Momentum & Collisions
Understanding momentum conservation and collision dynamics
Momentum is a measure of an object's motion, combining both mass and velocity. It's a vector quantity that points in the direction of velocity.
Momentum equals mass times velocity
Properties of Momentum:
- Vector quantity (has both magnitude and direction)
- Units: kg·m/s (kilogram-meters per second)
- Larger mass or higher velocity means greater momentum
- Direction of momentum is same as direction of velocity
Impulse is the change in momentum caused by a force acting over a time interval. It connects force and momentum.
Impulse equals force times time interval
Impulse equals change in momentum
This is the impulse-momentum theorem: the impulse applied to an object equals its change in momentum.
Real-World Application: Airbags work by increasing the time of collision, which decreases the force (since J = FΔt is constant). Same impulse, less force, safer collision!
Force-Time Graphs
The area under a force vs. time graph equals the impulse delivered. This is useful when force varies with time.
In an isolated system (no external forces), the total momentum before an interaction equals the total momentum after. This is one of the most fundamental laws in physics.
Total initial momentum equals total final momentum
Conservation for two-object system
Key Points:
- Applies to all collisions and explosions
- Works even when kinetic energy is not conserved
- Must consider momentum as a vector (direction matters)
- External forces can change total momentum
Elastic Collisions
Both momentum and kinetic energy are conserved. Objects bounce off each other without losing energy to heat or deformation.
Momentum conservation
Kinetic energy conservation
Examples: Collisions between billiard balls, atoms, or molecules (approximately elastic).
Inelastic Collisions
Momentum is conserved, but kinetic energy is not. Some energy is converted to heat, sound, or deformation.
Momentum conservation only
Examples: Car crashes, clay balls colliding, most real-world collisions.
Perfectly Inelastic Collisions
Objects stick together after collision, moving as one combined mass. Maximum kinetic energy is lost (while still conserving momentum).
Combined mass moves with common final velocity
Examples: Bullet embedding in wood, train cars coupling together.
When objects collide at angles, momentum is conserved in both x and y directions independently.
Momentum conservation in x-direction
Momentum conservation in y-direction
This means we can analyze the collision by breaking it into x and y components and applying conservation to each direction separately.
The center of mass is the average position of all the mass in a system. It moves as if all external forces act on it.
Center of mass position
Center of mass velocity
Important: In the absence of external forces, the center of mass moves with constant velocity, even during collisions or explosions!