Conservation of momentum is a core physics idea used to analyze collisions, explosions, recoil, and objects pushing off from each other. This cheat sheet helps students set up worked examples in a clear, repeatable way. It is especially useful when forces during an interaction are large but act for a short time.
The goal is to identify the system, write momentum before and after, and solve carefully with signs.
Key Facts
- Linear momentum is , where is momentum, is mass, and is velocity.
- The total momentum of an isolated system is conserved, so .
- For a two-object collision in one dimension, .
- In a perfectly inelastic collision, objects stick together and the final velocity is .
- In an elastic collision, both momentum and kinetic energy are conserved, so and .
- Impulse equals change in momentum, so for a constant net force.
- Momentum is a vector, so velocities in opposite directions must have opposite signs in equations.
- For an explosion or recoil starting from rest, the total final momentum is , so .
Vocabulary
- Momentum
- Momentum is the quantity of motion of an object, calculated by .
- Isolated system
- An isolated system is a group of objects with no net external force acting on it during the interaction.
- Impulse
- Impulse is the change in momentum caused by a force acting over a time interval, given by .
- Elastic collision
- An elastic collision is a collision in which total momentum and total kinetic energy are both conserved.
- Inelastic collision
- An inelastic collision is a collision in which momentum is conserved but kinetic energy is not conserved.
- Perfectly inelastic collision
- A perfectly inelastic collision is a collision in which the objects stick together and move with one shared final velocity.
Common Mistakes to Avoid
- Ignoring direction signs is wrong because momentum is a vector. Choose a positive direction first, then make velocities in the opposite direction negative.
- Using conservation of kinetic energy for every collision is wrong because kinetic energy is conserved only in elastic collisions. Momentum is conserved for isolated collisions, but kinetic energy may become heat, sound, or deformation.
- Forgetting to combine masses after objects stick together is wrong because a perfectly inelastic collision has one final velocity. Use on the final side.
- Treating external forces as always negligible is wrong because momentum is conserved only when the net external impulse is zero or small during the interaction. Always define the system and check the situation.
- Mixing initial and final velocities in the same term is wrong because each momentum term must describe one object at one time. Keep values on the initial side and values on the final side unless algebraically rearranging.
Practice Questions
- 1 A cart moving at collides and sticks to a cart at rest. Find their shared final velocity.
- 2 A ball moving at is stopped by a glove in . Find the ball's impulse and the average force on the ball.
- 3 A skater at rest throws a ball forward at . Find the skater's recoil velocity, ignoring friction.
- 4 Two identical carts collide on a nearly frictionless track. Explain why total momentum can remain constant even if one cart slows down or stops.
Understanding Conservation of Momentum Worked Examples
A conservation calculation depends on where the system boundary is drawn. If two carts collide on a smooth track, treating both carts together is usually useful. The contact forces between them are internal forces.
One cart pushes the other, while the other pushes back with an equal force in the opposite direction. These paired forces can change each cart’s momentum, but they do not change the total for the two cart system. Friction, a wall, or a person holding a cart are outside influences.
Their effect may be small during a brief impact, yet it becomes important when the interaction lasts longer. Good solutions state what belongs in the system before any numbers are used.
Choose one direction as positive and keep that choice for every velocity. On a straight track, right is often positive and left is negative. Write each object’s mass and velocity before the event, then write its values after the event.
This simple organization prevents many sign errors. A negative answer is useful information. It means the object moves opposite to the chosen positive direction.
If the answer gives zero final velocity for the combined system, the objects are at rest after the interaction. Check whether this fits the situation. A lighter cart hit by a heavier fast cart should not usually send the heavier cart backward at an unrealistically large speed.
Momentum conservation does not mean that kinetic energy always stays the same. During many everyday collisions, some kinetic energy becomes sound, heat, and energy stored in bent or crushed materials. A car crash, a clay ball hitting a wall, and two carts that lock together are examples.
The momentum calculation still works for a suitably isolated system, even though the collision looks messy. Elastic collisions are a special ideal case. They require no net loss of kinetic energy, which is hard to achieve exactly with real objects.
Air hockey pucks and some steel ball collisions can be close enough for school problems. In a one dimensional elastic collision, the objects move apart with the same relative speed that they had while approaching.
Impulse explains why the time of a collision matters. A force acting for a longer time produces a larger change in momentum when the force is constant. More generally, the area under a force versus time graph gives the impulse.
Helmets, airbags, padded mats, and crumple zones increase the stopping time. They reduce the average force needed to bring a person or object to rest. When checking a worked example, use kilograms for mass and meters per second for velocity.
Momentum then has units of kilogram meters per second. Compare total momentum before with total momentum after, including signs. For explosions that begin at rest, the parts must leave with opposite total momenta, so the smaller mass usually has the greater speed.