Inelastic Collision: When Objects Stick and Energy Transforms
The Core Principles of Collisions
In physics, a collision occurs when two or more bodies exert forces on each other for a relatively short time. Think of two bumper cars crashing at the fair or a baseball being hit by a bat. To analyze these events, we rely on two key conservation laws: the conservation of momentum and the conservation of energy. However, the type of collision determines which of these energies is conserved.
The total momentum before a collision equals the total momentum after the collision. This is true for ALL collisions, both elastic and inelastic.
$p_{initial} = p_{final}$
For two objects, this is written as:
$m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
Where $m$ is mass and $v$ is velocity. The "i" stands for initial and "f" for final.
The main difference between collision types lies in what happens to the kinetic energy ($KE$), which is the energy of motion, calculated as $\frac{1}{2}mv^2$.
| Collision Type | Is Momentum Conserved? | Is Kinetic Energy Conserved? | What Happens to the Objects? |
|---|---|---|---|
| Elastic | Yes | Yes | They bounce off each other perfectly. |
| Inelastic | Yes | No | They may stick together or separate with energy loss. |
| Perfectly Inelastic | Yes | No (Maximum loss) | They stick together and move as one object after the collision. |
Diving Deeper: Perfectly Inelastic Collisions
A perfectly inelastic collision is the most extreme and easiest to analyze type of inelastic collision. In this case, the colliding objects stick together after impact and move with a common final velocity. This is where the maximum amount of kinetic energy is lost to other forms of energy.
The conservation of momentum equation for a perfectly inelastic collision simplifies because the final velocities are the same ($v_f$).
$m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f$
You can solve for the final velocity $v_f$:
$v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2}$
Let's see this in action with a simple example. Imagine a 1000 kg car stopped at a red light. A 1500 kg truck, moving at 10 m/s, fails to brake and rear-ends the car, and their bumpers lock, causing them to stick together.
- Mass of car, $m_1 =$ 1000 kg (Initial velocity $v_{1i} =$ 0 m/s)
- Mass of truck, $m_2 =$ 1500 kg (Initial velocity $v_{2i} =$ 10 m/s)
To find their final velocity ($v_f$) as they move together:
$v_f = \frac{(1000 \times 0) + (1500 \times 10)}{1000 + 1500} = \frac{15000}{2500} =$ 6 m/s
They move together at 6 m/s. Now, let's check the kinetic energy.
- Initial $KE = \frac{1}{2}(1000)(0)^2 + \frac{1}{2}(1500)(10)^2 =$ 75,000 J
- Final $KE = \frac{1}{2}(1000+1500)(6)^2 = \frac{1}{2}(2500)(36) =$ 45,000 J
There is a clear loss of 30,000 J of kinetic energy. This energy didn't vanish; it was converted into sound (the crash noise), heat (from friction), and the work done to permanently deform the metal of the vehicles.
Real-World Applications and Examples
Inelastic collisions are everywhere once you know what to look for. They are far more common in everyday life than perfectly elastic collisions.
1. Vehicle Accidents: As shown in the calculation above, car crashes are classic examples of perfectly inelastic collisions. The crumple zones in modern cars are designed to deform in a controlled way during a collision, which increases the time over which the collision occurs. This reduces the force felt by the passengers (from Newton's second law, $F = \frac{\Delta p}{\Delta t}$) while converting the kinetic energy into deformation energy.
2. Ballistic Pendulum: This is a device used historically to measure the speed of a bullet. A bullet is fired into a wooden block. The bullet embeds itself into the block (a perfectly inelastic collision), and the combined system swings upward. By using conservation of momentum for the collision and conservation of energy for the swing, the initial speed of the bullet can be calculated.
3. Sports: When a baseball player catches a ball, the player's hand may give a little, making the collision more inelastic and reducing the force on their hand. Similarly, when a football player tackles an opponent, they often hold on, causing a perfectly inelastic collision to bring the play to a stop.
4. Asteroid Impacts and Planet Formation: In space, when two asteroids collide and stick together, it's a perfectly inelastic collision. This process is thought to be a fundamental mechanism in the early stages of planet formation, as dust and rock particles coalesced into larger bodies.
5. Hammer and Nail: When you hit a nail with a hammer, the hammer nearly comes to a stop, transferring its momentum to the nail. The loss of kinetic energy is used to drive the nail into the wood, overcoming friction and creating sound and heat.
Common Mistakes and Important Questions
Q: In an inelastic collision, is energy conserved or not?
Q: If two objects bounce apart, is the collision automatically elastic?
Q: Why is momentum always conserved in a collision, but kinetic energy is not?
Inelastic collisions are a pervasive and vital part of understanding the physical world. While they may seem more complex because kinetic energy is not conserved, they follow a simple and unwavering rule: momentum is always conserved. From the dramatic example of a car crash to the subtle action of catching a ball, these events show us how energy transforms from one type to another. Grasping the difference between elastic and inelastic collisions provides a powerful tool for analyzing motion, designing safer vehicles, and even understanding the cosmos. Remember, in an inelastic collision, the kinetic energy isn't destroyed; it simply changes its identity.
Footnote
1 KE (Kinetic Energy): The energy possessed by an object due to its motion. It is calculated with the formula $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity.
2 Momentum: A measure of the quantity of motion of a moving body, given by the product of its mass and velocity ($p = mv$). It is a vector quantity, meaning it has both magnitude and direction.
3 Macroscopic: Relating to large-scale objects that are visible to the naked eye, as opposed to microscopic objects like atoms.