Problem 13 - College Physics

Question

\(\cdot\) A bullet is fired into a large stationary absorber and comes to rest. Temperature measurements of the absorber show that the bullet lost 1960 \(\mathrm{J}\) of kinetic energy, and high-speed photos of the bullet show that it was moving at 965 \(\mathrm{m} / \mathrm{s}\) just as it struck the absorber. What is the mass of the bullet?

Step-by-Step Solution

Answer

The bullet's mass is approximately 0.00421 kg.

Step 1: Understand the Relationship

When the bullet comes to rest in the absorber, all of its kinetic energy is converted to thermal energy within the absorber. Hence, we can use the equation for kinetic energy loss to find the mass of the bullet.

Step 2: Recall the Kinetic Energy Formula

The kinetic energy (KE) of any object in motion is given by the formula: \[ KE = \frac{1}{2} mv^2 \]where \( m \) is the mass of the object and \( v \) is its velocity.

Step 3: Set Up the Equation

We know that the bullet loses 1960 J of kinetic energy when it comes to a stop. Therefore, we set the kinetic energy equation equal to 1960 J:\[ \frac{1}{2} m (965\, \mathrm{m/s})^2 = 1960 \]Our task is to solve for the mass \( m \).

Step 4: Solve for Mass

Rearrange the equation to solve for mass \( m \):\[ m = \frac{2 \times 1960}{965^2} \]Calculate the value by substituting in the numbers:\[ m = \frac{3920}{931225} \approx 0.00421 \text{ kg} \]Hence, the mass of the bullet is approximately 0.00421 kg.

Key Concepts

Thermal Energy

Thermal energy refers to the internal energy within a system due to the random motions of its molecules. When the bullet mentioned in the problem comes to rest in the absorber, its kinetic energy is converted into thermal energy. This conversion process is crucial in understanding energy transformations.
Thermal energy is a byproduct when kinetic energy dissipates, especially in scenarios where objects come to a stop. The increased molecular activity in the absorber results in a temperature rise, so measuring temperature changes can give clues about the energy initially present in the system.
In this example, the bullet's kinetic energy of 1960 Joules becomes thermal energy within the absorber, raising its temperature.

Mass Calculation

Determining the mass of an object involves understanding the relationship between energy and motion. In our example, the bullet's loss of kinetic energy helped determine its mass using known variables like velocity and energy loss.
Kinetic energy has a straightforward formula:

• \[ KE = \frac{1}{2} mv^2 \]

Where

• \( KE \) is kinetic energy,
• \( m \) is mass, and
• \( v \) is velocity.

Since kinetic energy and velocity are known, rearranging the formula allows solving for the mass \( m \). Plug the numbers into this equation:

• \[ m = \frac{2 \times KE}{v^2} \]
• \[ m = \frac{2 \times 1960}{965^2} \approx 0.00421 \text{ kg} \]

This calculation shows that the bullet's mass is about 0.00421 kg.

Motion Equations

Motion equations describe how objects move in space over time, and they're pivotal in problems involving kinetic energy. The main equation here is the kinetic energy equation:

• \[ KE = \frac{1}{2} mv^2 \]

This equation tells us how an object's mass and velocity contribute to its kinetic energy. In this exercise, it's crucial to converting kinetic energy into another form, highlighting energy conservation principles.
To solve motion problems, understand these components:

• Mass \( m \) and how it correlates to inertia, determining how an object responds to forces.
• Velocity \( v \), which reflects the object's speed in a specific direction.
• Using these, calculate kinetic energy to evaluate energy transformations and related mass, as seen with our bullet's mass determination.

These principles form the core of many physics problems, granting insights into the dynamics of objects in motion.