Problem 6 - College Physics

Question

A 15.0 - \(\mathrm{g}\) rubber bullet hits a wall with a speed of \(150 \mathrm{~m} / \mathrm{s}\). If the bullet bounces straight back with a speed of \(120 \mathrm{~m} / \mathrm{s}\), what is the change in momentum of the bullet?

Step-by-Step Solution

Answer

The change in momentum is -4.05 kg·m/s.

Step 1: Understand the Concept of Momentum

Momentum is defined as the product of an object's mass and its velocity. The formula for momentum (\( p \)) is given by \( p = m \cdot v \), where \( m \) is the mass and \( v \) is the velocity.

Step 2: Calculate Initial Momentum

The initial momentum of the bullet can be calculated using its mass and initial velocity. \[ p_{initial} = m \cdot v_{initial} = 0.015\, \text{kg} \cdot 150\, \text{m/s} = 2.25\, \text{kg} \cdot \text{m/s} \] We need to convert the mass from grams to kilograms by dividing by 1000.

Step 3: Calculate Final Momentum

The final momentum of the bullet can be calculated using its mass and final velocity. Note the direction change upon bouncing, making the velocity negative. \[ p_{final} = m \cdot v_{final} = 0.015\, \text{kg} \cdot (-120\, \text{m/s}) = -1.8\, \text{kg} \cdot \text{m/s} \]

Step 4: Determine Change in Momentum

To find the change in momentum, subtract the initial momentum from the final momentum: \[ \Delta p = p_{final} - p_{initial} = (-1.8\, \text{kg}\cdot\text{m/s}) - 2.25\, \text{kg}\cdot\text{m/s} = -4.05\, \text{kg}\cdot\text{m/s} \] The negative sign indicates a reversal in direction.

Key Concepts

Momentum Calculation

Momentum is a key concept in physics that helps us understand the movement of objects. It considers both the mass and the velocity of an object. To find an object's momentum, use the formula \( p = m \cdot v \). Here, \( m \) stands for mass and \( v \) represents velocity. The units of momentum are kilograms meters per second (\( \text{kg} \cdot \text{m/s} \)).For example, in the original exercise, the bullet's mass is given as 15 grams. First, we need to convert this into kilograms for the calculation. Remember to divide by 1000 to change grams to kilograms: \( \text{mass} = 15 \text{g}/1000 = 0.015 \text{kg} \). By using the initial velocity of 150 meters per second, you multiply the mass by the velocity: \( p_{\text{initial}} = 0.015 \text{kg} \cdot 150 \text{m/s} \), which gives a momentum of 2.25 \( \text{kg} \cdot \text{m/s} \). This is how you calculate the initial momentum of the bullet.

Physics Problem

Understanding physics problems often involves breaking down all given information and methodically working through calculations. In this case, our task was to determine the change in momentum when the bullet hits the wall and bounces back. It's essential to analyze the direction of movement during the collision. When the bullet initially moves towards the wall, the velocity is positive. When it bounces back, the velocity becomes negative. Physics problems like this one often contain steps such as:

• Identifying known values (e.g., initial and final velocities)
• Understanding the physical principles involved (conservation of momentum or energy)
• Performing calculations to find unknowns

These steps aid in systematically solving problems by converting word problems into mathematical expressions.

Collision Dynamics

Collision dynamics involves understanding the interaction between objects, like when a bullet rebounds off a wall. During a collision, the conservation of momentum principle can be applied. However, in non-elastic collisions, like the one in the exercise, some energy is lost. The conservation of momentum helps us understand how momentum is transferred or changed through interactions. When dealing with collision problems, we consider:

• The initial and final momenta
• The direction and magnitude of velocities involved
• Whether the collision is elastic or inelastic

In the exercise's context, the bullet’s velocity changes from positive (towards the wall) to negative (away from the wall). The change in direction is reflected in the final velocity's negative sign. Overall, understanding collision dynamics aids in quickly evaluating how an object's momentum changes when impacted by external forces.