Glossary

8.3 Collisions in One and Two Dimensions

2 min read•july 24, 2024

Collisions are key events in physics, where objects interact and exchange energy and momentum. Understanding elastic and inelastic collisions helps us analyze everything from billiard ball impacts to car crashes, revealing how energy and momentum are conserved or transformed.

One-dimensional collisions are simpler to solve, using conservation laws and equations. Two-dimensional collisions require , breaking motion into components. These principles apply to real-world scenarios, from particle physics experiments to astronomical events like comet impacts.

Types of Collisions

Elastic vs inelastic collisions

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  • Elastic collisions
    • Total kinetic energy and momentum conserved during impact
    • Objects bounce off each other with no deformation (billiard balls, atomic collisions)
  • Inelastic collisions
    • Momentum conserved but kinetic energy partially lost
    • Energy converted to heat, sound, or deformation (car crashes, clay balls sticking together)
  • Perfectly inelastic collisions
    • Objects combine into single mass after impact
    • Maximum kinetic while conserving momentum (bullet embedding in wood)

One-dimensional collision problem-solving

  • : p1+p2=p1′+p2′p_1 + p_2 = p_1' + p_2' applies to all collisions
  • Elastic collisions
    • Kinetic energy conserved: 12m1v12+12m2v22=12m1v1′2+12m2v2′2\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2
    • : v1−v2=−(v1′−v2′)v_1 - v_2 = -(v_1' - v_2') helps solve for final velocities
  • Inelastic collisions
    • Find final velocity using
    • Energy loss: ΔE=12m1v12+12m2v22−12(m1+m2)vf2\Delta E = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 - \frac{1}{2}(m_1 + m_2)v_f^2
    • Elasticity measure: e=v2′−v1′v1−v2e = \frac{v_2' - v_1'}{v_1 - v_2} ranges 0 to 1
    • 0 for perfectly inelastic, 1 for perfectly elastic collisions

Two-Dimensional Collisions

Vector analysis of two-dimensional collisions

  • Decompose vectors into x and y components
  • Apply momentum conservation separately in each direction
    • p1x+p2x=p1x′+p2x′p_{1x} + p_{2x} = p_{1x}' + p_{2x}' and p1y+p2y=p1y′+p2y′p_{1y} + p_{2y} = p_{1y}' + p_{2y}'
  • Use vector addition to find final momenta
  • Calculate with trigonometry (glancing collisions)

Conservation principles in collision problems

  • Vector momentum conservation: p1⃗+p2⃗=p1′⃗+p2′⃗\vec{p_1} + \vec{p_2} = \vec{p_1'} + \vec{p_2'} for all collisions
  • Apply kinetic energy conservation for elastic collisions
  • Solve simultaneous equations for unknown variables
    • Simplifies calculations, total momentum zero
    • Useful for analyzing particle collisions
  • 2D collision types include glancing and head-on impacts
  • Real-world applications in particle physics experiments and astronomical events (comet impacts)

Key Terms to Review (21)

Center of mass frame: The center of mass frame is a specific reference frame in which the center of mass of a system is at rest. This frame simplifies the analysis of collisions and interactions, as it allows for a clearer understanding of momentum and energy conservation, especially in one and two-dimensional collisions.
Change in momentum: Change in momentum refers to the difference in momentum of an object before and after an event, such as a collision or force application. Momentum, which is the product of an object's mass and its velocity, can change due to external forces or interactions between objects. Understanding this concept is crucial for analyzing motion and interactions, particularly in scenarios involving impulse and collisions.
Closed System: A closed system is defined as a physical system that does not exchange matter with its surroundings but can exchange energy in the form of work or heat. In this type of system, the total amount of mass remains constant, allowing for the analysis of energy transformations and momentum interactions. Understanding closed systems is crucial for analyzing the conservation principles, especially when studying how objects interact during collisions and how linear momentum is conserved.
Coefficient of restitution: The coefficient of restitution is a measure of the elasticity of collisions between two bodies, defined as the ratio of the relative speed of separation to the relative speed of approach. It indicates how much kinetic energy is conserved in a collision, ranging from 0 (perfectly inelastic collision) to 1 (perfectly elastic collision). This concept helps in understanding how objects behave upon impact in one and two-dimensional collisions.
Conservation of Momentum: Conservation of momentum is a fundamental principle stating that the total linear momentum of a closed system remains constant over time, provided that no external forces act upon it. This principle is crucial for analyzing interactions between objects, particularly during collisions and explosions, where momentum before an event equals momentum after. Understanding this concept allows for deeper insights into how objects move and interact in both one and two dimensions, as well as how systems of particles behave collectively.
Deflection Angles: Deflection angles refer to the angles formed between the original trajectory of an object and its new path after a collision with another object. These angles are crucial in understanding how momentum and energy are conserved in collisions, both elastic and inelastic, and they play a vital role in predicting the outcomes of collisions in one and two dimensions.
Elastic collision: An elastic collision is a type of collision where both momentum and kinetic energy are conserved. During such interactions, the colliding objects bounce off each other without any permanent deformation or generation of heat. This means that the total energy and total momentum before and after the collision remain unchanged, which is a key characteristic of elastic collisions in various physical scenarios.
Energy Loss: Energy loss refers to the reduction of mechanical energy in a system, often due to non-conservative forces such as friction or air resistance, which transform kinetic or potential energy into other forms of energy, like thermal energy. This phenomenon plays a crucial role in understanding how systems behave when subjected to forces that do not store energy, affecting both the efficiency of processes and the analysis of collisions.
Glancing Collision: A glancing collision is a type of impact where two objects collide at an angle rather than head-on, resulting in a change in direction for at least one of the objects involved. This type of collision often involves one object deflecting off another, which can affect their respective velocities and paths. Understanding glancing collisions is crucial in analyzing momentum and energy transfer during interactions between moving bodies.
Head-on impact: A head-on impact refers to a collision where two objects collide directly with their front surfaces facing each other. This type of impact is significant in understanding the conservation of momentum and energy transfer during collisions, particularly in one-dimensional scenarios where the motion is linear. Analyzing head-on impacts helps in solving problems involving elastic and inelastic collisions, as it illustrates how velocity and mass interact during the collision process.
Impact force: Impact force is the force exerted when two or more objects collide, which results from the change in momentum of the objects involved in the collision. This force is a crucial factor in understanding how energy is transferred during collisions, whether they are elastic or inelastic. The magnitude of the impact force depends on factors such as the speed of the objects, their mass, and the duration of the collision.
Impulse: Impulse is defined as the change in momentum of an object when a force is applied over a specific time interval. It connects directly to how forces affect an object's motion, showing that the greater the force applied or the longer it's applied, the greater the change in momentum. This concept is crucial for understanding collisions and how objects interact with one another in both one and two dimensions.
Inelastic collision: An inelastic collision is a type of collision in which kinetic energy is not conserved, although momentum is conserved. During such collisions, the objects involved may stick together or deform, resulting in some of the kinetic energy being transformed into other forms of energy like heat or sound. This loss of kinetic energy differentiates inelastic collisions from elastic collisions, where both momentum and kinetic energy are conserved.
Isolated system: An isolated system is a physical system that does not exchange matter or energy with its surroundings, effectively operating independently. This concept is crucial for understanding conservation laws, as it allows for the simplification of complex interactions by assuming that the total momentum and energy remain constant within the system. An isolated system provides a framework for analyzing collisions, the conservation of linear momentum, and angular momentum without external influences.
Momentum conservation: Momentum conservation refers to the principle stating that the total momentum of a closed system remains constant over time, provided no external forces act on it. This principle is fundamental in understanding interactions such as collisions, where the momentum before and after the event can be compared. It plays a critical role in analyzing both one-dimensional and two-dimensional collisions, helping us to predict the outcomes of these interactions based on the masses and velocities involved.
Perfectly inelastic collision: A perfectly inelastic collision is a type of collision where two objects collide and stick together, moving as a single combined mass after the impact. This type of collision maximizes the loss of kinetic energy, as some of it is transformed into other forms of energy like heat or sound, but the total momentum of the system is conserved. Understanding this concept is crucial for analyzing how objects interact in one or two dimensions, and it ties into the broader principle of conservation of linear momentum.
Rebound Velocity: Rebound velocity refers to the speed and direction at which an object moves after it has collided with another object and then bounces back. This term is crucial in understanding collisions, especially in analyzing how kinetic energy is transferred during elastic and inelastic collisions, where the outcome significantly depends on the masses and initial velocities of the colliding bodies.
Relative Velocity Reversal: Relative velocity reversal occurs when two objects in a collision exchange their velocities, leading to a change in their respective motions. This concept is crucial in understanding how momentum and energy are conserved during collisions, allowing us to analyze the outcome of interactions between objects in one or two dimensions.
Two-dimensional collision: A two-dimensional collision occurs when two objects collide and their velocities change in a plane, involving both x and y components. This type of collision can be analyzed using vector components to determine the final velocities of the objects involved, taking into account the principles of conservation of momentum and kinetic energy. Understanding these collisions is essential for solving problems related to interactions between moving bodies in various physical scenarios.
Vector Analysis: Vector analysis is a branch of mathematics that deals with quantities that have both magnitude and direction. It provides the tools to analyze and solve problems involving vectors, making it essential for understanding motion, forces, and other physical phenomena in both one and two dimensions.
Vector Components: Vector components are the projections of a vector along the axes of a coordinate system, typically broken down into horizontal and vertical parts. This concept is crucial for analyzing forces, velocities, and displacements, allowing complex vector quantities to be expressed as simpler one-dimensional vectors. Understanding vector components enables the resolution of forces in different directions and is foundational in problems involving motion, especially in collisions and interactions in multiple dimensions.
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