3cConservation of Momentum
Apply conservation of momentum to 1D and 2D collisions, explosions, and rocket propulsion. Analyze center of mass motion.
Conservation of momentum is a universal law that applies from subatomic particle collisions to galaxy mergers. It is the foundation of rocket science, collision analysis, and our understanding of fundamental symmetries in nature.
How do rockets accelerate in the vacuum of space — and how can a 2D collision be analyzed when objects scatter at angles?
Lesson Overview
The law of conservation of momentum is one of the most fundamental principles in physics. In an isolated system, the total momentum vector is constant — even if individual momenta change. This applies to 1D collisions (along a line), 2D collisions (where momentum is conserved independently in x and y directions), explosions (where objects push apart from rest), and rocket propulsion. The center of mass of an isolated system moves at constant velocity regardless of internal interactions.
Key Equations
Worked Examples
A 5 kg object moving at 4 m/s east collides with a 3 kg object moving at 2 m/s west. After the collision, the 5 kg object moves at 1 m/s east. Find the 3 kg object's final velocity.
A 10 kg bomb at rest explodes into two fragments: a 3 kg piece moving at 20 m/s east and a 7 kg piece. Find the velocity of the 7 kg piece.
A 2 kg ball moving at 5 m/s east collides with a 3 kg ball at rest. After the collision, the 2 kg ball moves at 30° north of east at 3 m/s. Find the x and y components of the 3 kg ball's final velocity.
Find the center of mass of a system: a 4 kg mass at x = 1 m and a 6 kg mass at x = 4 m.
A 3 kg mass moves at 2 m/s east and a 5 kg mass moves at 4 m/s west. Find the velocity of the center of mass.
Guided Problems
A 6 kg object at rest explodes into a 2 kg piece moving at 9 m/s east and a 4 kg piece. Find the 4 kg piece's velocity.
Hint: Initial momentum = 0. Use m₁v₁ + m₂v₂ = 0. Solve for v₂.
A 4 kg ball moving at 6 m/s east collides with a 2 kg ball moving at 3 m/s east. After the collision, the 4 kg ball moves at 4 m/s east. Find the 2 kg ball's final velocity.
Hint: Write Σp_before = Σp_after. Substitute known values and solve for the unknown velocity.
Find the center of mass of three masses: 2 kg at x = 0, 3 kg at x = 3 m, and 5 kg at x = 6 m.
Hint: x_cm = (m₁x₁ + m₂x₂ + m₃x₃)/(m₁+m₂+m₃). Multiply each mass by its position, sum, then divide by total mass.
A 1 kg ball moving at 4 m/s east collides with a 2 kg ball at rest. After the collision, the 1 kg ball moves at 1 m/s north. Find the x and y components of the 2 kg ball's final velocity.
Hint: Apply conservation of momentum separately in x and y directions. x: 1(4) = 1(0) + 2v₂x. y: 0 = 1(1) + 2v₂y.
A rocket (mass 1000 kg including fuel) ejects 10 kg of gas per second at 500 m/s backward. Find the thrust force and explain how the rocket accelerates in space.
Hint: Thrust = Δp/Δt = (mass ejected per second) × (exhaust speed). The rocket gains forward momentum as gas gains backward momentum.
Key Vocabulary
Law of Conservation of Momentum
The total momentum of an isolated system remains constant. For any interaction: Σp_before = Σp_after. This is a consequence of Newton's third law.
Example: In any collision or explosion, the total momentum vector of the system is unchanged, even though individual momenta change.
2D Collision
A collision in which objects move in two dimensions after impact. Momentum is conserved independently in the x and y directions: Σp_x = constant and Σp_y = constant.
Example: A billiard ball struck off-center scatters at an angle — both x and y momentum components are separately conserved.
Explosion
An event where a stationary object breaks apart. Since initial momentum is zero, the fragments must have equal and opposite momenta: m₁v₁ + m₂v₂ = 0.
Example: A cannon firing, a rocket ejecting exhaust, or a grenade exploding — all start from rest and conserve zero total momentum.
Center of Mass (CM)
The average position of mass in a system, weighted by mass: x_cm = Σmᵢxᵢ/Σmᵢ. The CM of an isolated system moves at constant velocity.
Example: For a 1 kg mass at x = 0 and a 3 kg mass at x = 4 m: x_cm = (1×0 + 3×4)/4 = 3 m (closer to the heavier mass).
Rocket Propulsion
A rocket accelerates by ejecting mass (exhaust) backward at high speed. By conservation of momentum, the rocket gains forward momentum equal to the backward momentum of the exhaust.
Example: A rocket ejecting 10 kg/s at 3000 m/s generates a thrust of F = 30,000 N — no external surface to push against is needed.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
A 4 kg object at rest explodes into a 1 kg piece at 12 m/s east and a 3 kg piece. What is the 3 kg piece's velocity?
In a 2D collision, momentum is conserved:
Find the center of mass of a 2 kg mass at x = 0 and a 6 kg mass at x = 4 m.
A rocket in space ejects exhaust backward. What happens to the rocket?
The center of mass of an isolated system:
Independent Practice
A 3 kg object at rest explodes into a 1 kg piece at 6 m/s east and a 2 kg piece. Find the 2 kg piece's velocity.
A 5 kg cart at 3 m/s east collides with a 3 kg cart at 1 m/s west. After the collision, the 5 kg cart moves at 1.5 m/s east. Find the 3 kg cart's final velocity.
Find the center of mass of: 3 kg at x = 0, 5 kg at x = 2 m, and 2 kg at x = 5 m.
A 2 kg ball at 6 m/s east collides with a 4 kg ball at rest. After the collision, the 2 kg ball moves at 2 m/s at 60° north of east. Find the x and y components of the 4 kg ball's final velocity.
★ A 1000 kg rocket (including 200 kg of fuel) ejects all fuel at 2000 m/s backward in one burst. Find the rocket's final velocity. (Hint: use conservation of momentum; initial state is at rest)
ChallengeCommon Mistakes
Applying conservation of momentum only in one direction for 2D collisions
In 2D, apply conservation separately: Σp_x before = Σp_x after AND Σp_y before = Σp_y after. Treat each direction independently
Forgetting that the center of mass position is mass-weighted, not just the average position
x_cm = Σmᵢxᵢ/Σmᵢ. The CM is always closer to the more massive object
Thinking a rocket needs something to push against in space
Rockets work by conservation of momentum — exhaust goes backward, rocket goes forward. No external surface is needed
Applying conservation of momentum when the system is not isolated
Check for external forces (friction, gravity components, normal forces) before applying conservation. Only isolated systems conserve momentum
Math Tips
For 2D collisions: break all velocities into x and y components. Apply Σp_x = constant and Σp_y = constant separately. Then recombine if needed.
For explosions from rest: m₁v₁ + m₂v₂ = 0 → v₂ = −m₁v₁/m₂. The fragments always move in opposite directions.
Center of mass: x_cm = Σmᵢxᵢ/Σmᵢ. For two masses: x_cm = (m₁x₁ + m₂x₂)/(m₁+m₂). The CM is always between the two masses, closer to the heavier one.
The CM of an isolated system moves at constant velocity. In the CM frame, total momentum = 0. This simplifies many collision problems.