Elastic collisions are collisions in which objects bounce apart without losing total kinetic energy to heat, sound, or deformation. They are a key model in physics because they let us predict motion using conservation laws instead of detailed contact forces. In one dimension, the objects move along a single line, so velocity signs show direction.
These equations are useful for carts on tracks, air-table pucks, gas molecules, and idealized ball collisions.
Key Facts
- Momentum is conserved: m1v1i + m2v2i = m1v1f + m2v2f.
- Kinetic energy is conserved: 1/2 m1v1i^2 + 1/2 m2v2i^2 = 1/2 m1v1f^2 + 1/2 m2v2f^2.
- General 1D elastic result: v1f = ((m1 - m2)/(m1 + m2))v1i + (2m2/(m1 + m2))v2i.
- General 1D elastic result: v2f = (2m1/(m1 + m2))v1i + ((m2 - m1)/(m1 + m2))v2i.
- Relative velocity reverses: v1i - v2i = -(v1f - v2f).
- For equal masses in a 1D elastic collision, the objects exchange velocities: v1f = v2i and v2f = v1i.
Vocabulary
- Elastic collision
- A collision in which total momentum and total kinetic energy are both conserved.
- Momentum
- The quantity p = mv that measures an object's motion using its mass and velocity.
- Kinetic energy
- The energy of motion given by KE = 1/2 mv^2 for a moving object.
- Center of mass
- The mass-weighted average position of a system, which moves at constant velocity when no external net force acts.
- Relative velocity
- The velocity of one object as measured from another object, found by subtracting their velocities.
Common Mistakes to Avoid
- Ignoring velocity signs is wrong because direction matters in one-dimensional collisions. Choose a positive direction and keep every velocity consistent with it.
- Using only momentum conservation is incomplete because many different final velocities can satisfy momentum alone. Elastic collisions also require kinetic energy conservation or the relative velocity rule.
- Treating speed and velocity as the same is wrong because speed has no direction. In collision equations, a negative velocity means the object moves in the opposite direction.
- Applying the equal-mass velocity swap to unequal masses is wrong because that shortcut works only when m1 = m2 in a 1D elastic collision. For unequal masses, use the full elastic collision equations.
Practice Questions
- 1 A 2.0 kg cart moving at 3.0 m/s hits a 2.0 kg cart at rest in a 1D elastic collision. Find the final velocity of each cart.
- 2 A 1.0 kg cart moving at 4.0 m/s collides elastically with a 3.0 kg cart initially at rest. Use the 1D elastic collision equations to find v1f and v2f.
- 3 A ball collides elastically with a very massive wall that is initially at rest. Explain why the ball reverses direction with nearly the same speed while the wall's speed changes by an extremely small amount.