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Chapter 7: Problem 49

Relate When a collision occurs, how are the initial and final momentums of the system related?

Short Answer

Expert verified
Initial and final momentums are equal due to conservation of momentum.

Step by step solution

01

Define Momentum

Momentum is defined as the product of an object's mass and velocity. Mathematically, it is expressed as \( p = mv \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.
02

Understand the Law of Conservation of Momentum

The law of conservation of momentum states that in a closed system with no external forces, the total momentum before an interaction (collision) is equal to the total momentum after the interaction. This can be expressed as \( p_{initial} = p_{final} \).
03

Analyze Initial and Final Momentums

When a collision occurs in a closed system, the sum of the individual momentums of all objects before the collision (initial momentum) will equal the sum after the collision (final momentum). This means \( m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} \) for two colliding objects.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum
Momentum is a fundamental concept in physics that combines both the mass and velocity of an object. It determines how much motion an object has, which is given by the equation: \( p = mv \). Here, \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. Momentum is important because it helps to quantify the amount of motion a body possesses. When discussing momentum, sometimes people refer to it as a measure of an object's inertia in motion.
Collision
Collisions involve two or more objects hitting each other and interacting over a short time period. They can be elastic, where objects bounce off with no loss of kinetic energy, or inelastic, where some kinetic energy is lost and objects may stick together. During any collision, forces exerted are equal and opposite, but what remains crucial is the momentum transfer between the bodies involved. Understanding collisions helps in predicting post-collision behaviors of objects.
Closed System
A closed system refers to an idealized physics system where no external forces are acting. This means that external influences like friction or air resistance are absent. Within such a system, interactions like collisions can occur without losing total momentum to the surroundings. The concept of a closed system is essential for applying the conservation of momentum, as it ensures that the total momentum remains constant pre- and post-interaction.
Initial Momentum
Initial momentum describes the momentum of a system before any interaction or collision has taken place. It involves summing up the products of mass and velocity for all objects involved. In formula representation, it's usually written as \( p_{initial} = m_1v_{1i} + m_2v_{2i} + \ldots + m_nv_{ni} \), where each term specifies the momentum of an individual object. Knowing the initial momentum is crucial for predicting outcomes after interactions based on the conservation principle.
Final Momentum
Final momentum refers to the momentum present in the system after an interaction or collision has occurred. In a closed system, the final momentum will equal the initial momentum due to momentum conservation. The equation used to describe this in a simple two-object system is \( p_{final} = m_1v_{1f} + m_2v_{2f} \). Identifying final momentum helps in understanding the changes and dynamics within the system post-collision.

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Most popular questions from this chapter

A cart of mass \(m=0.12 \mathrm{~kg}\) moves with a speed \(v=0.45 \mathrm{~m} / \mathrm{s}\) on a frictionless air track and collides with an identical cart that is stationary. The carts stick together after the collision. What are (a) the initial kinetic energy and (b) the final kinetic energy of the system?

A \(26-\mathrm{kg}\) dog is running northward at \(2.7 \mathrm{~m} / \mathrm{s}\), while a \(5.3-\mathrm{kg}\) cat is running eastward at \(3.0 \mathrm{~m} / \mathrm{s}\). Find the magnitude and direction of the total momentum for this system.

What two physical quantities are conserved in an elastic collision?

Two canoes are touching and at rest on a lake. The occupants push away from each other in opposite directions, giving canoe 1 a speed of \(0.58 \mathrm{~m} / \mathrm{s}\) and canoe 2 a speed of \(0.42 \mathrm{~m} / \mathrm{s}\). If the mass of canoe 1 is \(320 \mathrm{~kg}\), what is the mass of canoe 2 ?

Two ice skaters stand at rest in the center of an ice rink. When they push off against one another, the \(45-\mathrm{kg}\) skater acquires a speed of \(0.62 \mathrm{~m} / \mathrm{s}\). If the speed of the other skater is \(0.89 \mathrm{~m} / \mathrm{s}\), what is that skater's mass?
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