Lesson_2_-_Collisions

Collisions

Overview of Collisions

  • Collisions are events where two or more bodies exert forces on each other in a relatively short time.

Do Now

  • Calculate the momentum of a 1200-kilogram car traveling at 18 m/s due east.

Elastic Vs. Inelastic Collisions

  • Elastic Collision: Total kinetic energy remains the same before and after the collision (no energy wasted).

  • Inelastic Collision: Some kinetic energy is transformed into waste heat.

  • Purely Inelastic Collision: The colliding bodies stick together after the collision.

  • Real Life Context: Most collisions are inelastic, though some may be treated as elastic for calculations.

Conservation of Momentum

  • In an isolated system, total momentum remains constant.

  • Initial momentum = Final momentum!

  • Mathematical Representation:( p_{before} = p_{after} )( m_A v_{iA} + m_B v_{iB} = m_A v_{fA} + m_B v_{fB} )

Inelastic Collisions

  • Two objects collide and stick together.

  • Example: Big fish swallowing a small fish; both masses are added after collision.

  • Mathematical Representation:( m_A v_{iA} + m_B v_{iB} = (m_A + m_B)v_f )

Case 1: Recoil

  • Two objects at rest explode in opposite directions.

  • Total momentum before & after = 0 kg·m/s.

  • Calculation Example:( (2 ext{ kg})(8 ext{ m/s}) + (4 ext{ kg})(v) = 0 )( 16 ext{ kg·m/s} = -(4 ext{ kg})(v) ) Result: v = -4 m/s (indicating direction).

Case 2: Perfect Inelastic Collisions

  • Two objects collide and stick together; momentum is conserved but kinetic energy is not.

  • Example: A 50-kg child jumping onto a 10-kg sled at rest.

  • Calculate final velocity post-collision.

Case 3: Head-On Collision

  • Objects with equal and opposite momentum before the collision.

  • Both objects come to rest after collision (total momentum before = 0 kg·m/s).

Case 4: Elastic Collision

  • Objects bounce off each other; kinetic energy is conserved.

  • Example: Object A (1 kg at rest) and Object B (0.2 kg moving at 10 m/s).

  • Determine the final velocity of Object B after collision using conservation of momentum and energy.

Elastic Collision Example

  • Two rubber-band balls (A: 95 kg, B: 18 kg) collide elastically.

  • Initial velocities: A at 6 m/s, B at 8 m/s.

  • Determine final velocity of rubber-band ball B.

Example Problems

Example 1

  • 2 kg cart (6 m/s east) collides with 3 kg cart (initially west).

  • After collision, they stick and come to rest. Calculate initial speed of 3 kg cart.

Example 2

  • 0.180 kg cart (0.80 m/s right) collides with 0.100 kg cart (at rest).

  • Carts lock together. Calculate final velocity.

Example 3

  • 10 kg cart A at rest collides with 20 kg cart B moving 40 m/s.

  • Calculate cart B's velocity after collision with observational data.

Example 4

  • Similar to Example 3. Confirm findings and approach consistency.

Example 5

  • A man on ice fires a rifle (70 kg body mass, 10 g bullet).

  • Calculate man's speed post-firing with a bullet speed of 500 m/s.

Example 6

  • Strongman compresses springs between weights (2.3 kg and 5.3 kg).

  • Lighter weight shoots at 6.0 m/s. Determine speed of heavier weight using conservation of momentum.

Example 7

  • Frank (47 kg) and Marie (36 kg) on a trampoline; Marie grabs Frank at 4.1 m/s.

  • Calculate their combined speed after the grab.