AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/4/4-4)
Unit 4 · Topic 4.4
4.4 Elastic and Inelastic Collisions
Collisions are sorted by what happens to kinetic energy. In an elastic collision the total kinetic energy is the same before and after; in an inelastic collision some of it becomes thermal energy, sound or deformation. Momentum is conserved in both, and objects that stick together have a perfectly inelastic collision.
Key terms
- elastic collision
- inelastic collision
- perfectly inelastic collision
- kinetic energy
- conservation of momentum
Momentum versus kinetic energy in collisions
If no net external force acts, momentum is conserved in every collision. Kinetic energy is a separate question: it's conserved only in elastic collisions.
To classify a collision, calculate the system's total kinetic energy just before and just after. Equal totals mean elastic. A drop means inelastic.
Elastic collisions
In an elastic collision, the system's total kinetic energy after equals the total before. The individual objects' kinetic energies can still change; energy can move from one object to the other.
Nearly elastic examples: steel balls in a Newton's cradle, billiard balls, gliders with magnetic bumpers on an air track. Real collisions are never perfectly elastic, but these come close.
A useful special case: when a moving object hits an identical object at rest head-on and elastically, they swap velocities. The first stops and the second moves off with the first's original velocity. Both momentum and kinetic energy check out.
Inelastic collisions
In an inelastic collision, the system's total kinetic energy decreases. Nonconservative forces during the impact turn some of it into thermal energy, sound and permanent changes in shape. Car crashes, a ball of clay hitting a wall and most everyday collisions are inelastic.
In a perfectly inelastic collision, the objects stick together and move with the same velocity afterward. This loses the most kinetic energy that momentum conservation allows. Because the stuck-together objects move with the center-of-mass velocity, the energy that's left is just the kinetic energy of the center of mass.
Explosions
In an explosion, stored energy (chemical, or elastic in a compressed spring) turns into kinetic energy, so the system's kinetic energy increases. Momentum is still conserved: pieces flying apart from rest have total momentum zero.
| Type | Momentum (no net external force) | Total kinetic energy | Example |
|---|---|---|---|
| Elastic | conserved | same before and after | steel balls colliding |
| Inelastic | conserved | decreases | car bumpers crumpling |
| Perfectly inelastic | conserved | decreases the most possible | carts that stick together |
| Explosion | conserved | increases | spring-loaded carts pushing apart |
Combining momentum and energy
Some problems have two stages, such as a bullet that embeds in a hanging block, which then swings upward (a ballistic pendulum). Use momentum conservation for the collision, because kinetic energy isn't conserved in it. Then use energy conservation for the swing, because no collision happens during it. Mixing up the stages is one of the most common errors in this unit.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Is this collision elastic?
A 1.0 kg cart moving at +3.0 m/s hits a 2.0 kg cart at rest. Afterward, the 1.0 kg cart moves at −1.0 m/s and the 2.0 kg cart at +2.0 m/s. Check that momentum is conserved, and decide whether the collision is elastic.
Show the solutionHide the solution
- Step 1: Momentum before: (1.0)(3.0) + 0 = 3.0 kg·m/s. After: (1.0)(−1.0) + (2.0)(2.0) = −1.0 + 4.0 = 3.0 kg·m/s. ✓
- Step 2: Kinetic energy before: ½(1.0)(3.0)² = 4.5 J.
- Step 3: Kinetic energy after: ½(1.0)(1.0)² + ½(2.0)(2.0)² = 0.5 + 4.0 = 4.5 J.
- Step 4: Total kinetic energy is unchanged, so the collision is elastic, even though each cart's kinetic energy changed.
Answer: Momentum is conserved (3.0 kg·m/s); the collision is elastic (4.5 J before and after)
- Example 2Calculator allowed
Energy lost when carts stick
A 2.0 kg cart moving at 4.0 m/s hits a 3.0 kg cart at rest, and they stick together at 1.6 m/s (see 4.3). How much kinetic energy is transformed, and what fraction is that?
Show the solutionHide the solution
- Step 1: Before: K = ½(2.0)(4.0)² = 16 J.
- Step 2: After: K = ½(5.0)(1.6)² = 6.4 J.
- Step 3: Transformed into other forms: 16 − 6.4 = 9.6 J, which is 9.6 ÷ 16 = 60% of the original.
Answer: 9.6 J, or 60% of the initial kinetic energy
- Example 3Calculator allowed
Ballistic pendulum (classic trap)
A 0.010 kg bullet moving at 300 m/s embeds itself in a 1.99 kg block hanging at rest from strings. Use g = 9.8 m/s². How high does the block (with the bullet) swing?
Show the solutionHide the solution
- Step 1: Stage 1, the collision: momentum is conserved. (0.010)(300) = (2.00)v, so v = 3.0 ÷ 2.00 = 1.5 m/s just after impact.
- Step 2: Stage 2, the swing: mechanical energy of the block–bullet–Earth system is conserved. ½(2.00)(1.5)² = (2.00)(9.8)h, so h = (1.5)² ÷ (2 × 9.8) ≈ 0.11 m.
- Step 3: The trap is using energy conservation for the collision. The bullet's kinetic energy was ½(0.010)(300)² = 450 J, but the block and bullet have only ½(2.00)(1.5)² = 2.25 J just after. More than 99% became thermal energy and deformation.
Answer: About 0.11 m (11 cm)
Common mistakes
- Assuming kinetic energy is conserved because momentum is. Check the kinetic energy totals separately.
- Calling a collision elastic because one object's kinetic energy didn't change. The definition uses the system's total.
- Using energy conservation across a sticky collision, such as a ballistic pendulum. Use momentum for the collision and energy for what follows.
- Saying energy is destroyed in an inelastic collision. It's transformed into thermal energy, sound and deformation.
On the exam
- Expect a table of before-and-after velocities and a question asking whether the collision is elastic. Show both total kinetic energies.
- Lab questions often use carts on a track with motion sensors. You might be asked how to test whether momentum or kinetic energy is conserved, and what sources of error (like friction) would affect the result.
- In two-stage problems, say explicitly which principle you use for each stage and why it applies.
Connected topics
Videos
Check yourself
4 questions on 4.4 Elastic and Inelastic Collisions. Pick an answer to see if you got it, and why.
A 1.0 kg cart moving at 4.0 m/s hits an identical cart at rest. Afterward, the first cart moves at 1.0 m/s and the second at 3.0 m/s, both in the original direction. Which describes the collision?
A steel ball moving at speed v hits an identical steel ball at rest head-on, and the collision is elastic. What happens?
A 2.0 kg cart moving right at 3.0 m/s hits a 1.0 kg cart at rest on a frictionless track. There are no springs or explosives on the carts. Which set of final velocities, both to the right, is impossible?
A 2.0 kg cart moving at 6.0 m/s collides with a 4.0 kg cart at rest, and they stick together. How much kinetic energy is transformed into other forms?
0 of 4 answered