8.5 Inelastic Collisions in One Dimension

Inelastic Collisions

Elastic vs inelastic collisions

The difference between elastic and inelastic collisions comes down to one question: what happens to kinetic energy during the collision?

• Inelastic collisions do not conserve kinetic energy. Total energy is still conserved (it always is), but some kinetic energy converts to heat, sound, or permanent deformation of the objects. The objects may crumple, stick together, or change shape. Car crashes and clay balls smashing together are classic examples.
• Elastic collisions conserve kinetic energy. The total kinetic energy before and after the collision stays the same, and the objects bounce off each other without permanent deformation. Billiard ball collisions and collisions between atomic particles come close to this ideal.

Both types of collision conserve momentum. That's the unifying principle. The distinction is purely about what happens to kinetic energy.

Characteristics of perfectly inelastic collisions

A perfectly inelastic collision is the extreme case: the colliding objects stick together and move as one unit afterward. Think of a bullet embedding in a wooden block, or two football players tackling and falling together.

• This type of collision loses the maximum possible kinetic energy while still conserving momentum. No other collision type loses more.
• Conservation of momentum still holds:

$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$

where $v_f$ is the final velocity of the combined object.

• The velocity of the system's center of mass doesn't change during the collision. Before, during, and after impact, the center of mass moves at the same velocity.

Calculations for inelastic collisions

Finding the final velocity: Rearrange the momentum conservation equation to solve for $v_f$:

$v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$

This tells you both the speed and direction of the combined object after the collision. If $v_f$ comes out negative, the combined object moves in the direction you defined as negative.

Example: A 10 kg cart moving at 3 m/s collides with and sticks to a 5 kg cart initially at rest.

1. Write the momentum equation: $(10)(3) + (5)(0) = (10 + 5) v_f$
2. Solve: $30 = 15 v_f$, so $v_f = 2 \text{ m/s}$

Finding kinetic energy lost: Calculate the kinetic energy before and after, then take the difference:

$\Delta KE = \left(\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2\right) - \frac{1}{2}(m_1 + m_2) v_f^2$

Using the same example:

1. KE before: $\frac{1}{2}(10)(3)^2 + 0 = 45 \text{ J}$
2. KE after: $\frac{1}{2}(15)(2)^2 = 30 \text{ J}$
3. Energy lost: $45 - 30 = 15 \text{ J}$

That 15 J went into deformation, heat, and sound during the collision.

Real-world applications of collisions

Car crashes are inelastic collisions by design. Vehicles include crumple zones that deform on purpose, absorbing kinetic energy so that less force is transmitted to passengers. Seat belts and airbags extend the time of the collision, reducing the peak force (this connects to impulse, covered below).

Sports collisions fall somewhere between perfectly elastic and perfectly inelastic. When a baseball bat hits a ball, both compress briefly on impact, and some kinetic energy is lost to deformation and vibration. The coefficient of restitution (a number between 0 and 1) measures how "bouncy" a collision is. A tennis ball on a hard court has a high coefficient; a lump of clay has a coefficient near zero.

Ballistics problems often model a bullet embedding in a target as a perfectly inelastic collision. If you know the bullet's mass and speed, plus the target's mass, you can find the final velocity of the bullet-target system and figure out how much energy was deposited into the target.

Impulse and momentum transfer

Impulse connects force and momentum. It's defined as the product of the average force and the time interval over which it acts:

$J = F \Delta t = \Delta p$

This means the impulse on an object equals its change in momentum. A large force over a short time produces the same momentum change as a smaller force over a longer time.

In any collision, momentum transfers between the objects, but the total system momentum stays constant. Objects with greater mass (more inertia) experience smaller velocity changes for the same impulse, which is why a heavy truck barely slows down when it collides with a small car, while the car's velocity changes dramatically.