π₯Conservation of Momentum
The Law of Conservation of Momentum
For a closed system (no external forces), the total momentum is conserved:
p_total_before = p_total_after
Ξ£p_i = Ξ£p_f
This follows directly from Newton's Third Law: internal forces between objects come in equal-and-opposite pairs. Their impulses cancel, leaving total momentum unchanged.
"Closed system" means no net external forces β or more practically, if the internal forces are much larger than external forces (like during a brief collision), we can treat it as approximately closed.
Think About It
A cannon fires a cannonball. The cannon recoils backward. If the system starts at rest, what must be true about the cannon's momentum after firing?
βοΈ Worked Example: Explosion
Problem: A 10 kg cart at rest explodes into two pieces: 3 kg and 7 kg. The 3 kg piece flies off at 14 m/s left. Find the velocity of the 7 kg piece.
π Key Equations
Conservation of Momentum
Ξ£ p_before = Ξ£ p_afterm_1 v_1i + m_2 v_2i = m_1 v_1f + m_2 v_2fβ οΈ Common Mistakes
Misconception: Applying conservation of momentum when there IS a net external force.
β Correct thinking: Momentum is only conserved in a closed system β one with no net external forces. Gravity, friction, and normal forces can all be external forces.
Why: During brief collisions, internal collision forces are so large that external forces contribute negligible impulse, making conservation a good approximation.
Misconception: Treating momentum as a scalar and ignoring direction.
β Correct thinking: Momentum is a vector. In one dimension, carefully assign + and β signs. In two dimensions, conserve x- and y-components separately.
Why: A head-on collision where both objects stop has total momentum = 0 before AND after β this only works if you track direction.
Misconception: Confusing conservation of momentum with conservation of energy.
β Correct thinking: Momentum is always conserved in a closed system. Kinetic energy is only conserved in elastic collisions.
Why: Real collisions lose KE to heat and sound, but they still obey momentum conservation because Newton's Third Law still holds.
π Practice Problems
Try these problems. Check your answer when ready.
A 5 kg cart moving at 4 m/s collides with a stationary 3 kg cart. After the collision the 5 kg cart moves at 1 m/s in the same direction. Find the velocity of the 3 kg cart.
A 70 kg person standing on a frictionless ice rink throws a 2 kg ball at 10 m/s. Find the person's recoil velocity.
Two ice skaters (60 kg and 80 kg) push off each other from rest. The 60 kg skater moves at 3 m/s to the right. Find the velocity of the 80 kg skater.
A 1200 kg car moving east at 15 m/s collides with a 1800 kg SUV moving west at 10 m/s. After the collision the car moves west at 3 m/s. Find the SUV's final velocity.
A 0.1 kg bullet is fired into a 4.9 kg block resting on a frictionless surface. The bullet passes through, exiting at 100 m/s, and the block moves at 10 m/s. Find the initial bullet speed.
p_before = p_afterFinished reading through this lesson?