Conservation of Momentum in Collisions
Momentum is a vector quantity defined as the product of an object's mass and velocity: $p = mv$. The law of conservation of momentum states that in a closed, isolated system (no external forces), the total momentum before a collision is equal to the total momentum after the collision.
Mathematically, for two objects:
$$ m_1 v_{1,\text{initial}} + m_2 v_{2,\text{initial}} = m_1 v_{1,\text{final}} + m_2 v_{2,\text{final}} $$
This principle applies to all types of collisions: elastic, inelastic, and perfectly inelastic.
Worked Example
Problem:
Two carts collide on a frictionless track. Cart A ($m_1 = 2,\text{kg}$) moves at $3,text{m/s}$ toward stationary Cart B ($m_2 = 1,text{kg}$). After the collision, Cart A moves at $1,text{m/s}$ in the same direction. What is Cart B's velocity after the collision?
**Step 1: Write the conservation of momentum equation**
%%MATH_DISPLAY_1%%
**Step 2: Substitute known values**
- $m_1 = 2,text{kg}$
- $v_{1,text{initial}} = 3,text{m/s}$
- $m_2 = 1,text{kg}$
- $v_{2,text{initial}} = 0,text{m/s}$
- $v_{1,text{final}} = 1,text{m/s}$
- Let $v_{2,text{final}} = v$
%%MATH_DISPLAY_2%%
%%MATH_DISPLAY_3%%
**Step 3: Solve for $v$**
%%MATH_DISPLAY_4%%
**Answer:** Cart B moves at $4,text{m/s}$ after the collision.
Takeaways
- Momentum is always conserved in collisions if no external forces act.
- The total momentum before and after a collision is equal.
- Conservation of momentum applies to all collision types, regardless of whether kinetic energy is conserved.