KAHIBARO
Table of Contents
Core idea
An elastic collision is a collision in which the total momentum is conserved and the total kinetic energy is also conserved. This makes elastic collisions special, because many collisions conserve momentum, but only some also preserve kinetic energy.
If two objects collide and then separate without losing kinetic energy to heat, sound, deformation, or internal vibration, the collision is elastic. At the beginner level, elastic collisions are often studied for idealized objects such as smooth billiard balls, atoms, or particles.
In an elastic collision, two conservation laws apply at the same time:
$$\text{Total momentum before} = \text{Total momentum after}$$
$$\text{Total kinetic energy before} = \text{Total kinetic energy after}$$
What makes a collision elastic
During any collision, the objects exert large forces on each other for a short time. In an elastic collision, the objects may briefly deform, but they return to their original shape and the stored energy is given back as kinetic energy.
For two objects of masses $m_1$ and $m_2$, with initial velocities $u_1$ and $u_2$, and final velocities $v_1$ and $v_2$, the two key equations are
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
and
$$\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2
=
\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$$
These equations are especially useful in one dimensional collisions.
Relative speed property
A very important shortcut for one dimensional elastic collisions is the relative speed rule. In such a collision, the speed at which the objects approach each other equals the speed at which they separate.
If the motion is along one line, then
$$u_1 - u_2 = -(v_1 - v_2)$$
or equivalently,
$$u_1 + v_1 = u_2 + v_2$$
This relation comes from combining momentum conservation with kinetic energy conservation.
For a one dimensional elastic collision:
$$\text{relative speed of approach} = \text{relative speed of separation}$$
So,
$$|u_1 - u_2| = |v_1 - v_2|$$
Typical results in one dimension
When solving one dimensional elastic collisions, some important patterns appear.
If the second object is initially at rest, so $u_2 = 0$, then the final velocities are
$$v_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1$$
$$v_2 = \frac{2m_1}{m_1 + m_2} u_1$$
These formulas show how the masses control the outcome.
Equal masses
If $m_1 = m_2$ and the second object starts at rest, then
$$v_1 = 0, \qquad v_2 = u_1$$
The first object stops and the second moves away with the original speed. This is the familiar behavior of identical billiard balls in an ideal head on collision.
Very heavy target
If $m_2 \gg m_1$ and the second object is initially at rest, then approximately
$$v_1 \approx -u_1, \qquad v_2 \approx 0$$
The lighter object bounces back, and the heavy object hardly moves.
Very light target
If $m_1 \gg m_2$ and the second object is initially at rest, then approximately
$$v_1 \approx u_1, \qquad v_2 \approx 2u_1$$
The heavy object keeps moving almost unchanged, while the lighter one shoots forward faster.
Visualizing a head on elastic collision
Comparing elastic and inelastic collisions
The key difference is what happens to kinetic energy.
| Type of collision | Momentum conserved | Kinetic energy conserved |
|---|---|---|
| Elastic | Yes | Yes |
| Inelastic | Yes | No |
| Perfectly inelastic | Yes | No, maximum loss consistent with momentum conservation |
In a real world collision, perfect elasticity is rare. Still, many microscopic collisions, such as collisions between gas molecules in simple models, are treated as elastic because that approximation works well.
Why elastic collisions matter
Elastic collisions are important because they provide clean, solvable examples of motion. They appear in atomic physics, gas theory, scattering experiments, and mechanics problems. Since both momentum and kinetic energy are conserved, they give strong mathematical constraints, which often allow the final velocities to be found exactly.
To identify an elastic collision, check both conditions:
$$\sum \vec{p}_{\text{before}} = \sum \vec{p}_{\text{after}}$$
and
$$\sum K_{\text{before}} = \sum K_{\text{after}}$$
If kinetic energy is not conserved, the collision is not elastic.
A simple example
Suppose a $2\,\text{kg}$ object moves at $6\,\text{m/s}$ toward a stationary $1\,\text{kg}$ object. Assume a one dimensional elastic collision.
Using the formulas for $u_2 = 0$,
$$v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1
= \frac{2-1}{2+1}\cdot 6 = 2\,\text{m/s}$$
$$v_2 = \frac{2m_1}{m_1 + m_2}u_1
= \frac{2\cdot 2}{2+1}\cdot 6 = 8\,\text{m/s}$$
So after the collision, the heavier object continues forward more slowly, and the lighter object moves forward faster.
We can check momentum:
$$p_{\text{before}} = 2\cdot 6 + 1\cdot 0 = 12$$
$$p_{\text{after}} = 2\cdot 2 + 1\cdot 8 = 12$$
We can also check kinetic energy:
$$K_{\text{before}} = \frac{1}{2}(2)(6^2) = 36\,\text{J}$$
$$K_{\text{after}} = \frac{1}{2}(2)(2^2) + \frac{1}{2}(1)(8^2) = 4 + 32 = 36\,\text{J}$$
Both are conserved, so the collision is elastic.
Final observation
Elastic collisions are the ideal case where motion changes, but total kinetic energy does not. In one dimension, they are governed by conservation of momentum together with conservation of kinetic energy, or equivalently by the relative speed rule. These ideas make elastic collisions one of the most elegant and useful topics in classical mechanics.