In an elastic collision, total momentum and total kinetic energy are conserved. What this tells us is, for an isolated system, the combined momentum before the collision equals the combined momentum after the collision, and the total kinetic energy of the objects also remains the same. The phrase “in an elastic collision what is conserved” usually points to these two key ideas: conservation of momentum and conservation of kinetic energy Less friction, more output..
Introduction
An elastic collision is a special type of collision in physics where objects bounce off each other without losing kinetic energy to heat, sound, permanent deformation, or other forms of internal energy. In real life, perfectly elastic collisions are rare, but they are useful in physics because they help explain the motion of objects such as gas molecules, billiard balls, and atoms under ideal conditions.
When studying collisions, students often ask: what exactly stays the same before and after the collision? The answer depends on the type of collision. In an elastic collision, the most important conserved quantities are:
- Total linear momentum
- Total kinetic energy
- Total energy of the system
- Angular momentum, if no external torque acts on the system
The first two are the defining features of an elastic collision Not complicated — just consistent. Turns out it matters..
What Is an Elastic Collision?
An elastic collision is a collision in which the total kinetic energy of the system is the same before and after the collision.
Before collision:
[ K_{\text{initial}} = K_{\text{final}} ]
After collision:
[ \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 ]
Where:
- (m_1) and (m_2) are the masses of the two objects
- (u_1) and (u_2) are their initial velocities
- (v_1) and (v_2) are their final velocities
In simple terms, the objects may change direction or speed, but the total amount of kinetic energy in the system does not decrease.
As an example, if two identical balls collide head-on elastically, one ball may stop while the other moves away with the same speed. The kinetic energy has not disappeared; it has simply been transferred from one object to another.
Momentum Is Conserved in an Elastic Collision
The first major quantity conserved in an elastic collision is linear momentum.
Momentum is defined as:
[ p = mv ]
Where:
- (p) is momentum
- (m) is mass
- (v) is velocity
For two objects colliding, conservation of momentum means:
[ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 ]
This equation works because the forces between the colliding objects are internal forces. According to Newton’s third law, the force that object 1 exerts on object 2 is equal in magnitude and opposite in direction to the force that object 2 exerts on object 1 Surprisingly effective..
Because these internal forces cancel out within the system, the total momentum does not change, assuming there is no net external force acting on the system.
This is true not only for elastic collisions but also for inelastic collisions. Put another way, momentum is conserved in both elastic and inelastic collisions, as long as the system is isolated Practical, not theoretical..
Kinetic Energy Is Conserved in an Elastic Collision
The second major quantity conserved in an elastic collision is kinetic energy.
Kinetic energy is the energy an object has because of its motion:
[ K = \frac{1}{2}mv^2 ]
In an elastic collision:
[ K_{\text{initial}} = K_{\text{final}} ]
This is what makes an elastic collision different from an inelastic collision. In an inelastic collision, some kinetic energy is transformed into other
Kinetic Energy Is Conserved in an Elastic Collision
The second major quantity conserved in an elastic collision is kinetic energy.
For two objects moving along a straight line, the condition of energy conservation reads
[ \frac12 m_1u_1^{2}+\frac12 m_2u_2^{2} ;=; \frac12 m_1v_1^{2}+\frac12 m_2v_2^{2}; . ]
When the masses and velocities are known, this single scalar equation, together with the momentum conservation equation derived earlier, is sufficient to solve for the two unknown final velocities (v_1) and (v_2). The algebraic solution yields the familiar “exchange of velocities” result for equal masses, but the general formulas are
[ \begin{aligned} v_1 &= \frac{(m_1-m_2)u_1 + 2m_2u_2}{m_1+m_2},\[4pt] v_2 &= \frac{(m_2-m_1)u_2 + 2m_1u_1}{m_1+m_2}. \end{aligned} ]
These expressions automatically satisfy both momentum and kinetic‑energy constraints, guaranteeing that the collision is truly elastic Simple, but easy to overlook..
One‑Dimensional vs. Multi‑Dimensional Cases
In one dimension the direction of motion is fixed, so the signs of the velocities already encode any change of direction. In two or three dimensions the vectors must be treated with care: the conservation equations become vector equations for momentum and scalar equations for kinetic energy. The vector nature of momentum allows the post‑collision directions to be determined by the geometry of the impact, while the scalar energy equation still forces the magnitudes of the velocities to satisfy the same energy budget. Because of this, an elastic collision in higher dimensions can involve deflections at arbitrary angles, as seen in billiard‑ball collisions.
Coefficient of Restitution
Real‑world collisions rarely meet the ideal elastic condition exactly. To quantify how “elastic’’ a collision is, physicists introduce the coefficient of restitution (e), defined as the ratio of relative speed after impact to relative speed before impact along the line of impact:
[ e = \frac{(v_2 - v_1)\cdot\hat{n}}{(u_1 - u_2)\cdot\hat{n}}, ]
where (\hat{n}) is a unit vector pointing along the contact normal. For a perfectly elastic impact (e = 1); for a completely inelastic one (e = 0). The algebraic treatment of an elastic collision corresponds to setting (e = 1) and solving the resulting system of equations.
Total Energy of the System
While kinetic energy is the portion of energy that is readily observable as motion, the total energy of a mechanical system also contains internal forms:
- Potential energy associated with the positions of the particles (gravitational, elastic, electrostatic, etc.).
- Internal energy arising from microscopic motions—vibrational, rotational, and translational modes of the constituent atoms or molecules.
- Thermal energy, which is essentially the random component of internal energy.
In an idealized, perfectly isolated mechanical system, the sum
[E_{\text{total}} = K + U + E_{\text{internal}} ]
remains constant. In an elastic collision the kinetic part of this total is conserved, but the internal energy may still change if microscopic degrees of freedom are excited (e.g., lattice vibrations in a solid). In the simplest textbook model, however, we assume that the objects are rigid, structureless point masses, so no internal energy is created or destroyed, and the entire energy budget reduces to the conservation of kinetic energy Simple, but easy to overlook. Practical, not theoretical..
And yeah — that's actually more nuanced than it sounds.
Angular Momentum Conservation
When the colliding bodies are not merely point masses but have finite size and possibly rotational degrees of freedom, another quantity becomes relevant: angular momentum. If the system is isolated and no external torques act about a chosen origin, the total angular momentum vector (\mathbf{L}) is conserved:
[ \mathbf{L}{\text{initial}} = \mathbf{L}{\text{final}}. ]
For two point particles, the angular momentum about a point (O) is[ \mathbf{L} = \sum_i \mathbf{r}_i \times m_i \mathbf{v}_i, ]
where (\mathbf{r}_i) is the position vector of particle (i) relative to (O). In a head‑on collision along the line joining the centers, the external torque about the center of mass is zero, so (\mathbf{L}) remains unchanged. In off‑center (glancing) collisions, the impact can transfer some of the linear momentum into rotational motion, altering the translational velocities while preserving the combined translational + rotational angular momentum The details matter here..
The conservation of angular momentum is especially important when analyzing collisions involving extended bodies such as spinning tops, rotating cylinders, or planetary encounters. It provides an additional constraint that, together with momentum and energy conservation, uniquely determines the outcome of the interaction.
Putting It All Together
An elastic collision is defined by two simultaneous, strict conservation laws:
- Linear momentum remains unchanged because internal forces
The impulse delivered duringthe brief contact period is equal to the change in each particle’s linear momentum, (\Delta\mathbf{p}= \mathbf{J}), where (\mathbf{J}) is the integral of the contact force over time. Because the forces are internal, the impulse on one body is exactly opposite to the impulse on the other, guaranteeing that the vector sum of all momenta before the encounter equals the vector sum after it. This invariance holds irrespective of the direction of the impact, the magnitude of the forces, or whether the bodies are spherical, elongated, or irregularly shaped; the only requirement is that no external torque or force acts about the chosen reference point Most people skip this — try not to. But it adds up..
When the collision is not perfectly head‑on, the line of impact may be offset from the center of mass, producing a couple that can initiate rotation. In such cases the change in linear momentum of each body is still matched by an opposite change in the other, while the simultaneous exchange of angular momentum ensures that the total moment about the reference point stays constant. The combination of these two vector conservations—linear and angular—provides a complete kinematic description of the post‑collision states for extended bodies And that's really what it comes down to..
In practical applications, engineers often introduce a coefficient of restitution (e) to quantify how “bouncy” a collision is. For a one‑dimensional impact, (e) relates the relative speed of separation to the relative speed of approach:
[ e = -\frac{(\mathbf{v}_2' - \mathbf{v}_1')\cdot\hat{\mathbf{n}}} {(\mathbf{v}_2 - \mathbf{v}_1)\cdot\hat{\mathbf{n}}}. ]
When (e = 1) the collision is perfectly elastic, kinetic energy is conserved, and the velocities follow the analytic solutions derived from momentum and energy conservation. So for (0 < e < 1) the interaction dissipates some kinetic energy into internal modes (heat, sound, deformation), yet the momentum and angular momentum remain strictly conserved. When (e = 0) the bodies stick together, forming a perfectly inelastic collision; the final combined mass moves with a velocity that satisfies momentum conservation alone Small thing, real impact..
It sounds simple, but the gap is usually here.
Thus, the physics of elastic collisions can be summarized as follows:
- Linear momentum is conserved because internal forces occur in equal and opposite pairs, leaving the total vector momentum unchanged.
- Kinetic energy may be fully conserved (ideal elastic case) or partially diverted to internal energy in real‑world impacts.
- Angular momentum is conserved when no external torque acts, allowing the distribution between translational and rotational motion to shift while the total remains fixed.
- The coefficient of restitution provides a convenient parameter to bridge the idealized elastic limit and practical, partially inelastic scenarios.
All in all, the rigorous framework of conservation laws—linear momentum, angular momentum, and, when appropriate, kinetic energy—offers a universal language for predicting the outcomes of collisions across scales, from subatomic particle scattering to celestial mechanics. On the flip side, by applying these principles, one can accurately forecast the trajectories, rotations, and energy distributions that emerge from any impact, thereby linking microscopic interactions to macroscopic observable phenomena. This integrated understanding underscores why conservation laws are not merely abstract statements but indispensable tools for engineers, physicists, and scientists seeking to model and manipulate the dynamics of the physical world.
