8.1 Linear Momentum, Force, and Impulse
3 min read•june 24, 2024
is the product of and velocity, representing an object's quantity of motion. It's closely tied to , which is the force applied over time that changes . These concepts are crucial for understanding how objects move and interact.
The links these ideas, showing how forces change an object's motion. Newton's second law, rewritten in terms of momentum, further illuminates this relationship. These principles are key to analyzing collisions, explosions, and other dynamic scenarios.
Linear Momentum and Its Relationship to Force and Impulse
Concept of linear momentum
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- Linear momentum () is the product of an object's mass () and its velocity () calculated using the formula
- Represents the quantity of motion an object possesses, considering both its mass and velocity
- Vector quantity with both magnitude and direction, meaning it has a specific value and points in a specific direction (north, east)
- Impulse () is the product of the net force () acting on an object and the time interval () over which the force acts, calculated using the formula
- Also a vector quantity, having both magnitude and direction (45 N·s, 60° from horizontal)
- Impulse is equal to the change in momentum () of an object, represented by the equation , where is final momentum and is initial momentum
- Impulse can be thought of as the "kick" or "punch" that changes an object's momentum over a period of time (baseball bat hitting a ball)
Impulse-momentum theorem applications
- The impulse-momentum theorem states that the impulse applied to an object equals the change in its momentum, expressed as
- To solve problems using the impulse-momentum theorem, follow these steps:
- Identify the initial and final velocities ( and ) or momenta ( and ) of the object (car traveling at 20 m/s, coming to a stop)
- Determine the mass () of the object (1,500 kg car)
- Calculate the change in momentum () using the given information (mass and change in velocity)
- If the force () and time interval () are known, calculate the impulse () and equate it to the change in momentum (force of brakes over 5 seconds)
- Applying the theorem helps analyze situations involving collisions, explosions, or any scenario where forces act on objects over time (rocket launch, two billiard balls colliding)
- The can be used to characterize the elasticity of collisions in impulse-momentum problems
Newton's law and momentum change
- states that the net force acting on an object equals the product of its mass and acceleration, expressed as
- The law can be rewritten in terms of momentum change: , meaning the net force acting on an object is equal to the rate of change of its momentum
- Implications of Newton's second law in terms of momentum include:
- If there is no net force acting on an object, its momentum remains constant, known as the (spacecraft in frictionless space)
- A net force acting on an object will cause a change in its momentum over time ( from a jet engine changes aircraft momentum)
- The greater the net force, the greater the rate of change of momentum (higher force from a tennis racket causes a faster change in ball momentum)
- Newton's second law connects force, mass, and acceleration to explain changes in motion and momentum (understanding car crashes, designing safer sports equipment)
- The relates the work done on an object to its change in kinetic energy, which is closely linked to momentum
Extended Concepts in Momentum
- is the rotational analog of linear momentum, describing the quantity of rotational motion of an object
- represents the flow of momentum through a surface, important in fluid dynamics and electromagnetic theory
- These concepts extend our understanding of momentum beyond linear motion to rotational systems and continuous media
Key Terms to Review (28)
Angular Momentum: Angular momentum is a measure of the rotational motion of an object around a fixed axis. It is the product of an object's moment of inertia and its angular velocity, and it is a conserved quantity in the absence of external torques.
Center of Mass: The center of mass is a point within an object or system of objects where the object's entire mass can be considered to be concentrated. It is the point at which the object's weight is evenly distributed and acts as the object's effective point of application for any external forces acting on it.
Coefficient of Restitution: The coefficient of restitution is a measure of the elasticity of a collision between two objects. It quantifies the amount of kinetic energy lost during the collision and is used to determine the outcome of collisions in various contexts, including physics, engineering, and sports.
Conservation of Momentum: Conservation of momentum is a fundamental principle in physics which states that the total momentum of a closed system remains constant unless an external force acts upon it. This principle is a direct consequence of Newton's laws of motion and is applicable to both elastic and inelastic collisions.
Elastic Collision: An elastic collision is a type of collision between two objects in which there is no net loss of kinetic energy. The total kinetic energy of the colliding objects before the collision is equal to the total kinetic energy after the collision, and the objects may bounce off each other.
F: F is a fundamental physical quantity that represents the interaction between two objects, causing one or both to change their motion. It is a vector quantity, meaning it has both magnitude and direction, and is a key concept in the study of mechanics and dynamics.
F_{net}: F_{net}, or the net force, is the vector sum of all the forces acting on an object. It represents the overall force that determines the object's acceleration and motion according to Newton's second law of motion.
Friction: Friction is a force that opposes the relative motion between two surfaces in contact. It arises due to the roughness and irregularities of the surfaces, causing them to resist sliding or rolling past one another. Friction is a crucial concept in understanding the behavior of objects and the forces acting upon them in various physics topics.
Impulse: Impulse is the change in momentum of an object caused by the application of a force over a period of time. It is a vector quantity that combines the magnitude of the force and the duration of its application, providing a measure of the total effect of the force on the object's motion.
Impulse-Momentum Theorem: The impulse-momentum theorem establishes a fundamental relationship between the impulse exerted on an object and the change in its momentum. It is a crucial concept in the study of linear momentum, force, and the conservation of momentum.
Inelastic Collision: An inelastic collision is a type of collision in which the total kinetic energy of the colliding objects is not conserved. In an inelastic collision, the colliding objects stick together or undergo a change in shape, resulting in a loss of kinetic energy that is converted into other forms of energy, such as heat or sound.
Isaac Newton: Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher who is widely regarded as one of the most influential scientists of all time. His groundbreaking work in the fields of mechanics, optics, and astronomy laid the foundation for our modern understanding of the physical world.
J: J is a fundamental physical quantity that represents the amount of change in the linear momentum of an object due to the application of a force over a period of time. It is a vector quantity, meaning it has both magnitude and direction, and is a crucial concept in the understanding of linear momentum, force, and impulse.
Kg⋅m/s: kg⋅m/s, or kilogram-meter per second, is a unit of linear momentum, which is a measure of an object's motion and its ability to resist changes in that motion. It represents the product of an object's mass and its velocity, and is a fundamental concept in the study of physics, particularly in the topics of linear momentum, force, and impulse.
Linear Momentum: Linear momentum is a fundamental concept in physics that describes the quantity of motion possessed by an object. It is defined as the product of an object's mass and its velocity, and it is a vector quantity, meaning it has both magnitude and direction.
Mass: Mass is a fundamental property of an object that quantifies the amount of matter it contains. It is a measure of the object's resistance to changes in its motion, and is a key concept in understanding Newton's Second Law of Motion and the principles of linear momentum and impulse.
Momentum: Momentum is a vector quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity, and it represents the amount of motion or inertia possessed by the object. Momentum is a fundamental concept in classical mechanics and is closely related to the concepts of force, impulse, and linear momentum.
Momentum Flux: Momentum flux is the rate of change of momentum per unit area, representing the flow of momentum through a given surface or boundary. It is a fundamental concept in the study of fluid dynamics and the analysis of forces acting on objects in motion.
N⋅s: N⋅s, or newton-seconds, is a unit that represents the change in linear momentum. It is the product of force and the time over which that force is applied, and it is a fundamental concept in the study of linear momentum, force, and impulse.
Newton's Second Law of Motion: Newton's Second Law of Motion is a fundamental principle in classical mechanics that describes the relationship between an object's acceleration, the net force acting upon it, and the object's mass. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
P_f: The final momentum of an object, which represents the momentum of the object after a force has been applied to it for a certain period of time. It is a fundamental concept in the study of linear momentum, force, and impulse.
P_i: The initial momentum of an object, also known as the initial linear momentum, is a fundamental concept in classical mechanics that describes the quantity of motion an object possesses at a specific point in time. It is a vector quantity, meaning it has both magnitude and direction, and is a crucial factor in understanding the behavior of objects under the influence of forces.
Thrust: Thrust is the force that propels an object forward, providing the necessary momentum to overcome resistance and achieve motion. It is a fundamental concept in the fields of linear momentum, force, and impulse, governing the dynamics of various systems and applications.
V_f: The final velocity, or v_f, is the velocity of an object at the end of a motion or event. It is a fundamental concept in the study of linear momentum, force, and impulse, as it represents the speed and direction of an object after the application of a force or the occurrence of an impulse.
V_i: v_i, or the initial velocity, is a fundamental concept in physics that describes the speed and direction of an object at the start of a motion or event. It is a crucial parameter in the analysis of linear momentum, force, and impulse, as it helps determine the object's behavior and the changes it undergoes over time.
Work-Energy Theorem: The work-energy theorem states that the work done on an object is equal to the change in the object's kinetic energy. This fundamental principle connects the concepts of work, force, and energy, and is a crucial tool for analyzing the motion and energy transformations of objects in various physical systems.
Δp: Δp, or change in momentum, is a fundamental concept in physics that describes the change in an object's linear momentum over a given time period. It is the product of an object's mass and the change in its velocity, and is a key factor in understanding the relationship between force, impulse, and the motion of objects.
Δt: Δt, or delta t, represents the change in time between two events or the duration of a time interval. It is a fundamental concept in various areas of physics, including the study of linear momentum, force, impulse, and the zeroth law of thermodynamics.