Collisions in one dimension are events where two objects interact while moving along a single straight line. They are important because the total momentum of an isolated system stays constant, even when the objects bounce, stick, or exchange speeds. This makes collisions a powerful way to predict motion after an impact using information from before the impact.
Carts on a frictionless track are a classic model because outside forces along the track can be made very small.
The main idea is to choose a positive direction, assign signs to velocities, and write momentum conservation for the whole system. In an elastic collision, both momentum and kinetic energy are conserved, which gives enough information to solve for both final velocities. In a perfectly inelastic collision, the objects stick together and move with one common final velocity.
Real collisions often fall between these extremes, so understanding the ideal cases helps students analyze experiments and estimate outcomes.
Key Facts
- Momentum of one object: p = mv
- Momentum conservation in 1D: m1v1i + m2v2i = m1v1f + m2v2f
- Kinetic energy: K = 1/2 mv^2
- Elastic collision condition: 1/2 m1v1i^2 + 1/2 m2v2i^2 = 1/2 m1v1f^2 + 1/2 m2v2f^2
- Perfectly inelastic final velocity: vf = (m1v1i + m2v2i) / (m1 + m2)
- 1D elastic relative speed rule: v1i - v2i = -(v1f - v2f)
Vocabulary
- Momentum
- Momentum is the product of an object's mass and velocity, and it includes direction.
- Impulse
- Impulse is the change in momentum caused by a force acting over a time interval.
- Elastic collision
- An elastic collision is a collision in which total momentum and total kinetic energy are both conserved.
- Inelastic collision
- An inelastic collision is a collision in which total momentum is conserved but kinetic energy is not conserved.
- Perfectly inelastic collision
- A perfectly inelastic collision is one where the objects stick together and move with the same final velocity.
Common Mistakes to Avoid
- Ignoring velocity signs, which is wrong because momentum is a vector and direction matters in one dimension.
- Assuming kinetic energy is always conserved, which is wrong because only elastic collisions conserve total kinetic energy.
- Using mass in grams instead of kilograms, which gives incorrect SI units and can make momentum and energy calculations inconsistent.
- Solving each object separately without using the system, which is wrong because momentum conservation applies to the combined isolated system during the collision.
Practice Questions
- 1 A 2.0 kg cart moving at +3.0 m/s collides with a 1.0 kg cart initially at rest. If they stick together, what is their final velocity?
- 2 A 0.50 kg cart moving at +4.0 m/s elastically collides with a 0.50 kg cart initially at rest. What are the final velocities of both carts?
- 3 Two carts collide on a nearly frictionless track. After the collision, the total kinetic energy is smaller than before, but the total momentum is unchanged. What type of collision is this, and why can momentum still be conserved?