Conservation of Momentum

Problem-Solving Approach: Conservation of Momentum Problems

The law of conservation of momentum allows us to relate the velocities and momenta of objects before and after an interaction, provided the system is isolated or external forces are negligible during the interaction.

• System Definition: Clearly define the system. If possible, choose an isolated system (where ) or one where external forces can be ignored for the duration of the interaction (impulse approximation).

• Segmented Analysis: If an isolated system cannot be chosen for the entire motion, divide the problem into segments where momentum is conserved, using Newton's laws or conservation of energy for other segments.

Mathematical Representation

• The law of conservation of momentum is expressed as: where is mass, is initial velocity, and is final velocity.

• For multiple objects, sum the momenta before and after the interaction.

Example: Jack and the Skateboard

• Jack (75 kg with skateboard) catches a 3.0 kg ball thrown at 4.0 m/s. Find the final speed after the catch.

• Solution:

  • Initial momentum: kg·m/s (all from the ball).

  • Final mass: kg.

  • Final velocity: m/s.

Choosing the System

• Choose the system so that momentum is conserved (i.e., net external force is zero).

• For a person and sled, if both are included in the system, the net force is zero and momentum is conserved.

Example: Bailey and the Sled

• Bailey (26 kg) runs at 4 m/s and jumps onto a 5.9 kg sled. Find the sled's speed just after landing.

• Solution:

  • Initial momentum: kg·m/s.

  • Final mass: kg.

  • Final velocity: m/s.

Explosions

An explosion is an event where particles of a system move apart after a brief, intense interaction. It is the opposite of a collision. If the system is isolated, the total momentum is conserved because the forces are internal.

• Example: A rocket expelling exhaust gases; the rocket moves forward as gases move backward, conserving total momentum.

QuickCheck Example

• Two boxes (1 kg and 2 kg) on a frictionless surface are pushed apart by an explosion. If the 1 kg box moves at 4 m/s, the 2 kg box moves at 2 m/s in the opposite direction (by conservation of momentum).

Recoil Example: Toy Rifle

• A 30 g ball is fired from a 1.2 kg toy rifle at 15 m/s. Find the recoil speed of the rifle.

• Solution:

  • Initial momentum: 0 (system at rest).

  • After firing:

  • Solving for rifle's velocity: m/s.

  • The negative sign indicates the rifle moves in the opposite direction to the ball.

Inelastic Collisions

Perfectly Inelastic Collisions

A perfectly inelastic collision is one in which two objects stick together and move with a common final velocity after colliding. Momentum is conserved, but kinetic energy is not.

• Examples: Clay hitting the floor and sticking, a bullet embedding in wood.

QuickCheck Example

• Two boxes (1 kg and 2 kg) sliding on a frictionless surface collide and stick together. If the 1 kg box moves at 4 m/s and the 2 kg box is at rest, after collision both move at 0 m/s (at rest), since their momenta cancel.

Example: Railroad Cars

• Two railroad cars (2.0 × 104 kg and 4.0 × 104 kg) couple after collision. The lighter car moves at 1.5 m/s, and after collision, both move at -0.25 m/s (to the left). Find the initial speed of the heavier car.

• Solution:

  • Conservation of momentum:

  • Solving for :

  • Plug in values: m/s

  • The negative sign indicates the heavier car was moving to the left initially.

Example: Bullet and Block

• A 10 g bullet is fired into a 1.0 kg wood block and lodges in it. The block slides 4.0 m across a floor with . Find the bullet's speed.

• Solution:

  • Deceleration due to friction: m/s2

  • Use with , m: m/s

  • Total momentum after collision: kg m/s kg·m/s

  • Initial bullet speed: m/s

Summary Table: Types of Collisions

Key Equations:

• Conservation of momentum:

• Frictional deceleration:

• Kinematic equation for stopping distance:

Additional info: In all examples, the principle of conservation of momentum is applied to isolated systems or during short interactions where external forces are negligible. In inelastic collisions, kinetic energy is not conserved, but momentum always is.