Glossary

Momentum and collisions in two dimensions add complexity to our understanding of motion. We'll explore how momentum behaves as a vector quantity, with components in both x and y directions. This builds on our previous knowledge of one-dimensional motion.

Conservation of momentum applies separately to both x and y components in two-dimensional collisions. We'll analyze elastic and inelastic collisions, using conservation equations to solve problems involving objects moving at angles. This expands our toolkit for understanding real-world interactions.

Momentum and Collisions in Two Dimensions

Momentum as vector quantity

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  • Momentum is a vector quantity has both magnitude and direction
  • In two dimensions, momentum is represented by a vector with components in the x and y directions
    • px=mvxp_x = mv_x where mm is mass and vxv_x is velocity in x-direction
    • py=mvyp_y = mv_y where vyv_y is velocity in y-direction
  • is sum of its components p⃗=pxi^+pyj^\vec{p} = p_x \hat{i} + p_y \hat{j}
    • i^\hat{i} and j^\hat{j} are unit vectors in x and y directions respectively
  • Magnitude of momentum vector given by ∣p⃗∣=px2+py2|\vec{p}| = \sqrt{p_x^2 + p_y^2}
  • Direction of momentum vector given by angle θ=tan⁡−1(pypx)\theta = \tan^{-1}(\frac{p_y}{p_x})
  • Examples:
    • Car moving northeast has momentum components in both x and y directions
    • Billiard ball struck off-center has initial momentum at an angle to the x-axis

Conservation of momentum in components

  • Law of conservation of momentum states total momentum of closed system remains constant
    • In two dimensions, both x and y components of total momentum are conserved separately
  • For system of two colliding objects, are:
    • x-component: m1v1x,i+m2v2x,i=m1v1x,f+m2v2x,fm_1v_{1x,i} + m_2v_{2x,i} = m_1v_{1x,f} + m_2v_{2x,f}
    • y-component: m1v1y,i+m2v2y,i=m1v1y,f+m2v2y,fm_1v_{1y,i} + m_2v_{2y,i} = m_1v_{1y,f} + m_2v_{2y,f}
    • Subscripts ii and ff denote initial and final velocities respectively
  • To solve problems using conservation of momentum in two dimensions:
    1. Identify initial and final velocities of each object in x and y directions
    2. Write conservation of momentum equations for both x and y components
    3. Solve equations simultaneously to find unknown velocities
  • Examples:
    • Two ice skaters pushing off each other at an angle
    • Projectile fired from a cannon mounted on a moving cart

Analysis of two-dimensional collisions

  • Elastic collisions:
    • is conserved in elastic collisions
    • In two dimensions, both momentum and kinetic energy are conserved
    • Equations for conservation of kinetic energy in two dimensions:
      • 12m1(v1x,i2+v1y,i2)+12m2(v2x,i2+v2y,i2)=12m1(v1x,f2+v1y,f2)+12m2(v2x,f2+v2y,f2)\frac{1}{2}m_1(v_{1x,i}^2 + v_{1y,i}^2) + \frac{1}{2}m_2(v_{2x,i}^2 + v_{2y,i}^2) = \frac{1}{2}m_1(v_{1x,f}^2 + v_{1y,f}^2) + \frac{1}{2}m_2(v_{2x,f}^2 + v_{2y,f}^2)
    • To solve problems, use conservation of momentum equations along with conservation of kinetic energy equation
    • Examples: Two billiard balls colliding, subatomic particle collisions
  • Inelastic collisions:
    • Kinetic energy is not conserved in inelastic collisions
    • In two dimensions, only momentum is conserved
    • Objects may stick together after collision (perfectly inelastic) or separate with different velocities (partially inelastic)
      • For perfectly inelastic collisions, final velocities of objects are equal: v1x,f=v2x,fv_{1x,f} = v_{2x,f} and v1y,f=v2y,fv_{1y,f} = v_{2y,f}
    • To solve problems, use conservation of momentum equations and any additional information given about final velocities
    • Examples: Two lumps of clay colliding and sticking, car crashes

Additional Concepts in Two-Dimensional Collisions

  • is conserved in collisions without external torques
  • relates the work done by forces to changes in kinetic energy during collisions
  • ensures that forces between colliding objects are equal and opposite
  • choice can simplify collision analysis (e.g., )
  • between colliding objects determines the nature of the collision

Problem-Solving Strategies for Two-Dimensional Collisions

Steps to solve two-dimensional collision problems

  1. Identify type of collision (elastic or inelastic)
  2. Choose convenient coordinate system (x-y axes)
  3. Write down given information (masses, initial velocities, angles)
  4. If necessary, break initial velocities into x and y components using trigonometry
  5. Write conservation of momentum equations for x and y components
  6. If collision is elastic, also write conservation of kinetic energy equation
  7. Solve equations simultaneously to find unknown quantities (final velocities, angles)
  8. Check answers for consistency and plausibility
  • Examples:
    • Two cars colliding at an intersection
    • Puck struck by a hockey stick at an angle

Key Terms to Review (28)

Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Center of Mass Frame: The center of mass frame, also known as the center of momentum frame, is a reference frame in which the total momentum of a system is zero. This frame is particularly useful in the analysis of collisions and other interactions involving multiple objects or particles.
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.
Conservation of Momentum Equations: The conservation of momentum is a fundamental principle in physics that states the total momentum of a closed system is constant unless an external force acts on the system. The conservation of momentum equations describe the mathematical relationships that govern the momentum of objects in a collision or other interaction.
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.
Glancing Collision: A glancing collision is a type of collision where two objects interact with each other at a shallow angle, resulting in a deflection of their trajectories rather than a direct, head-on impact. This type of collision is commonly observed in the context of collisions in multiple dimensions.
Head-on Collision: A head-on collision occurs when two objects, typically vehicles, collide while traveling in opposite directions along the same path. This type of collision is often one of the most severe and dangerous types of impact, as the combined kinetic energy of the two objects can result in catastrophic damage and serious injuries.
Impact Parameter: The impact parameter is a measure of the closest distance of approach between two objects or particles in a collision. It is a crucial concept in the study of collisions, particularly in the context of multiple dimensions.
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.
Inertial reference frame: An inertial reference frame is a frame of reference in which an object remains at rest or moves at a constant velocity unless acted upon by an external force. It is crucial for the formulation of Newton's laws of motion.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Lab Frame: The lab frame, also known as the laboratory frame or the stationary frame, is a reference frame that is fixed in space and is used to describe the motion and interactions of objects in a physical system. It serves as the primary frame of reference for analyzing and understanding the dynamics of collisions in multiple dimensions.
Momentum Conservation: Momentum conservation is a fundamental principle in physics that states the total momentum of a closed system remains constant unless an external force acts upon it. This principle is crucial in understanding the dynamics of collisions and the behavior of objects in motion.
Newton's Third Law: Newton's Third Law, also known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This fundamental principle of physics describes the relationship between forces acting on interacting objects.
Partially Inelastic Collision: A partially inelastic collision is a type of collision where the colliding objects stick together after the collision, but some of the initial kinetic energy is lost as other forms of energy, such as heat or sound. In this type of collision, the total momentum is conserved, but the total kinetic energy is not conserved.
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.
Reference Frame: A reference frame is a coordinate system used to describe the motion and position of objects in space. It provides a frame of reference from which observations and measurements can be made, allowing for the consistent and meaningful analysis of physical phenomena.
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.
Scattering Angle: The scattering angle is the angle between the initial direction of a particle or wave and its final direction after it has been scattered or deflected by an interaction with another particle or medium. It is a fundamental concept in the study of collisions and the behavior of particles and waves.
Total Momentum Vector: The total momentum vector is a vector quantity that represents the total amount of momentum possessed by a system of objects. It is the vector sum of the individual momentum vectors of all the objects in the system, and its magnitude and direction describe the overall momentum of the system.
Vector components: Vector components are the projections of a vector along the axes of a coordinate system. They simplify vector calculations by breaking vectors into perpendicular directions.
Vector Components: Vector components are the individual parts or projections of a vector along specific coordinate axes. They represent the magnitude and direction of a vector in a given reference frame and are essential for analyzing and manipulating vectors in various physics and mathematics applications.
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.
X-Component of Momentum: The x-component of momentum is the portion of an object's total momentum that is directed along the x-axis in a two- or three-dimensional coordinate system. It represents the horizontal component of an object's momentum and is a crucial factor in analyzing collisions in multiple dimensions.
Y-component of momentum: The y-component of momentum refers to the portion of an object's total momentum that is directed along the vertical (y) axis. This is an important concept in the analysis of collisions in multiple dimensions, as the y-component of momentum must be conserved during the collision process.
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