Conservation of Momentum
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
If the net external force on a system is zero, its total momentum stays the same: Σpbefore = Σpafter. Internal forces, however large, only move momentum around inside the system. In 2-D, conserve the x and y components separately.
1. Momentum adds as vectors
2. Choosing the system
Pick a system that includes everything that pushes on everything else. Then the big forces are internal and cancel.
Friction or gravity act during a collision too, but over a very short time they deliver almost no impulse compared with the collision forces.
Recoil: a gun and bullet, or a skater throwing a ball, start with zero momentum, so afterward m1v1 = −m2v2.
Adding mass: sand dropped straight down into a moving cart adds mass but no horizontal momentum, so the cart slows.
3. Example, then try it
A 4.0 kg rifle fires a 0.010 kg bullet at 400 m/s. Find the rifle’s recoil speed.
0 = (0.010)(400) + (4.0)v, so v = −4.0 ÷ 4.0 = 1.0 m/s backward.
Car A (1000 kg) goes east at 18 m/s and car B (1500 kg) goes north at 16 m/s. They collide and lock together. Find their velocity just after, and how much kinetic energy is lost.
Momenta: px = (1000)(18) = 18,000; py = (1500)(16) = 24,000 kg·m/s.
Total: √(18,000² + 24,000²) = 30,000 kg·m/s, so v = 30,000 ÷ 2500 = 12 m/s.
Direction: tan θ = 24,000/18,000, so θ ≈ 53° north of east.
Energy: before ½(1000)(18)² + ½(1500)(16)² = 354 kJ; after ½(2500)(12)² = 180 kJ. About half is lost.
Answer: 12 m/s at 53° north of east; about 174 kJ of kinetic energy lost.
4. Common mistakes
A person standing on a boat at rest on still water dives off the front. What does the boat do?
Total momentum stays zero, so the boat moves opposite to the diver. It’s heavier, so its speed is smaller: mboatvboat = mdivervdiver.
Ready to practice?
5 questions on Conservation of Momentum, each with a hint and an explanation for every wrong answer.