Problem 17 - University Physics with Modern Physics

Question

The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A 30 -caliber bullet has mass 0.00720 \(\mathrm{kg}\) and a speed of 601 \(\mathrm{m} / \mathrm{s}\) relative to the muzzle when fired from a rifle that has mass 2.80 \(\mathrm{kg}\) . The loosely held rifle recoils at a speed of 1.85 \(\mathrm{m} / \mathrm{s}\) relative to the earth. Find the momentum of the propellant gases in a coordinate system attached to the earth as they leave the muzzle of the rifle.

Step-by-Step Solution

Answer

Momentum of gases is 0.8528 kg⋅m/s.

Step 1: Define Conservation of Momentum

The conservation of momentum states that the total momentum before firing is equal to the total momentum after firing. Since initially everything is at rest, the total initial momentum is zero. Therefore, the total final momentum of the bullet, rifle, and gases must also be zero.

Step 2: Calculate Momentum of the Bullet

The momentum of the bullet can be calculated using the formula: \( p_{ ext{bullet}} = m_{ ext{bullet}} \cdot v_{ ext{bullet}} \). Substituting the values: \( p_{ ext{bullet}} = 0.00720 \cdot 601 = 4.3272 \, \text{kg}\cdot\text{m/s} \).

Step 3: Calculate Momentum of the Rifle

The momentum of the rifle is given by: \( p_{ ext{rifle}} = m_{ ext{rifle}} \cdot v_{ ext{rifle}} \). Substituting the values: \( p_{ ext{rifle}} = 2.80 \cdot (-1.85) = -5.18 \, \text{kg}\cdot\text{m/s} \). Note the negative sign indicates the rifle's momentum opposes the bullet's motion.

Step 4: Apply Conservation of Momentum to Solve for Gas Momentum

According to conservation of momentum, the sum of the momenta of the bullet, rifle, and gases should be zero. Therefore, the momentum of the gases is given by: \( p_{ ext{gases}} = -(p_{ ext{bullet}} + p_{ ext{rifle}}) \). Substituting the momenta calculated earlier: \( p_{ ext{gases}} = -(4.3272 - 5.18) = 0.8528 \, \text{kg}\cdot\text{m/s} \).

Key Concepts

Momentum

Momentum is a fundamental concept in physics that is defined as the product of an object's mass and its velocity. This concept is crucial because it allows us to understand how objects interact in motion. In our exercise, the momentum of various components—bullet, rifle, and gases—was analyzed. Each component has its momentum described by the formula:

• Momentum (\( p \)) = Mass (\( m \)) \( \times \) Velocity (\( v \))

Different objects have different masses and velocities, thus contributing differently to the system's total momentum. It's important to remember:

• Momentum is a vector quantity, meaning it has both magnitude and direction.
• In a closed system, momentum is conserved, meaning the total momentum before an event equals the total momentum after.

In our example, since initially the system (rifle, bullet, and gases) was at rest, the initial momentum was zero. Hence, the final momentum also equals zero.

Recoil

Recoil is the backwards movement experienced by an object when it ejects another object forward. This phenomenon is commonly observed in firearms. Here, when the bullet is fired forward, the rifle experiences a backward recoil.
The concept of recoil can be explained through Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. When a bullet is fired:

• The bullet moves forward with a certain momentum.
• In reaction, the rifle moves backward with an equal and opposite momentum.

The backward velocity of the rifle is what we call recoil velocity. In this exercise, the rifle had a recoil speed of 1.85 m/s, highlighting how recoil is a direct result of the momentum conservation principle.

Propellant Gases

Propellant gases play a significant role in the firing mechanism of firearms. When the bullet is fired, these gases are expelled along with it. Their momentum contributes significantly to the backward recoil and needs to be accounted for in momentum calculations.
In our solution, the momentum of the propellant gases was calculated last, highlighting their role in ensuring the conservation of momentum. Despite not being visible, they have:

• A notable mass, even though it is always less than the bullet.
• A high velocity, which is why their momentum is not negligible.

The conservation of momentum allows us to calculate the momentum of these gases by accounting for the momentum of the rifle and bullet. This ensures that all components contribute accurately to the conservation equations.

Coordinate System

In analyzing physical problems, particularly in mechanics, it's critical to establish a reference frame or coordinate system. This allows observers to consistently analyze and interpret motion relative to a fixed point.
For this problem, the coordinate system is key in defining velocities and directions:

• Since we consider a coordinate system attached to the earth, all velocities and momenta are measured relative to the earth.
• The negative momentum of the rifle indicates that its recoil is in the opposite direction to the bullet's motion.

Using a consistent coordinate system allows physicists to apply laws of motion uniformly and derive accurate results, such as determining the direction and magnitude of the recoil or ensuring momentum is conserved throughout various events.