Linear momentum is a measure of how much motion an object has. It’s defined as the product of mass and velocity:
\$p = m\,v\$
The direction of momentum is the same as the direction of velocity. Think of it like a moving skateboard: the heavier the skateboarder (larger \$m\$) or the faster they’re going (larger \$v\$), the more momentum they carry.
In an isolated system (no external forces), the total momentum before an event equals the total momentum after:
\$\sum mi v{i,\text{initial}} = \sum mi v{i,\text{final}}\$
This is true whether the objects stick together, bounce off, or break apart. It’s like a dance where the total energy of all dancers stays the same even if they change partners.
An elastic collision is a special case where both momentum and kinetic energy are conserved. The key points are:
\$|v{1i} - v{2i}| = |v{1f} - v{2f}|\$
Imagine two ice skaters standing still. If they push off each other, they move in opposite directions. The force they exert on each other is equal and opposite (Newton’s Third Law), so the total momentum stays zero. If they push hard enough, they’ll both speed up, but the total kinetic energy also stays the same because no energy is lost to friction or sound—just like an elastic collision.
A cue ball (mass \$mc\$) strikes a stationary object ball (mass \$mo\$). If the collision is perfectly elastic:
This is why a well‑hit cue ball can send the object ball flying across the table with the same speed it had before the collision.
| Quantity | Formula |
|---|---|
| Momentum | \$p = m\,v\$ |
| Total Momentum (before) | \$\displaystyle \sum mi v{i,\text{initial}}\$ |
| Total Momentum (after) | \$\displaystyle \sum mi v{i,\text{final}}\$ |
| Kinetic Energy (before) | \$\displaystyle \tfrac12 \sum mi v{i,\text{initial}}^2\$ |
| Kinetic Energy (after) | \$\displaystyle \tfrac12 \sum mi v{i,\text{final}}^2\$ |
| Relative Speed Equality | \$|v{1i} - v{2i}| = |v{1f} - v{2f}|\$ |