Section 4.3: Conservation of Momentum
The law of conservation of momentum states that in an isolated system, where no external forces act, the total momentum remains constant.
\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]
This principle is widely applied in collisions, explosions, and interactions between objects in a closed system.
Example 1: Elastic Collision
A 2 kg ball moving at 3 m/s collides elastically with a 1 kg ball at rest. Find the velocities after collision.
Using conservation of momentum and kinetic energy:
\( v_{1f} = 1 \, \text{m/s}, \quad v_{2f} = 5 \, \text{m/s} \)
Velocities after collision: 1 m/s and 5 m/s.
Example 2: Inelastic Collision
A 3 kg cart moving at 4 m/s collides with a 2 kg cart at rest. They stick together. Find the final velocity.
\( v_f = \frac{3\cdot4 + 2\cdot0}{3+2} = \frac{12}{5} = 2.4 \, \text{m/s} \)
The combined carts move at 2.4 m/s.
Practice Problems
- A 5 kg ball moving at 2 m/s collides with a 3 kg ball at rest. Find their velocities after an elastic collision.
- Two ice skaters push off each other. Skater A has mass 50 kg, skater B 70 kg. If skater A moves at 1.2 m/s, find the velocity of skater B.
- A 4 kg cart moving at 3 m/s collides inelastically with a 6 kg cart at rest. Find the final velocity after sticking together.
- Two vehicles of masses 800 kg and 600 kg collide head-on. Calculate final velocities if they stick together.
- A ball of mass 0.5 kg moving at 6 m/s collides elastically with a 0.5 kg ball at rest. Determine their final velocities.