State the law of conservation of momentum and use it to calculate an unknown velocity after a collision when no external force acts (MA STE Introductory Physics, HS-PS2-2).

What this topic is asking

Conservation of momentum is one of the most powerful ideas in physics, and the Massachusetts Introductory Physics standard HS-PS2-2 asks you to use it. You must state the law and calculate an unknown velocity after a collision when no external force acts. The skill is setting up the equation total momentum before = total momentum after, then solving. This is the practice of using mathematics, applied to a system, and it rests on the crosscutting concept that systems can be analyzed by tracking what is conserved.

The law of conservation of momentum

Why is momentum conserved? It follows from Newton's third law. In a collision, the two objects push on each other with equal and opposite forces for the same time, so they receive equal and opposite impulses. One object gains exactly the momentum the other loses, so the total does not change. The phrase "no external force" matters: if an outside force (like friction or a wall) acts, it can change the total. In the brief moment of a collision, the internal forces between the objects are usually far larger than friction, so momentum is conserved to a good approximation.

Setting up the equation

Every conservation-of-momentum problem uses the same structure:

Written out for two objects:

The method:

1. Choose a positive direction (usually the direction of the first object's motion).
2. Write the total momentum before, adding $mv$ for each object, with signs for direction.
3. Write the total momentum after, again summing $mv$ with signs. If objects stick together, use their combined mass and a single velocity.
4. Set before equal to after and solve for the unknown.

A negative answer simply means that object moves in the negative direction.

Common situations

• Objects that stick together (inelastic): after the collision they move as one combined mass with a single velocity. Two carts that couple together are the classic case.
• Recoil or explosion: the objects start together (or at rest) and push apart, like a skater throwing a ball or a gun firing. The total momentum stays at its starting value (often zero), so the two objects move in opposite directions with equal and opposite momenta.
• Objects that bounce apart (elastic): they separate with their own velocities; momentum is still conserved, and you set up the same before-equals-after equation.

Reference-sheet note

The reference sheet gives momentum as $p = mv$ but does not print the conservation law $p_{before} = p_{after}$. You must recall the conservation principle and apply it. This is one of the few relationships on the MCAS that is not handed to you, so it is worth memorizing the before-equals-after structure and practicing the sign bookkeeping for head-on and recoil situations.

Try this

Q1. Two objects collide and stick together. What quantity is the same before and after, assuming no external force? [1]

• Cue. The total momentum of the system.

Q2. A stationary $40$ kg object explodes into two pieces. One piece moves left with $120$ kg m/s of momentum. State the momentum of the other piece. [2]

• Cue. $120$ kg m/s to the right, so that the total stays zero (momentum is conserved and started at zero).