Problem 9 - Applied Physics

Question

One cart of mass \(12.0 \mathrm{~kg}\) is moving \(6.00 \mathrm{~m} / \mathrm{s}\) to the right on a frictionless track and collides with a cart of mass \(4.00 \mathrm{~kg}\) moving in the opposite direction \(3.00 \mathrm{~m} / \mathrm{s}\). Find the final velocity of the carts that become stuck together after the collision.

Step-by-Step Solution

Answer

The final velocity of the carts stuck together is 3.75 m/s to the right.

Step 1: Understand the Conservation of Momentum

In collisions, the total linear momentum before the collision is equal to the total linear momentum after the collision, assuming no external forces. This can be expressed as \( m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f \), where \(v_f\) is the final velocity after collision.

Step 2: Identify Initial Velocity and Mass Information

Here, the mass of the first cart \(m_1\) is 12.0 kg and it has an initial velocity \(v_{1i}\) of 6.00 m/s to the right. The second cart, \(m_2\), has a mass of 4.00 kg and an initial velocity \(v_{2i}\) of -3.00 m/s (opposite direction).

Step 3: Substitute Known Values into the Momentum Equation

Substitute the known values into the equation of conservation of momentum: \( 12.0 \, \text{kg} \times 6.00 \, \text{m/s} + 4.00 \, \text{kg} \times (-3.00 \, \text{m/s}) = (12.0 \, \text{kg} + 4.00 \, \text{kg}) \times v_f \).

Step 4: Compute the Momentum Values

Calculate the momentum of each cart and the total momentum before collision: \( 12.0 \, \text{kg} \times 6.00 \, \text{m/s} = 72.0 \, \text{kg} \cdot \text{m/s} \) and \( 4.00 \, \text{kg} \times -3.00 \, \text{m/s} = -12.0 \, \text{kg} \cdot \text{m/s} \). Add these values to find the total: \( 72.0 - 12.0 = 60.0 \, \text{kg} \cdot \text{m/s} \).

Step 5: Solve for the Final Velocity \(v_f\)

Now solve for the final velocity: \( 60.0 \, \text{kg} \cdot \text{m/s} = 16.0 \, \text{kg} \times v_f \). Divide both sides by 16.0 kg to find \( v_f = \frac{60.0 \, \text{kg} \cdot \text{m/s}}{16.0 \, \text{kg}} = 3.75 \, \text{m/s} \).

Key Concepts

Elastic and Inelastic Collisions

Collisions are interactions where two or more bodies exert forces on each other for a brief time. A crucial concept in these scenarios is the distinction between elastic and inelastic collisions.
In an **elastic collision**, both momentum and kinetic energy are conserved. This means no energy is lost to sound, heat, or deformation. The bodies involved "bounce" off each other, each retaining most of their motion and energy.
In contrast, an **inelastic collision** involves the bodies sticking together, resulting in the loss of kinetic energy. Energy might be transformed into other forms such as heat or sound, but the total momentum is conserved. Our original exercise is an example of a perfectly inelastic collision since the carts stick together post-collision.
Understanding these collision types is essential as they dictate how you solve problems in physics. For instance:

• For elastic collisions: Use both conservation of momentum and kinetic energy equations.
• For inelastic collisions: Focus solely on conservation of momentum.

By identifying the nature of the collision, you can easily determine the appropriate methods and formulas to apply.

Linear Motion

Linear motion refers to movement along a straight path. It's one of the simplest forms of motion, characterized by uniform or constantly changing velocity. In our exercise, both carts move along a straight, frictionless track, a perfect scenario to explore linear motion principles.
The key aspects of linear motion include:

• **Velocity**: A measure of how fast an object is moving in a particular direction. It can be positive when moving to the right and negative when moving to the left, as indicated by the opposite directions of the carts.
• **Acceleration**: The rate at which an object's velocity changes with time. In our problem, since the collision happens very quickly, the acceleration is not explicitly considered.

Understanding linear motion is foundational for analyzing vehicle dynamics, projectiles, and collision scenarios, as consistent direction and speed significantly simplify calculations.

Problem Solving in Physics

Problem-solving in physics often means applying theoretical knowledge to practical situations. Properly understanding the scenario and precisely applying conservation laws are key to success.
Here's a step-by-step approach for tackling physics problems, using our original exercise as a guide:

• **Identify the Physical Principles**: Recognize which laws of physics apply to the problem. In our example, the conservation of momentum is the central principle.
• **Gather Known Information**: Make a list of given values such as masses, velocities, and any constraints. This involves identifying the direction and magnitude of velocities accurately.
• **Use Appropriate Equations**: Substitute the known values into the relevant equations, such as those from the conservation laws.
• **Perform Calculations**: Solve for the unknowns. This might involve algebraic manipulation to isolate variables.
• **Check the Results**: Consider if the results make physical sense and check units. The result should be consistent with the conditions of the problem.

By following these steps, you ensure clarity and accuracy in your solutions, enhancing your understanding and problem-solving skills in physics.