Problem 25 - Chemistry The Central Science In Si Units

Question

(a) Calculate the kinetic energy, in joules, of a 15 -g bullet moving at \(120 \mathrm{~m} / \mathrm{s}\). (b) Convert this energy to calories. (c) When the bullet is stopped by a bulletproof vest, which form of energy does the kinetic energy of the bullet convert to?

Step-by-Step Solution

Answer

(a) 108 J; (b) 25.8 cal; (c) Thermal energy.

Step 1: Convert mass to kilograms

The mass of the bullet is given in grams. To use the kinetic energy formula, we need to convert it to kilograms. Since there are 1000 grams in a kilogram, divide the mass in grams by 1000. Mass in kilograms: \[ m = \frac{15 \text{ g}}{1000} = 0.015 \text{ kg} \]

Step 2: Calculate Kinetic Energy

Use the formula for kinetic energy, which is given by \[ KE = \frac{1}{2} m v^2 \]where \( m \) is the mass in kilograms and \( v \) is the velocity in meters per second. Substitute the given values:\[ KE = \frac{1}{2} \times 0.015 \times (120)^2 = 108 \text{ J} \]

Step 3: Convert joules to calories

To convert energy from joules to calories, use the conversion factor where 1 calorie = 4.184 joules.\[ \text{Energy in calories} = \frac{108 \text{ J}}{4.184 \text{ J/cal}} \approx 25.8 \text{ cal} \]

Step 4: Analyze energy conversion

When the bullet is stopped by a bulletproof vest, the kinetic energy is primarily converted into thermal energy (heat) due to energy absorption, deformation of the vest, and sound. The material in the vest dissipates energy as heat, helping to stop the bullet.

Key Concepts

Energy Conversion

Energy conversion is a fundamental concept in physics that describes the process of transforming energy from one form to another. In the context of the given exercise, the kinetic energy of the bullet is converted into another form when the bullet is halted by a bulletproof vest. When the bullet makes contact with the vest, its kinetic energy doesn’t just vanish; instead, it's transformed into:

• Thermal Energy: Due to the friction and deformation of materials, heat is generated during the impact, raising the temperature of both the bullet and the vest.
• Sound Energy: The collision creates noise as the materials clash, converting some of the kinetic energy into sound waves.
• Deformation Energy: The vest absorbs some energy by undergoing a temporary shape change, helping to redistribute the energy and slow the bullet down.

This transformation from kinetic to other forms of energy illustrates the principle of energy conservation, which states energy cannot be created or destroyed, only changed from one type to another.

Mass and Velocity

Mass and velocity are key components in calculating kinetic energy. Kinetic energy (\( KE \)) depends on these two factors, as seen in the formula \( KE = \frac{1}{2} m v^2 \).

• Mass (\( m \)): Represents the amount of matter present in the object, measured in kilograms (kg). In our exercise, the bullet's mass is 0.015 kg.
• Velocity (\( v \)): The speed of the object in a particular direction, measured in meters per second (m/s). Here, the bullet moves at a speed of 120 m/s.

Because kinetic energy is proportional to the square of velocity, even a small increase in velocity results in a much larger increase in kinetic energy. This is evident in high-speed projectiles like bullets, where rapid movement translates into substantial kinetic energy. Therefore, both mass and velocity are crucial for understanding how much energy an object has in motion.

Thermal Energy

Thermal energy refers to the internal energy in a system due to its temperature. When the high-speed bullet impacts the bulletproof vest, its kinetic energy is mostly converted into thermal energy. Several factors contribute to this conversion:

• Friction: The contact between the bullet and the vest generates significant friction, which produces heat.
• Material Deformation: The vest absorbs energy as it changes shape, with some of this absorbed energy becoming heat.
• Energy Dissipation: As the vest halts the bullet, it spreads and transforms the kinetic energy, increasing the temperature at the point of impact and around it.

This process underscores how energy transformations can lead to increases in thermal energy, often perceived as heat. Understanding these transitions is essential in materials science and safety design, as it helps develop materials that can withstand and dissipate energy effectively, like the bulletproof vest.