9.3 Conservation of Linear Momentum
3 min read•june 24, 2024
Linear momentum conservation is a fundamental principle in physics. It states that the total momentum of a remains constant, regardless of interactions within it. This concept applies to , , and systems with internal forces, providing a powerful tool for analyzing motion.
Understanding momentum conservation helps predict outcomes in various scenarios. From billiard ball collisions to , this principle explains how objects behave during interactions. It's crucial for solving problems involving velocities and masses in both elastic and inelastic collisions.
Conservation of Linear Momentum
Conservation of momentum fundamentals
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- states the total momentum of a remains constant over time
- Closed experiences no net external forces acting on it (isolated from surroundings)
- Vector sum of momenta for all objects in the system stays the same before and after interactions
- Applies to various scenarios involving interactions between objects
- Collisions between two or more objects (billiard balls, car crashes)
- Explosions where initially stationary objects break apart and move in different directions (fireworks, fragmenting asteroids)
- Systems with internal forces canceling out resulting in no net external force (rocket expelling fuel, of a gun)
Mathematical expressions of momentum conservation
- For a system of two objects before and after an interaction:
- , represent masses of objects 1 and 2
- , denote initial velocities of objects 1 and 2 before the interaction
- , denote final velocities of objects 1 and 2 after the interaction
- For a system with objects:
- represents mass of the -th object
- represents velocity of the -th object
- Sum of the products of mass and velocity for all objects remains constant
- Momentum and velocity are , requiring consideration of both magnitude and direction
Momentum conservation in collisions and explosions
- In collisions, use conservation of momentum to find unknown velocities or masses
- Consider initial and final velocities and masses of colliding objects
- Substitute known values into momentum conservation equation and solve for unknowns (pool ball collisions, hockey pucks)
- In explosions, objects initially at rest () break apart with different velocities
- Sum of momenta for all fragments after the explosion equals zero ()
- Useful for analyzing exploding fireworks, fragmenting asteroids, or separating stages of a rocket
- The of a system follows a straight-line path unless acted upon by external forces
Analysis of linear momentum conservation
- Check if the system is closed with no net external forces
- Presence of external forces (friction, air resistance) can lead to momentum not being conserved
- Verify vector sum of momenta is equal before and after the interaction
- Unequal sums indicate linear momentum is not conserved
- Consider objects entering or leaving the system during the interaction
- Objects entering or leaving can change the total momentum of the system (baseball being caught by a glove, a car leaving a collision)
Elastic vs inelastic collisions
- Elastic collisions conserve both kinetic energy and linear momentum
- Total kinetic energy remains the same before and after the collision ()
- Objects separate after the collision (bouncing balls, atomic and subatomic particle collisions)
- Inelastic collisions conserve linear momentum but not kinetic energy
- Some kinetic energy converts to other forms (heat, deformation)
- Objects may stick together after the collision, known as a (two lumps of clay colliding and merging)
- Total kinetic energy after the collision is less than the total kinetic energy before the collision (car crashes, bullet hitting a block of wood)
- The characterizes the elasticity of a collision
Impulse and momentum change
- is the change in momentum of an object due to a force applied over a time interval
- ###-Momentum_Theorem_0###: The impulse experienced by an object equals its change in momentum
- between objects is crucial in analyzing collisions and determining post-collision velocities
Key Terms to Review (26)
Center of mass: The center of mass is the point in an object or system where all its mass can be considered to be concentrated for the purpose of analyzing translational motion. It is the weighted average position of all the mass in the system.
Center of Mass: The center of mass is the point at which an object's entire mass can be considered to be concentrated. It is the average position of the mass of an object, and it is the point around which the object's rotation and motion can be analyzed.
Closed system: A closed system is a physical system that does not exchange matter with its surroundings, but can exchange energy. In mechanics, it is often used to analyze conservation laws such as the conservation of linear momentum.
Closed System: A closed system is a thermodynamic system that does not exchange matter with its surroundings, but may exchange energy. It is an idealized model used to understand the behavior of various physical and chemical processes, particularly in the context of energy conservation and momentum conservation.
Coefficient of Restitution: The coefficient of restitution is a measure of the elasticity of a collision between two objects. It quantifies the ratio of the relative speed of the objects after the collision to the relative speed before the collision, and is a key factor in determining the outcomes of various types of collisions.
Collisions: A collision is an event in which two or more objects interact for a short period of time, during which the objects' velocities change. Collisions are a fundamental concept in the study of mechanics, particularly in the context of conservation of linear momentum.
Conservation of Linear Momentum: Conservation of linear momentum is a fundamental principle in physics that states the total linear momentum of a closed system is constant unless an external force acts on the system. This means the total momentum before an event is equal to the total momentum after the event, as long as no external forces are involved.
Elastic Collision: An elastic collision is a type of collision between two objects where the total kinetic energy of the system is conserved. In an elastic collision, there is no net loss of kinetic energy, and the objects simply exchange momentum without any deformation or change in internal energy.
Explosions: An explosion is a rapid and violent release of energy that creates a shockwave and can cause significant damage. In the context of conservation of linear momentum, explosions are an important phenomenon to understand as they involve the transfer of momentum between objects.
Impulse: Impulse is the product of the average force and the time interval over which it acts on an object. It is equal to the change in momentum of the object.
Impulse: Impulse is a quantity that describes the change in momentum of an object over a given time interval. It is the product of the net force acting on an object and the time interval during which that force is applied. Impulse is a fundamental concept in physics that connects the ideas of force, time, and momentum, and is essential for understanding topics such as solving problems in physics, forces, Newton's laws, and collisions.
Impulse-Momentum Theorem: The impulse-momentum theorem states that the impulse, or the change in momentum, of an object is equal to the net force acting on the object multiplied by the time over which the force acts. This theorem establishes a fundamental relationship between the concepts of impulse and momentum, which are crucial in understanding the dynamics of collisions and the conservation of linear momentum.
Inelastic Collision: An inelastic collision is a type of collision where the colliding objects stick together after the collision, or undergo a deformation, resulting in a loss of kinetic energy. In an inelastic collision, the total momentum of the system is conserved, but the total kinetic energy is not.
Isolated System: An isolated system is a physical system that does not exchange any matter with its surroundings, but may exchange energy. It is a closed system that is completely separated from its environment, allowing for the study of the system's internal processes and the conservation of certain physical quantities.
Kg⋅m/s: kg⋅m/s, or kilogram-meter per second, is a unit that represents linear momentum. It is the product of an object's mass (in kilograms) and its velocity (in meters per second), and it quantifies the amount of motion an object possesses.
Newton: Newton is the standard unit of force in the International System of Units (SI), named after the renowned English physicist and mathematician, Sir Isaac Newton. It is a fundamental unit that is essential in understanding and describing the behavior of objects under the influence of various forces, as well as in the study of mechanics, dynamics, and other related areas of physics.
Perfectly Inelastic Collision: A perfectly inelastic collision is a type of collision where the colliding objects stick together after impact, resulting in a single object with a combined mass and a shared velocity. In this type of collision, the total momentum of the system is conserved, but the kinetic energy is not.
Philae: Philae is a robotic lander that accompanied the Rosetta spacecraft. It successfully landed on comet 67P/Churyumov-Gerasimenko to conduct scientific experiments.
Recoil: Recoil is the backward movement or force that occurs when an object is propelled forward, such as the backward motion of a gun when it is fired. This concept is closely related to Newton's Third Law of Motion, Linear Momentum, and the Conservation of Linear Momentum.
Relative Velocity: Relative velocity is the velocity of one object as observed from the frame of reference of another object. It describes the motion of an object relative to another object, rather than in an absolute sense.
Rocket Propulsion: Rocket propulsion is the use of a rocket engine to generate thrust and propel a vehicle, such as a spacecraft or missile, through the vacuum of space or the Earth's atmosphere. It is a fundamental principle of physics that enables the movement of objects by the rearward acceleration of a propellant.
System: A system is a defined collection of objects considered for studying interactions and conservation laws. It can be isolated, closed, or open depending on the exchange of energy and matter.
Two-body Collision: A two-body collision is a type of collision event where two objects or particles interact and exchange momentum and energy. This concept is fundamental in the understanding of conservation of linear momentum, a key principle in classical mechanics.
Vector Quantities: Vector quantities are physical quantities that have both magnitude and direction, distinguishing them from scalar quantities, which only have magnitude. In physics, understanding vector quantities is crucial for analyzing motion and forces, as they provide essential information about how objects move and interact in space.
Work-energy theorem: The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, it is expressed as $W_{net} = \Delta KE$.
Work-Energy Theorem: The work-energy theorem is a fundamental principle in physics that states the change in the kinetic energy of an object is equal to the net work done on that object. It establishes a direct relationship between the work performed on an object and the resulting change in its kinetic energy, providing a powerful tool for analyzing and solving problems involving energy transformations.