformed into other forms, mainly heat, sound, and sometimes light if sparks are created upon impact.

- **Specific Heat Capacity**: This is a physical property of materials that measures how much heat energy is required to raise the temperature of a certain mass of a substance by a degree Celsius. Different materials have different specific heat capacities. For example, water has a high specific heat capacity, meaning it takes a lot of energy to raise its temperature; metals like lead have lower specific heat capacities, so they heat up more easily.

An interesting point for students to consider is that not all the kinetic energy would be converted to thermal energy. Some energy will also be used in deforming the bullet and the target, and in creating sound. However, for the purpose of this exercise, we assume a simplified scenario where the only significant energy conversion is into thermal energy.