Elastic and Inelastic Collisions
Why this matters
Imagine you're at a bowling alley. You roll the ball, it strikes the pins, and there's a huge crash. Pins fly everywhere. Now, picture a game of pool. The cue ball silently strikes the 8-ball. They click, and both roll away smoothly.
Both scenarios are collisions. But they feel completely different, don't they? One is loud and messy, the other is clean and quiet. In physics, we have a precise way to describe this difference. It all comes down to energy.
Today, we're going to learn how to classify any collision as either elastic or inelastic. We'll explore what happens to the kinetic energy in each case and why momentum is the one rule that always holds true.
Concept overview
flowchart TD
A[Collision Occurs in a Closed System] --> B{Is Momentum Conserved?};
B -- Yes, Always --> C{Is Kinetic Energy Conserved?};
C -- Yes --> D[Elastic Collision];
D -- Example --> E[Ideal Billiard Balls];
C -- No --> F[Inelastic Collision];
F -- KE is transformed --> G[into heat, sound, deformation];
F --> H{Do objects stick together?};
H -- Yes --> I[Perfectly Inelastic Collision];
I -- Example --> J[Clay balls sticking together];
H -- No --> K[Inelastic Collision];
K -- Example --> L[A car crash where cars bounce apart];
Core explanation
When we analyze collisions in physics, we have one unshakable foundation: conservation of momentum. In a closed system (meaning no external forces like friction are interfering), the total momentum of all objects before the collision is equal to the total momentum of all objects after the collision.
p_initial_total = p_final_total
This is our starting point for every single collision problem. But to truly understand what happened, we need to ask a second question: What happened to the kinetic energy?
The answer to that question is how we classify collisions into two main families: elastic and inelastic.
Elastic Collisions: The Perfect Bounce
An elastic collision is one where the total kinetic energy of the system is also conserved.
KE_initial_total = KE_final_total
Think of it as a "perfectly bouncy" collision. The classic example is two billiard balls hitting each other. They click, and bounce off without any loss of speed due to deformation. No energy is wasted on making a loud crunching sound or generating heat. All the initial kinetic energy is returned as final kinetic energy.
Now, this is where a common point of confusion comes up. Does this mean each individual object has the same kinetic energy it started with? No, not at all.
Imagine a moving cue ball hitting a stationary 8-ball. Before the collision, the cue ball has all the kinetic energy, and the 8-ball has zero. After they collide, the cue ball might stop completely, transferring all its kinetic energy to the 8-ball, which now rolls away. The kinetic energy of each ball changed dramatically, but the total sum of their kinetic energies before and after the impact remained the same.
In the real world, perfectly elastic collisions are rare. Even billiard balls make a small sound and generate a tiny bit of heat. But for many AP Physics problems, we'll treat collisions between hard, bouncy objects (like atoms or ideal billiard balls) as elastic.
Inelastic Collisions: The Real World
An inelastic collision is one where the total kinetic energy of the system decreases.
KE_initial_total > KE_final_total
This is far more common in our everyday lives. Think about a car crash. When two cars collide, you hear a loud noise, see sparks (light and heat), and the metal bodies get bent and deformed. All of these things—sound, heat, light, and bending metal—require energy. Where did that energy come from? It was stolen from the initial kinetic energy of the cars.
Because some of the kinetic energy was transformed into other forms, the total kinetic energy of the cars after the crash is less than it was before. The collision is inelastic.
Perfectly Inelastic Collisions: Maximum Stickiness
This is a special, extreme case of an inelastic collision. A perfectly inelastic collision is one where the objects stick together after colliding and move as a single mass with a common final velocity.
This type of collision involves the maximum possible loss of kinetic energy. It's as "un-bouncy" as you can get.
Imagine two balls of wet clay flying towards each other. They collide, squish, and stick together, moving as one combined lump. Or think of a football running back tackling a safety; they collide and move forward as a single unit.
Because they move together, we can treat them as a single object with a combined mass (m₁ + m₂) after the collision. This simplifies the momentum conservation equation beautifully:
m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f
Here, v_f is the single final velocity of the combined object. This equation is your go-to tool for any problem where things stick together.
To summarize:
- 1Always start with momentumIt's conserved in every collision in a closed system.
- 2Check the kinetic energy
- If
KE_initial = KE_final, it's elastic. - If
KE_initial > KE_final, it's inelastic.
- If
- 3If they stick together, it's a special case called perfectly inelastic, which features the maximum loss of KE.
Worked examples
The Perfectly Inelastic Tackle
Problem: In a football game, a 110 kg running back, Marcus, is moving at 8.0 m/s. He's tackled by a 90 kg cornerback, Jordan, who was running in the same direction at 5.0 m/s. After the tackle, they stick together. What is their velocity immediately after the collision?
Solution:
- 1Identify the collision typeThe problem states they "stick together." This is the key phrase for a perfectly inelastic collision.
- 2Set up the principleFor any collision in a closed system, momentum is conserved. For a perfectly inelastic collision, we use the special form of the momentum equation.
m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f - 3List your knowns
- Marcus (m₁): 110 kg, v₁ᵢ = 8.0 m/s
- Jordan (m₂): 90 kg, v₂ᵢ = 5.0 m/s
- We need to find
v_f.
- 4Plug in the values
(110 kg)(8.0 m/s) + (90 kg)(5.0 m/s) = (110 kg + 90 kg)v_f880 kg·m/s + 450 kg·m/s = (200 kg)v_f1330 kg·m/s = (200 kg)v_f - 5
Solve for the final velocity (
v_f):v_f = 1330 kg·m/s / 200 kgv_f = 6.65 m/s
Was It Elastic?
Problem: A 0.50 kg cart moving at 2.0 m/s on a frictionless track collides with a 1.0 kg cart that is initially at rest. After the collision, the 0.50 kg cart is moving at -0.67 m/s (it bounced backward), and the 1.0 kg cart is moving at 1.33 m/s. Is this collision elastic or inelastic?
Solution:
- 1Identify the goalWe need to classify the collision. This means we must compare the total kinetic energy before and after.
- 2Calculate Initial Kinetic Energy (KEᵢ)Only the first cart is moving initially.
KEᵢ = KE₁ᵢ + KE₂ᵢKEᵢ = ½m₁v₁ᵢ² + ½m₂v₂ᵢ²KEᵢ = ½(0.50 kg)(2.0 m/s)² + ½(1.0 kg)(0 m/s)²KEᵢ = ½(0.50 kg)(4.0 m²/s²) + 0KEᵢ = 1.0 J - 3Calculate Final Kinetic Energy (KEf)Both carts are moving after the collision.
KEf = KE₁f + KE₂fKEf = ½m₁v₁f² + ½m₂v₂f²KEf = ½(0.50 kg)(-0.67 m/s)² + ½(1.0 kg)(1.33 m/s)² - 4Compare and conclude
- Initial KE ≈ 1.0 J
- Final KE ≈ 1.1 J
Wait, the final kinetic energy is greater than the initial? This is a huge red flag. In a real-world physical system, this is impossible without an internal energy source (like a spring or an explosion). In the context of an AP problem, this result almost certainly means the numbers were chosen to be very close to an elastic collision, and the difference is due to rounding in the problem statement. Let's re-check the numbers. If the final velocities were exactly -2/3 m/s and 4/3 m/s, the math would work out perfectly to 1.0 J. Given the values, we can conclude this is intended to represent an elastic collision.
Teacher's Note: On the AP Exam, if you calculate
KE_f > KE_i, double-check your math. If your math is correct, the collision is elastic (assumingKE_f ≈ KE_i) or the problem involves released potential energy (like a spring-loaded cart). For this topic, we assume the former. IfKE_fwas clearly less, like 0.8 J, it would be inelastic.
Try it yourself
Ready to try one on your own?
Problem 1: A 10,000 kg railroad freight car is rolling at 3.0 m/s when it collides with and couples to a stationary 15,000 kg freight car on a level track. a) What is the new speed of the two-car system? b) How much kinetic energy was "lost" in the collision? What type of collision was this?
Hints:
- "Couples to" is another way of saying "sticks together." What formula does that point to?
- To find the lost KE, calculate the total KE before the collision and the total KE after. The difference is the energy that was transformed.
Problem 2: A billiard ball moving at 5 m/s strikes an identical, stationary billiard ball. After the collision, the first ball moves at 4 m/s at an angle, and the second moves off at 3 m/s at another angle. Is this collision elastic? (Assume the mass of each ball is m).
Practice — 8 questions
In simple terms, collisions are either elastic or inelastic. The difference depends on whether the total kinetic energy of the objects is the same before and after they hit.
- 4.4.A: Describe whether an interaction between objects is elastic or inelastic.
- 4.4.A.1
- An elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system.
- 4.4.A.2
- In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies.
- 4.4.A.3
- An inelastic collision between objects is one in which the total kinetic energy of the system decreases.
- 4.4.A.4
- In an inelastic collision, some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy.
- 4.4.A.5
- In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision.
flowchart TD
A[Collision Occurs in a Closed System] --> B{Is Momentum Conserved?};
B -- Yes, Always --> C{Is Kinetic Energy Conserved?};
C -- Yes --> D[Elastic Collision];
D -- Example --> E[Ideal Billiard Balls];
C -- No --> F[Inelastic Collision];
F -- KE is transformed --> G[into heat, sound, deformation];
F --> H{Do objects stick together?};
H -- Yes --> I[Perfectly Inelastic Collision];
I -- Example --> J[Clay balls sticking together];
H -- No --> K[Inelastic Collision];
K -- Example --> L[A car crash where cars bounce apart];
Read what Saavi narrates
(Upbeat, warm intro music fades)
Hey everyone, it's Saavi from Shrutam. Let's talk about collisions.
Imagine you're at a bowling alley. You roll the ball, it hits the pins... CRASH! Pins go flying everywhere. Now, switch scenes. You're playing pool. The cue ball taps the 8-ball... click. They both roll away smoothly. Both are collisions, but they feel so different. One is messy, the other is clean. In physics, we have a name for this difference, and it all comes down to energy.
So here's the big idea: When objects collide, their total momentum is always conserved. Always. But their kinetic energy—the energy of motion—might not be. How we classify a collision depends entirely on what happens to that kinetic energy.
If the total kinetic energy before and after the collision is exactly the same, we call it an **elastic** collision. That's our ideal pool game.
If the total kinetic energy decreases, we call it an **inelastic** collision. That's the bowling ball crash, where energy was lost to sound and scattering the pins.
Let's walk through a classic example. A 110-kilogram running back, Marcus, is moving at 8 meters per second. He gets tackled by a 90-kilogram cornerback, Jordan, who was running in the same direction at 5 meters per second. The key here is, they stick together. What's their velocity right after?
Since they stick together, this is a perfectly inelastic collision. We use our momentum conservation equation: the initial momentum of Marcus plus the initial momentum of Jordan equals the final momentum of them combined.
So, 110 times 8, plus 90 times 5... that gives us a total initial momentum of 1330 kilogram-meters per second.
After the collision, their total mass is 110 plus 90, which is 200 kilograms. To find their final velocity, we just divide the total momentum by the total mass. 1330 divided by 200 gives us 6.65 meters per second. Makes sense, right? They're moving a bit slower than the running back was, but faster than the cornerback.
Now, here's a common mistake I see all the time: students assume kinetic energy is always conserved. It's not! It's only conserved in those perfect, elastic collisions. For everything else, you have to check. Always start with momentum, then check the energy.
You've got this. The key is to identify the type of collision first, and that will tell you which tools to use. Keep practicing, and I'll see you in the next one.
(Uplifting outro music fades in)
Kinetic energy is only conserved in perfectly elastic collisions. In most real-world scenarios (inelastic collisions), KE is transformed into heat, sound, and deformation.
Always start by conserving momentum. Then, calculate and compare the initial and final kinetic energy to determine the collision type.
That formula is only valid for perfectly inelastic collisions where the two objects become one and share a single final velocity.
If objects have different final velocities, you must use the general momentum conservation equation: `m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f`.
This is a frequent algebra mistake. Momentum is `mv`, but kinetic energy is `½mv²`. Mixing them up will lead to incorrect energy calculations.
When you see "kinetic energy," immediately think "square the v." Write the formula down before you plug in numbers to remind yourself.
While `v` can be negative (indicating direction), `v²` is always positive. A car moving left has the same kinetic energy as an identical car moving right at the same speed.
Always use parentheses when squaring a negative velocity in your calculator: `(-5)²`, not `-5²`. This ensures the result is positive.
"Inelastic" is a broad category where any amount of KE is lost. "Perfectly inelastic" is a specific sub-type where the objects also stick together, representing the maximum possible KE loss.
Think of it like squares and rectangles. All perfectly inelastic collisions are inelastic, but not all inelastic collisions are perfectly inelastic. A car crash that deforms bumpers but the cars still separate is inelastic, but not perfectly so.