2.4.4.2. Inelastic Collisions

Table of Contents

What makes a collision inelastic

When two objects collide, momentum is still a central idea, but not all collisions behave the same way with energy. An inelastic collision is a collision in which the total kinetic energy of the system is not conserved. Some of the initial kinetic energy is transformed into other forms, such as heat, sound, deformation, or internal vibration.

This does not mean energy disappears. Total energy is always conserved. What changes is that kinetic energy, the energy of motion, is partly converted into non mechanical forms.

In an inelastic collision, total momentum is conserved if external forces are negligible, but total kinetic energy is not conserved.

A simple example is a lump of clay hitting a wall or two soft carts bumping and slightly deforming. After the collision, the objects may move more slowly, or one may stop, because some of the motion energy has gone into changing shape or warming the objects.

Momentum and kinetic energy in inelastic collisions

To understand inelastic collisions, it is important to separate two different conservation ideas.

Momentum is a vector quantity, so its direction matters. If the system is isolated during the short collision, then

$$
\vec{p}_{\text{before}} = \vec{p}_{\text{after}}
$$

For two objects, this becomes

$$
m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}
$$

Kinetic energy is different. In an inelastic collision,

$$
K_{\text{after}} < K_{\text{before}}
$$

where

$$
K = \frac{1}{2}mv^2
$$

The missing kinetic energy has been converted into other forms of energy.

Do not use conservation of kinetic energy for an inelastic collision. Use conservation of momentum, then interpret the kinetic energy change separately.

Common physical signs of inelastic behavior

In real life, many collisions are at least partly inelastic. Signs of inelastic behavior include visible deformation, a sound during impact, heating, and objects sticking or bouncing less than expected.

If a tennis ball strikes a surface and rebounds with a smaller speed than it had before impact, then some kinetic energy was lost from the ball surface system. If a car bumper crumples in a crash, the crumpling absorbs energy and reduces the kinetic energy of the cars.

This loss of kinetic energy is often useful. Safety devices such as helmets, crumple zones, and padding are designed to make collisions more inelastic so that energy is transferred into deformation instead of remaining in dangerous motion.

Comparison with elastic collisions

An elastic collision is an ideal case in which both momentum and kinetic energy are conserved. An inelastic collision is more general and more common in everyday life.

Feature	Elastic collision	Inelastic collision
Momentum conserved	Yes	Yes, if isolated
Kinetic energy conserved	Yes	No
Deformation possible	Usually negligible	Often present
Heat and sound produced	Small	Often noticeable

An inelastic collision can still involve bouncing. The objects do not need to stick together. Sticking together is a special case, covered separately as a perfectly inelastic collision.

Kinetic energy loss

A useful quantity is the change in kinetic energy:

$$
\Delta K = K_{\text{after}} - K_{\text{before}}
$$

For an inelastic collision, this value is negative.

If two objects move along one line, then

$$
K_{\text{before}} = \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2
$$

and

$$
K_{\text{after}} = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2
$$

The difference between these tells us how much kinetic energy was converted into other forms.

For an isolated system in an inelastic collision,
$$
m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}
$$
but generally
$$
\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 \neq \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2
$$
and usually
$$
K_{\text{after}} < K_{\text{before}}
$$

A simple one dimensional example

Suppose a cart of mass $2\,\text{kg}$ moves at $4\,\text{m/s}$ and collides with a cart of mass $1\,\text{kg}$ moving at $1\,\text{m/s}$ in the same direction. After the collision, imagine the first cart slows to $2\,\text{m/s}$ and the second speeds up to $5\,\text{m/s}$.

Before the collision, the total momentum is

$$
p_{\text{before}} = (2)(4) + (1)(1) = 9\,\text{kg m/s}
$$

After the collision, the total momentum is

$$
p_{\text{after}} = (2)(2) + (1)(5) = 9\,\text{kg m/s}
$$

So momentum is conserved.

Now compare kinetic energy. Before the collision,

$$
K_{\text{before}} = \frac{1}{2}(2)(4^2) + \frac{1}{2}(1)(1^2) = 16 + 0.5 = 16.5\,\text{J}
$$

After the collision,

$$
K_{\text{after}} = \frac{1}{2}(2)(2^2) + \frac{1}{2}(1)(5^2) = 4 + 12.5 = 16.5\,\text{J}
$$

In this numerical case, kinetic energy happens to be unchanged, so this specific example is actually elastic, not inelastic. That makes it a useful warning. Conserving momentum alone is not enough to decide the type of collision. We must also check kinetic energy.

Now change the final speeds. Suppose after collision the carts move at $3\,\text{m/s}$ and $3\,\text{m/s}$.

Then

$$
p_{\text{after}} = (2)(3) + (1)(3) = 9\,\text{kg m/s}
$$

Momentum is still conserved. But now

$$
K_{\text{after}} = \frac{1}{2}(2)(3^2) + \frac{1}{2}(1)(3^2) = 9 + 4.5 = 13.5\,\text{J}
$$

Since

$$
13.5\,\text{J} < 16.5\,\text{J}
$$

this collision is inelastic, and $3.0\,\text{J}$ of kinetic energy has been transformed into other forms.

Visualizing an inelastic collision

Rendering scientific sketch…

Before and after an inelastic collision

The picture shows that the objects continue moving after impact, but with a different distribution of speeds. The total momentum is the same before and after, while the total kinetic energy is smaller after the collision.

Why inelastic collisions matter

Inelastic collisions are important because most real collisions are not perfectly elastic. Materials bend, compress, vibrate, and heat up. Understanding this helps explain why colliding objects often do not rebound with their original speeds.

This idea is also essential in engineering and safety. A collision that reduces kinetic energy by deformation can lower the force transmitted to passengers or structures. In sports, construction, vehicle design, and materials science, controlled inelastic behavior is often desirable.

Key idea to remember

The defining feature of an inelastic collision is not whether the objects stick together, but whether kinetic energy decreases.

An inelastic collision is any collision for which momentum is conserved, in an isolated system, but kinetic energy is not conserved.

When solving problems, first apply momentum conservation. Then compare kinetic energies before and after to determine how much kinetic energy was transformed into other forms.

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