Question

How much internal energy is generated when a \(20.0-\mathrm{g}\) lead bullet, traveling at \(7.00 \times 10^{2} \mathrm{m} / \mathrm{s},\) comes to a stop as it strikes a metal plate?

Short answer

The internal energy generated is 4900 J.

Step-by-step solution

Identify the Given Information

From the problem,1. The mass of the lead bullet, \( m = 20.0 \, \text{g} = 0.0200 \, \text{kg} \) (since we need to convert grams to kilograms for this calculation),2. The initial velocity of the bullet, \( v = 700 \, \text{m/s} \),3. The final velocity of the bullet, \( v_f = 0 \, \text{m/s} \), as it comes to a stop.

Apply the Conservation of Energy Principle

The internal energy generated when the bullet strikes the plate and comes to a stop is due to the conversion of kinetic energy to internal energy. The change in kinetic energy is equal to the internal energy generated. The kinetic energy of the bullet before striking the plate is given by \[ KE = \frac{1}{2}mv^2 \] where \( KE \) is the kinetic energy, \( m \) is the mass, and \( v \) is the velocity.

Substitute the Values and Calculate

Plug the known values into the kinetic energy equation: \[ KE = \frac{1}{2} \times 0.0200 \, \text{kg} \times (700 \, \text{m/s})^2 \]Calculate the kinetic energy:\[ KE = 0.0100 \times 490000 \]\[ KE = 4900 \, \text{J} \]Therefore, the internal energy generated is \( 4900 \, \text{J} \).

Key concepts

Kinetic Energy

The concept of kinetic energy centers around the motion of objects. Kinetic energy is the energy that an object possesses due to its motion. This type of energy can be calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)
• Here, \( m \) represents the mass of the object, and \( v \) is its velocity.

For instance, in the case of the lead bullet traveling at 700 m/s, the kinetic energy is a measure of how much energy it has while moving at that speed. As velocity plays a critical role, even slight increases in velocity result in significant increases in kinetic energy, because velocity is squared in the formula. In our example, all the kinetic energy of the bullet is converted to internal energy when it stops, making it crucial in understanding this problem.

Conservation of Energy

The conservation of energy principle states that energy can neither be created nor destroyed; it can only be converted from one form to another. This fundamental principle underpins many physical processes, including the one described in the exercise. When the bullet hits the metal plate and stops, its kinetic energy doesn't just vanish. Instead, it is transformed into internal energy, which may manifest as heat, sound, or deformation of the bullet or plate. This conversion illustrates the conservation of energy:

• Initial kinetic energy = Internal energy generated
• In our example, the initial kinetic energy calculated is 4900 J, so this amount of energy is conserved as internal energy.

Energy Conversion

Energy conversion is the transition of energy from one form to another. This concept is constantly at work in numerous situations, including mechanical systems, thermal processes, and chemical reactions.

• In the context of the exercise, energy conversion takes place when the bullet's kinetic energy changes into internal energy as it comes to a stop.
• This type of conversion can involve several physical changes, like heat production, sound, and deformation. Specifically, the kinetic energy from the moving bullet is converted entirely into internal energy upon impact.

Recognizing these conversions helps to understand how energy transitions work in various scenarios, demonstrating how efficiency and energy loss can occur in real-life applications.

Mechanical Energy

Mechanical energy combines both potential energy and kinetic energy within a system or object. It is expressed mathematically as the sum of potential and kinetic energies:

• \( ME = PE + KE \)
• In some cases, like the one with our bullet, potential energy is not a significant factor, meaning mechanical energy is solely represented by kinetic energy before conversion.

Once the bullet hits the plate and stops, the mechanical energy that was initially kinetic is converted into other forms of energy such as heat and sound. This highlights how mechanical energy allows objects to perform work and undergo energy transitions, playing a vital role in understanding energy dynamics in physical systems. This helps illustrate how mechanical systems store and transfer energy efficiently, central to various engineering and physics applications.