Problem 30 - Classical Mechanics

Question

Conservation laws, such as conservation of momentum, often give a surprising amount of information about the possible outcome of an experiment. Here is perhaps the simplest example: Two objects of masses \(m_{1}\) and \(m_{2}\) are subject to no external forces. Object 1 is traveling with velocity \(\mathbf{v}\) when it collides with the stationary object \(2 .\) The two objects stick together and move off with common velocity \(\mathbf{v}^{\prime}\). Use conservation of momentum to find \(\mathbf{v}^{\prime}\) in terms of \(\mathbf{v}, m_{1},\) and \(m_{2}\)

Step-by-Step Solution

Answer

\(\mathbf{v}' = \frac{m_1 \mathbf{v}}{m_1 + m_2}\)

Step 1: Understand the Conservation of Momentum Principle

The principle of conservation of momentum states that in the absence of external forces, the momentum of a closed system remains constant. Since no external forces act on the system of these two colliding objects, their total momentum before the collision must equal their total momentum after the collision.

Step 2: Determine Initial Momentum

Calculate the total initial momentum of the system. Initially, object 1 with mass \(m_1\) is moving with velocity \(\mathbf{v}\) while object 2 is stationary. Thus, the initial momentum \(\mathbf{p}_{\text{initial}}\) is given by:\[\mathbf{p}_{\text{initial}} = m_1 \mathbf{v}\]

Step 3: Express Final Momentum After Collision

After the collision, both objects stick together and move as a single object with a common velocity \(\mathbf{v}'\). Thus, the final momentum \(\mathbf{p}_{\text{final}}\) can be expressed as:\[\mathbf{p}_{\text{final}} = (m_1 + m_2) \mathbf{v}'\]

Step 4: Set Initial Momentum Equal to Final Momentum

Apply the conservation of momentum by setting the initial momentum equal to the final momentum:\[m_1 \mathbf{v} = (m_1 + m_2) \mathbf{v}'\]

Step 5: Solve for \(\mathbf{v}'\)

Rearrange the equation to solve for \(\mathbf{v}'\):\[\mathbf{v}' = \frac{m_1 \mathbf{v}}{m_1 + m_2}\]

Key Concepts

Understanding Collisions

When two objects, such as two cars or two balls, come into contact and exert forces on each other, we describe this event as a collision. In physics, collisions can be categorized into different types based on how the total kinetic energy of the system changes during the event. However, regardless of the type of collision, the principle of momentum conservation applies as long as no external forces are acting on the system.

Collisions are broadly classified into:

• **Elastic Collisions**: Here, both momentum and kinetic energy are conserved. The objects bounce off each other without losing any kinetic energy during the collision.
• **Inelastic Collisions**: In these collisions, momentum is conserved, but kinetic energy is not. A common example is when colliding cars crumple together, converting some of the kinetic energy into sound, heat, or deformation energy.
• **Perfectly Inelastic Collisions**: This is an extreme case of an inelastic collision where the two objects stick together after impact, moving as a single object after the collision, sharing a common velocity.

In the exercise given, the objects experience a perfectly inelastic collision, where both objects stick together after colliding.

The Role of Momentum

Momentum is a crucial quantity in physics defined as the product of an object's mass and its velocity. It is depicted by the symbol \(\mathbf{p}\), where \(\mathbf{p} = m \mathbf{v}\).

The principle of conservation of momentum states that in a closed system (a system not influenced by external forces), the total momentum before an event like a collision remains the same as the total momentum after. This core concept of momentum conservation helps us understand and predict the outcome of collision events.

In our exercise, we're asked to use the conservation of momentum to determine the final velocity after a collision. The core idea here is to equate the momentum before the collision to the momentum after the collision:

• **Initial Momentum**: The moving object (mass \(m_1\)) has initial momentum \(m_1 \mathbf{v}\) since it's moving with velocity \(\mathbf{v}\), while the stationary object has zero momentum.
• **Final Momentum**: After the collision, both masses stick and share a common velocity \(\mathbf{v}'\). Thus, their total mass \((m_1 + m_2)\) moves with the velocity \(\mathbf{v}'\).
• **Momentum Conservation Equation**: Therefore, we write \(m_1 \mathbf{v} = (m_1 + m_2) \mathbf{v}'\).

This allows us to find the final velocity \(\mathbf{v}'\) after rearranging the equation.

Initial and Final Velocity

Velocity describes how fast an object is moving and in which direction. In collision problems, such as the one in this exercise, understanding initial and final velocities is key to applying the conservation of momentum principle.

Let's break down the significance of initial and final velocities:

• **Initial Velocity (\(\mathbf{v}\))**: This is the velocity that object 1 has before the collision. It's crucial because it gives us the initial momentum of the system along with the mass of object 1.
• **Final Velocity (\(\mathbf{v}'\))**: After the collision, both objects move together with this velocity. Although energy might not be conserved, as the objects stick together, the momentum is still conserved, allowing us to calculate this final velocity using the conservation equation.

From the solution derived through conservation of momentum, we find that the final velocity \(\mathbf{v}'\) is given by:

\[\mathbf{v}' = \frac{m_1 \mathbf{v}}{m_1 + m_2}\]

Therefore, this formula shows how the combined mass after collision influences the final shared velocity of the system.