Elastic collision
An elastic collision is a collision between two or more bodies in which both the total linear momentum and the total kinetic energy of the system are conserved, with no conversion of kinetic energy into other forms such as heat, sound, or deformation.[1] This idealized process assumes no dissipative forces act during the interaction, allowing the objects to separate with the same total kinetic energy they possessed before colliding.[2] In elastic collisions, the conservation laws enable precise predictions of post-collision velocities, particularly in one-dimensional scenarios involving two objects.[3] For instance, the final velocities v_1' and v_2' of two masses m_1 and m_2 with initial velocities v_1 and v_2 can be derived from the equations m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' (momentum conservation) and \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 (v_1')^2 + \frac{1}{2} m_2 (v_2')^2 (kinetic energy conservation).[2] Unlike inelastic collisions, where kinetic energy is not conserved, elastic collisions maintain the relative speed of approach and separation between the objects in their center-of-mass frame.[1] Elastic collisions are rare in macroscopic systems due to inevitable energy losses but are fundamental in microscopic and idealized contexts, such as particle physics and gas kinetics.[3] Examples include electron-nucleus interactions, collisions in ideal gases, and Rutherford scattering experiments, where subatomic particles behave nearly elastically.[1] Practical approximations occur in laboratory setups like carts with spring bumpers on air tracks or steel balls in Newton's cradle, and applications extend to spacecraft slingshot maneuvers using planetary gravity for velocity boosts without energy loss.[2]Basic Concepts
Definition and Characteristics
An elastic collision is defined as an interaction between two or more bodies in which both the total kinetic energy and the total momentum of the system remain conserved before and after the collision.[1] This conservation implies that no net energy is dissipated into other forms during the process.[2] Key characteristics of elastic collisions include the absence of permanent deformation in the colliding bodies and no conversion of kinetic energy into heat, sound, or other irreversible losses.[4] In the center-of-mass frame, the bodies rebound such that the magnitude of their relative velocity after the collision equals that before, preserving the total speed of approach and separation.[5] These properties make elastic collisions an idealization in classical mechanics, particularly suitable for modeling perfectly reversible interactions, such as those between atoms or molecules in gases where energy dissipation is minimal.[3] The concept was formalized in the 17th century within classical mechanics by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica and contemporaries like Christiaan Huygens, who analyzed collisions to establish foundational principles contrasting with typical inelastic events observed in daily life.[6] Intuitive approximations of elastic collisions include the impact of billiard balls on a frictionless table or the high rebound of a superball dropped from height, where kinetic energy is largely preserved.[7]Conservation Laws Involved
In elastic collisions, the conservation of linear momentum is a fundamental principle, stating that the total vector momentum of the system remains unchanged before and after the interaction. This arises from Newton's third law of motion, which dictates that the forces exerted between the colliding objects are equal and opposite, resulting in no net change in the system's overall momentum when external influences are absent.[8][9] Momentum, denoted as a vector quantity \vec{p} = m \vec{v} where m is mass and \vec{v} is velocity, ensures that for a system of particles, \sum \vec{p}_i = \sum \vec{p}'_i, with primes indicating post-collision values.[10] Complementing momentum conservation, elastic collisions also preserve the total translational kinetic energy of the system, meaning the sum of \frac{1}{2} m v^2 terms for all objects remains equal before and after the collision. This conservation distinguishes elastic processes, where no kinetic energy is converted into other forms such as heat, sound, or internal deformation energy, unlike in inelastic scenarios.[8][10] Kinetic energy is a scalar quantity, and its invariance in elastic collisions implies that the objects rebound without dissipating mechanical energy through non-conservative internal forces.[11] These conservation laws apply under specific prerequisites, primarily that the system must be isolated, with the vector sum of external forces acting on it being zero to prevent any net impulse that could alter momentum.[12] During the brief duration of the collision, external forces like gravity or friction are often negligible compared to the strong internal impulses between the objects, allowing the approximations to hold effectively.[7][8] Mathematically, momentum conservation is expressed in vector form to account for directionality in multi-dimensional cases, while kinetic energy uses the scalar sum of individual contributions, providing the dual constraints necessary to fully characterize elastic interactions without additional parameters.[10][8]Distinction from Inelastic Collisions
In inelastic collisions, the total linear momentum of the system is conserved, but kinetic energy is not, as some of it is transformed into other forms such as internal energy from deformation, heat, or sound.[13] This category encompasses a range of outcomes, including perfectly inelastic collisions in which the colliding bodies adhere to each other after impact, leading to the maximum possible dissipation of kinetic energy while still conserving momentum.[14] The degree of elasticity in collisions is quantified by the coefficient of restitution, e, which is the ratio of the magnitude of the relative velocity of separation to the relative velocity of approach along the line of impact.[15] Perfectly elastic collisions have e = 1, perfectly inelastic collisions have e = 0, and general inelastic collisions fall in the range $0 < e < 1.[16] In practice, perfectly elastic collisions are rare, as most real-world interactions involve some energy loss due to factors like friction and material deformation.[1] The outcomes of elastic and inelastic collisions differ markedly in terms of post-collision motion: elastic collisions permit full rebound, potentially with reversal of velocity components, whereas inelastic collisions result in either adhesion or reduced rebound speeds.[17] Inelastic collisions predominate in macroscopic scenarios, such as car crashes, where vehicles deform significantly upon impact.[18]One-Dimensional Newtonian Elastic Collisions
Velocity Formulas
In one-dimensional elastic collisions under Newtonian mechanics, the final velocities of two colliding particles are determined by the conservation of both linear momentum and kinetic energy. These collisions are assumed to be head-on, involving point particles or rigid bodies with no rotational effects, and occur at non-relativistic speeds where relativistic corrections are negligible.[7] The notation used here denotes initial velocities as v_1 and v_2 for particles of masses m_1 and m_2, respectively, with post-collision velocities marked by primes: v_1' and v_2'.[7] The general formulas for the final velocities are: v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2 m_2}{m_1 + m_2} v_2 v_2' = \frac{2 m_1}{m_1 + m_2} v_1 + \frac{m_2 - m_1}{m_1 + m_2} v_2 These expressions apply to any initial velocities in a one-dimensional setup.[7] For the special case of equal masses (m_1 = m_2), the formulas simplify such that the particles exchange velocities: v_1' = v_2 and v_2' = v_1. This velocity exchange occurs in head-on collisions between identical masses.[19] When the second particle is initially stationary (v_2 = 0), the final velocity of the first particle becomes v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1, while the second particle acquires v_2' = \frac{2 m_1}{m_1 + m_2} v_1. In the subcase of equal masses and a stationary target, the incident particle stops (v_1' = 0), and the target moves with the initial velocity of the incident particle (v_2' = v_1).[7][20]Derivation from Conservation Principles
Consider two particles of masses m_1 and m_2 undergoing a one-dimensional elastic collision, with initial velocities v_1 and v_2, and final velocities v_1' and v_2', respectively. The conservation of linear momentum gives the equationm_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'.
This follows from Newton's third law and the absence of external forces in the direction of motion. The conservation of kinetic energy for an elastic collision yields
\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 (v_1')^2 + \frac{1}{2} m_2 (v_2')^2.
Multiplying through by 2 simplifies it to
m_1 v_1^2 + m_2 v_2^2 = m_1 (v_1')^2 + m_2 (v_2')^2.
This holds because no kinetic energy is dissipated in an elastic collision. To solve algebraically, first express v_2' from the momentum equation:
v_2' = \frac{m_1 (v_1 - v_1') + m_2 v_2}{m_2}. [21] Substitute this into the kinetic energy equation:
m_1 v_1^2 + m_2 v_2^2 = m_1 (v_1')^2 + m_2 \left( \frac{m_1 (v_1 - v_1') + m_2 v_2}{m_2} \right)^2. Let P = m_1 v_1 + m_2 v_2 denote the total initial momentum (conserved). Then v_2' = (P - m_1 v_1') / m_2, and the substitution becomes
m_1 v_1^2 + m_2 v_2^2 = m_1 (v_1')^2 + \frac{(P - m_1 v_1')^2}{m_2}. Multiplying both sides by m_2 to clear the denominator gives
m_2 (m_1 v_1^2 + m_2 v_2^2) = m_1 m_2 (v_1')^2 + (P - m_1 v_1')^2. Expanding the squared term yields
(P - m_1 v_1')^2 = P^2 - 2 P m_1 v_1' + m_1^2 (v_1')^2. The full equation is now
m_2 (m_1 v_1^2 + m_2 v_2^2) = m_1 m_2 (v_1')^2 + P^2 - 2 P m_1 v_1' + m_1^2 (v_1')^2. Rearranging all terms to one side results in a quadratic equation in v_1':
(m_1^2 + m_1 m_2) (v_1')^2 - 2 P m_1 v_1' + P^2 - m_2 (m_1 v_1^2 + m_2 v_2^2) = 0. This is of the form a (v_1')^2 + b v_1' + c = 0, where a = m_1 (m_1 + m_2), b = -2 P m_1, and c = P^2 - m_1 m_2 v_1^2 - m_2^2 v_2^2. Solving using the quadratic formula v_1' = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} produces two solutions: one corresponding to no collision (v_1' = v_1, v_2' = v_2) and the physical post-collision solution.[22] The discriminant simplifies such that the non-trivial solution is
v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2 m_2}{m_1 + m_2} v_2.
Similarly,
v_2' = \frac{2 m_1}{m_1 + m_2} v_1 + \frac{m_2 - m_1}{m_1 + m_2} v_2. [23] To verify, substitute these back into the original momentum and kinetic energy equations. For momentum:
m_1 v_1' + m_2 v_2' = m_1 \left( \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2 m_2}{m_1 + m_2} v_2 \right) + m_2 \left( \frac{2 m_1}{m_1 + m_2} v_1 + \frac{m_2 - m_1}{m_1 + m_2} v_2 \right) = m_1 v_1 + m_2 v_2,
as the coefficients balance to recover the initial momentum. For kinetic energy, the squared terms and cross products similarly confirm equality, ensuring both principles hold.
Coefficient of Restitution Application
The coefficient of restitution, denoted by e, quantifies the elasticity of a collision between two bodies and is defined as the negative ratio of their relative velocity after the collision to the relative velocity before the collision:e = -\frac{v_2' - v_1'}{v_2 - v_1},
where v_1 and v_2 are the pre-collision velocities of the first and second bodies, respectively, and v_1' and v_2' are the post-collision velocities. The negative sign ensures e is positive by accounting for the reversal in the direction of relative motion during separation.[24][21] In elastic collisions, e = 1, indicating that the magnitude of the relative velocity after collision equals that before, but with reversed direction. This exact reversal, combined with conservation of linear momentum, leads to full conservation of kinetic energy. Substituting the post-collision velocities derived from momentum and energy conservation into the definition of e confirms that it equals 1 for such cases.[24][25] The requirement e = 1 is mathematically equivalent to kinetic energy conservation, provided linear momentum is conserved during the collision. For inelastic collisions, where $0 < e < 1, kinetic energy is not conserved, and the post-collision velocities incorporate e to account for energy dissipation; for instance, the velocity of the first body is given by
v_1' = \frac{m_1 - e m_2}{m_1 + m_2} v_1 + \frac{(1 + e) m_2}{m_1 + m_2} v_2,
assuming the second body has initial velocity v_2.[24][21] This parameter was first introduced by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (1687) as part of his experimental law of impacts and remains a fundamental tool in modern impact mechanics for analyzing and predicting collision outcomes across engineering and physics applications.[26][27]