KAHIBARO
Table of Contents
Sticking Together After Impact
A perfectly inelastic collision is a collision in which the objects stick together after they collide and then move as a single combined object. This is the defining feature of this type of collision. The objects may be carts with Velcro, lumps of clay, or train cars that lock together on impact.
The key idea is that although the objects stick together, the total momentum of the system is still conserved if external forces are negligible during the short collision time. What changes is the kinetic energy. In a perfectly inelastic collision, the loss of kinetic energy is as large as possible among collisions where the objects still satisfy momentum conservation.
For a perfectly inelastic collision, the objects stick together after impact.
Momentum is conserved:
$$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$$
Kinetic energy is not conserved:
$$K_i \neq K_f$$
Final Common Velocity
Suppose two objects of masses $m_1$ and $m_2$ move initially with velocities $u_1$ and $u_2$ along one straight line. After the collision they stick together and move with a common final velocity $v$.
Using conservation of momentum,
$$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$$
so the final velocity is
$$v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$
This formula shows that the final velocity is a mass weighted average of the initial velocities.
In a perfectly inelastic collision, both objects have the same final velocity:
$$v_1' = v_2' = v$$
What Happens to Kinetic Energy
Before the collision, the total kinetic energy is
$$K_i = \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$$
After the collision, since the two objects move together,
$$K_f = \frac{1}{2}(m_1 + m_2)v^2$$
Usually,
$$K_f < K_i$$
The missing kinetic energy is transformed into other forms of energy, such as heat, sound, deformation, or internal energy of the objects.
This does not mean energy is destroyed. Total energy is always conserved, but mechanical kinetic energy is not.
A Useful Expression for Energy Loss
For motion in one dimension, the kinetic energy lost in a perfectly inelastic collision can be written in a very useful form:
$$\Delta K = K_f - K_i = -\frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(u_1 - u_2)^2$$
Since the square term is always nonnegative, this expression is always zero or negative. That means the final kinetic energy can never exceed the initial kinetic energy in a perfectly inelastic collision.
If we want the amount of kinetic energy lost as a positive quantity, we write
$$K_i - K_f = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(u_1 - u_2)^2$$
The kinetic energy lost in a perfectly inelastic collision is
$$K_i - K_f = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(u_1 - u_2)^2$$
This loss depends on the relative speed $|u_1 - u_2|$.
Simple Example
Imagine a cart of mass $2\,\text{kg}$ moving at $4\,\text{m/s}$ collides with a stationary cart of mass $3\,\text{kg}$. They stick together.
Using momentum conservation,
$$v = \frac{(2)(4) + (3)(0)}{2 + 3} = \frac{8}{5} = 1.6\,\text{m/s}$$
The initial kinetic energy is
$$K_i = \frac{1}{2}(2)(4^2) + \frac{1}{2}(3)(0^2) = 16\,\text{J}$$
The final kinetic energy is
$$K_f = \frac{1}{2}(5)(1.6^2) = 6.4\,\text{J}$$
So the kinetic energy lost is
$$16 - 6.4 = 9.6\,\text{J}$$
The carts move together, and a significant part of the original kinetic energy has been converted into other forms.
Comparison with Other Collisions
It helps to compare perfectly inelastic collisions with other common cases.
| Collision type | Momentum conserved | Kinetic energy conserved | Objects stick together |
|---|---|---|---|
| Elastic | Yes | Yes | No |
| Inelastic | Yes | No | Not necessarily |
| Perfectly inelastic | Yes | No, maximum loss | Yes |
A perfectly inelastic collision is a special case of an inelastic collision.
Physical Interpretation
When two objects stick together, their relative motion after impact becomes zero. To make that happen, deformation or internal interaction must absorb energy. Soft materials, clay, putty, and damaged vehicles are common examples where sticking and deformation occur together.
A perfectly inelastic collision often represents an idealized model. Real collisions may come close to this behavior, but exact sticking is a simplified description that makes analysis easier.
Momentum Diagram
Center of Mass View
The final velocity in a perfectly inelastic collision is exactly the velocity of the center of mass before the collision. This is why the combined object continues moving at that value after impact, provided no significant external impulse acts on the system.
This gives a helpful interpretation of the formula
$$v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$
The stuck-together object simply moves with the original center-of-mass velocity.
Special Cases
If one object is initially at rest, $u_2 = 0$, then
$$v = \frac{m_1 u_1}{m_1 + m_2}$$
The combined object moves more slowly than the original moving object because the same momentum is now shared by a larger total mass.
If the two masses are equal and move toward each other with equal speeds, then the total initial momentum is zero. After they stick together, the final velocity is also zero.
$$m u + m(-u) = 0 \Rightarrow v = 0$$
They remain at rest after the collision, while the initial kinetic energy is converted into internal energy.
Main Points to Remember
A perfectly inelastic collision is identified by sticking. The most important equation comes from conservation of momentum, not conservation of kinetic energy. After impact, both objects share one final velocity, and kinetic energy decreases.
To solve a perfectly inelastic collision:
- Write conservation of momentum.
- Set the same final velocity for both objects.
- Do not set initial and final kinetic energies equal.