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Conservation of Momentum in One Dimension: Fundamental Physics Concept
Explore the core principles of momentum conservation in 1D physics. Learn how to analyze collisions, apply vector addition, and solve real-world problems with confidence.

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Now Playing:Conservation of momentum in one dimension – Example 0a
Intros
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  1. Introduction to conservation of momentum in one dimension:
  2. Introduction to conservation of momentum in one dimension:
    Meaning of conservation of momentum
  3. Introduction to conservation of momentum in one dimension:
    Deriving conservation of momentum from Newton's third law
Examples
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  1. ∑p⃗i=∑p⃗f:\bold{\sum\vec{p}_i = \sum\vec{p}_f}: Objects that bounce apart after collision
    1. A 0.200 kg billiard ball travelling 7.00 m/s to the right collides with a stationary 0.0450 kg golf ball. The velocity of the billiard ball after collision is 4.43 m/s [right]. What is the velocity of the golf ball after collision?

    2. A 2.20 kg ball rolling 1.50 m/s [E] collides head-on with a ball rolling 2.40 m/s [W]. The 2.20 kg ball rolls away at 1.00 m/s [W] after the collision, and the second ball rolls 1.80 m/s [E]. What is the mass of the second ball?

    3. 4500 kg train car A travels at 6 m/s [E]. It collides with 3750 kg train car B which is travelling 3.2 m/s [E] on the same track. After the collision, train car A travels at 3.5 m/s [E]. Find the final velocity of train car B.

    4. A rifle points [E] at a 3.40 kg wood block. It fires a 22.0 g bullet travelling at 460.0 m/s, which passes through the block and continues at 299 m/s. Find the velocity of the wood block when the bullet leaves.

Momentum and motion
Jump to:NotesConceptFAQsPrerequisites
Notes

In this lesson, we will learn:

  • Meaning of conservation of momentum
  • Deriving conservation of momentum from Newton's third law
  • Meaning of closed/open systems and identifying conservation of momentum problems
  • Problem solving with conservation of momentum in 1 dimension

Notes:

  • Momentum is a conserved quantity: the total momentum of a set of objects before an event (like a collision or explosion) is equal to total momentum after.
  • Conservation of momentum (∑p⃗i=∑p⃗f\sum\vec{p}_i = \sum\vec{p}_f)is a result of Newton's 3rd
    • Law (F⃗A=−F⃗B\vec{F}_A = -\vec{F}_B).
  • When talking about conservation of momentum, system is a name given to an object or set of objects that are being examined.
    • A closed system is an object or set of objects that are not affected by external forces (i.e. their motion is not changing due to forces from outside the system). Momentum is conserved in a closed system.
      • A system can have external forces acting on it while still being closed, as long as those forces are balanced. A ball rolling across a frictionless table would be considered a closed system even though gravity and normal force act on the ball. Gravity and normal force are balanced, so they are not causing the ball to accelerate, and momentum is conserved.
    • An open system is an object or set of objects that are affected by external forces (i.e. their motion does change due to forces from outside the system). Momentum is not conserved in an open system.
      • If the ball from the earlier example rolls off the edge of the frictionless table and is accelerated by gravity towards the earth, then it is no longer considered a closed system. The system is open and gaining momentum from the acceleration due to gravity.

Momentum

p⃗=mv⃗:\vec{p} = m \vec{v}: momentum, in kilogram meters per second (kg∙m/s)

m:m: mass, in kilograms (kg)

v⃗:\vec{v}: velocity, in meters per second (m/s)


Conservation of Momentum

∑p⃗i=∑p⃗f\sum\vec{p}_i = \sum\vec{p}_f

p⃗i:\vec{p}_i: initial momentum, in kilogram meters per second (kg·m/s)

p⃗f:\vec{p}_f: final momentum, in kilogram meters per second (kg·m/s)


Impulse

J⃗=F⃗Δt=Δp⃗=m(v⃗f−v⃗i\vec{J} = \vec{F} \Delta t = \Delta \vec{p} = m( \vec{v}_f - \vec{v}_i)

J⃗:\vec{J}: impulse, in newton seconds (N∙s)

F⃗:\vec{F}: net force acting on an object, in newtons (N)

Δt:\Delta t: the length of time for which the force acts, in seconds (s)

Δp⃗:\Delta \vec{p}: change in momentum of an object, in kilgram meters per second (kg∙m/s)

Concept

Introduction to Conservation of Momentum in One Dimension

Conservation of momentum in one dimension is a fundamental principle in physics that plays a crucial role in understanding collisions and other interactions between objects. The introduction video provides a clear and engaging overview of this concept, making it easier for students to grasp its significance. Momentum, defined as the product of an object's mass and velocity, is a conserved quantity in closed systems. This means that the total momentum before and after a collision remains constant, assuming no external forces are acting on the system. In one-dimensional collisions, objects move along a straight line, simplifying the analysis. The conservation of momentum principle applies to various types of collisions, including elastic and inelastic collisions. By studying this concept, students can predict the outcomes of collisions and gain insights into the behavior of objects in motion. Understanding conservation of momentum is essential for solving problems in mechanics and forms the basis for more advanced topics in physics.

FAQs

  1. What is the conservation of momentum in one dimension?

    Conservation of momentum in one dimension is a fundamental principle in physics stating that the total momentum of a closed system remains constant before and after a collision or interaction, assuming no external forces are acting on the system. In one-dimensional scenarios, objects move along a straight line, and their momentum is calculated as the product of mass and velocity (p = mv).

  2. How does the conservation of momentum apply to collisions?

    In collisions, the conservation of momentum principle ensures that the total momentum of the system before the collision equals the total momentum after the collision. This applies to both elastic collisions (where kinetic energy is conserved) and inelastic collisions (where kinetic energy is not conserved). The principle allows us to predict the final velocities of objects involved in collisions when given their initial velocities and masses.

  3. What is the difference between elastic and inelastic collisions in terms of momentum conservation?

    Both elastic and inelastic collisions conserve momentum. In elastic collisions, kinetic energy is also conserved, and objects bounce off each other without deformation. In inelastic collisions, kinetic energy is not conserved (some is converted to heat or deformation), but momentum is still conserved. The key difference is that in perfectly inelastic collisions, the objects stick together after collision, moving with a common final velocity.

  4. How can I apply the conservation of momentum principle to solve physics problems?

    To apply the conservation of momentum principle: 1. Identify the system and ensure it's closed (no external forces). 2. Determine the masses and initial velocities of objects involved. 3. Write the conservation of momentum equation: m1v1 + m2v2 = m1v1' + m2v2' (where primed velocities are after collision). 4. Solve for unknown velocities using the equation and given information. 5. Check your answer for reasonableness and correct units.

  5. Are there any limitations to the conservation of momentum principle?

    While the conservation of momentum is a fundamental principle, it has limitations in practical applications: 1. It only applies to closed systems with no external forces. 2. In real-world scenarios, factors like friction and air resistance can affect momentum conservation over time. 3. For complex systems or rotational motion, additional principles may be needed for a complete analysis. 4. In relativistic scenarios (near light speed), classical momentum formulas need to be adjusted, though the conservation principle still holds.

Prerequisites

Understanding the conservation of momentum in one dimension is a crucial concept in physics, but to fully grasp its significance, it's essential to have a solid foundation in several prerequisite topics. These fundamental concepts provide the necessary context and tools to comprehend the intricacies of momentum conservation.

At the core of this topic lies Newton's second law of motion, which establishes the relationship between force, mass, and acceleration. This law is fundamental to understanding how momentum changes over time and forms the basis for deriving the conservation of momentum principle. By mastering Newton's second law, students can better appreciate the forces at play during momentum conservation scenarios.

Another critical prerequisite is the concept of elastic and inelastic collisions. These types of collisions are prime examples of momentum conservation in action. Understanding the differences between elastic collisions, where kinetic energy is conserved, and inelastic collisions, where it is not, helps students apply the conservation of momentum principle to real-world situations.

The rate of change is another essential concept, particularly when considering the rate of change of momentum. This mathematical tool allows students to analyze how momentum varies over time, which is crucial for understanding impulse and its relationship to momentum change.

Speaking of which, the relationship between momentum and impulse is a key prerequisite. Impulse, being the change in momentum, directly ties into the conservation principle. Grasping this connection helps students understand how forces applied over time can affect an object's momentum while still adhering to the conservation law.

Lastly, while not directly related to linear momentum, understanding rotational kinetic energy and angular momentum provides a broader perspective on energy and momentum concepts. This knowledge allows students to draw parallels between linear and rotational systems, enhancing their overall comprehension of momentum conservation principles.

By thoroughly studying these prerequisite topics, students build a strong foundation for understanding the conservation of momentum in one dimension. Each concept contributes to a more comprehensive view of how objects interact and move in the physical world, making the study of momentum conservation not just more accessible, but also more meaningful and applicable to real-world scenarios.