Question

In the laboratory, a particle of mass \(3.16 \mathrm{~kg}\) moving at \(15.6 \mathrm{~m} / \mathrm{s}\) to the left collides head-on with a particle of mass \(2.84 \mathrm{~kg}\) moving at \(12.2 \mathrm{~m} / \mathrm{s}\) to the right. Find the velocity of the center of mass of the system of two particles after the collision.

Short answer

The velocity of the center of mass after the collision is the division between the total initial momentum and total mass of the system. In all answers make sure the sign for the direction of the velocity is maintained.

Step-by-step solution

Determine the Direction

It's important to first identify the direction of the movement. In this case, let's assume movement to the right is positive and to the left is negative. Therefore, the particle of mass \(3.16 \mathrm{~kg}\) moving to the left at \(15.6 \mathrm{~m/s}\) has a velocity of \(-15.6 \mathrm{~m/s}\), and the particle of mass \(2.84 \mathrm{~kg}\) moving to the right at \(12.2 \mathrm{~m/s}\) has a velocity of \(+12.2 \mathrm{~m/s}\).

Calculate Initial Momentum

Momentum (p) can be calculated as the product of mass (m) and velocity (v), i.e., \(p = m \cdot v\). Let's find the initial momentum for both particles. For the first particle it's \(3.16 \mathrm{~kg} \cdot -15.6 \mathrm{~m/s}\) and for the second one, it's \(2.84 \mathrm{~kg} \cdot 12.2 \mathrm{~m/s}\). Now, add these momenta to get the total initial momentum of the system.

Calculate the Mass of the System

The total mass of the system is the sum of the individual masses i.e., \(3.16 \mathrm{~kg} + 2.84 \mathrm{~kg}\).

Calculate the Velocity of the Center of Mass after the Collision

According to conservation of momentum, the total momentum of the system remains unchanged before and after the collision. Therefore, the velocity of the center of mass after the collision is the total initial momentum divided by total mass of the system.

Key concepts

Conservation of Momentum

The principle of conservation of momentum is crucial in understanding collision scenarios. It states that in a closed system with no external forces, the total momentum remains constant. This is because momentum is a conserved quantity, much like energy, during interactions between objects.

When we analyze a collision between two particles, we can use the conservation of momentum to predict the outcome. In its simplest formula, \( \text{Total Initial Momentum} = \text{Total Final Momentum} \). Thus, if we know the momentum of each object before the collision, we can calculate the overall momentum after the collision, assuming the system is closed and there are no external forces acting on it. This is a fundamental concept not only in physics but also in various engineering applications.

Momentum Calculation

Momentum is defined as the product of an object's mass and its velocity, represented by \( p = m \cdot v \). It’s a vector quantity, meaning it has both magnitude and direction. For our calculation purposes, we can assign a positive or negative sign to indicate direction, based on an agreed frame of reference.

To calculate the momentum of each particle in our exercise, we multiply the mass of the particle by its velocity, taking into account the direction. For instance, if a particle is moving to the left (which we've designated as negative), its velocity will be a negative value. It's the combination of these individual momenta that gives us the total momentum of the system, a crucial step before we can apply the conservation of momentum principle after a collision occurs.

Collision Physics

Collision physics deals with analyzing the behaviors of objects when they collide. There are two main types of collisions: elastic and inelastic. In an elastic collision, both momentum and kinetic energy are conserved, while in an inelastic collision, only momentum is conserved.

The problem we're dealing with does not specify the type of collision. However, for the purpose of calculating the velocity of the center of mass, we only need to focus on the conservation of momentum. During a collision, even if the individual velocities of the particles change, the physics of collisions tells us the system's center of mass will move with a velocity that reflects the conservation of the total system's momentum.

Mass and Velocity

Mass and velocity are both integral components of momentum. The mass of an object is a measure of its quantity of matter and is constant irrespective of its location and speed. Velocity, however, is the speed of the object in a given direction and can be variable.

In the context of the exercise, each particle has a given mass and velocity. Mass being a scalar and velocity being a vector give rise to momentum's vector nature. When we talk about the system's center of mass velocity, we're referring to a weighted average velocity where each mass contributes to the overall motion based on how much mass and how fast each particle is moving. By accurately understanding the components of each, the combined knowledge serves as the foundation for solving problems related to momentum, such as finding the center of mass velocity after a collision.