Momentum and collisions

TL;DR:

• Momentum is mass times velocity, p = mv, and it has a direction.

• In a collision with no outside net force, the total momentum before equals the total momentum after.

• Kinetic energy is only conserved in elastic collisions. Sticking together loses some to heat, sound and deformation.

Two lab carts sit on a low-friction track. One rolls toward the other, they hit, and Velcro on the ends makes them stay joined. How fast do they move afterwards? You do not need to know anything about the forces inside the crash. One conservation law answers the question.

The idea in plain words

Momentum is mass times velocity: p = mv, measured in kg·m/s. It captures how hard something is to stop. A slow truck and a fast bicycle can carry the same momentum. Because velocity has a direction, momentum does too, so in one dimension you give it a sign, positive to the right and negative to the left.

The law is simple to state. If no net outside force acts on a system, its total momentum stays constant. When two objects collide, whatever momentum one loses the other gains.

Why momentum is conserved

When cart A pushes on cart B, Newton's third law says B pushes back on A with an equal and opposite force, for exactly the same duration. The change in momentum equals force times time, so A's change is the exact negative of B's. The sum does not change.

The catch is the word outside. Friction with the track or a hand pushing from outside would change the total. Collisions are usually so brief that these outside forces have almost no time to act, which is why the rule works so well just before and after an impact.

Worked example: two carts that stick together

Cart A has mass 2 kg and moves right at 3 m/s. Cart B has mass 1 kg and is at rest. They stick together. Find their common velocity, then the energy lost. (These are example numbers.)

1. Momentum before: p = (2 × 3) + (1 × 0) = 6 kg·m/s to the right.

2. After sticking, the combined mass is 3 kg moving at v: p = 3v.

3. Conservation: 3v = 6, so v = 2 m/s to the right.

4. Kinetic energy before: ½ × 2 × 3² = 9 J.

5. Kinetic energy after: ½ × 3 × 2² = 6 J.

6. Energy lost: 9 − 6 = 3 J, turned into heat, sound and squashing the Velcro.

Elastic versus inelastic

Sticking together is a perfectly inelastic collision: momentum is conserved, kinetic energy is not. In an elastic collision both are conserved, as happens (very nearly) with hard steel balls. Most real collisions fall between the two.

To see the contrast, repeat the example with an elastic bounce. For a moving mass m₁ hitting a stationary m₂ in one dimension, the final speeds are v₁ = (m₁ − m₂)/(m₁ + m₂) × u and v₂ = 2m₁/(m₁ + m₂) × u, with u the initial speed. Here that gives v₁ = (1/3) × 3 = 1 m/s and v₂ = (4/3) × 3 = 4 m/s.

How to check with numbers

• Momentum. Add up m × v for everything after the collision and compare with before. Both were 6 above.

• Direction. If the signs are mixed, such as one cart heading left, keep the minus signs. A head-on collision can give an answer of zero total momentum.

• Energy never increases. A collision with no explosion cannot make kinetic energy grow. If your final energy is higher than the initial, the arithmetic is wrong.

• Plausibility. The stuck pair moves slower than the moving cart alone (2 m/s versus 3 m/s) because the same momentum is spread over more mass.

Common mistakes

• Conserving velocity or speed instead of momentum.

• Assuming kinetic energy is conserved in every collision. It is conserved only in the elastic case.

• Dropping the sign of a velocity moving the opposite way.

• Applying the law when a big outside force, such as a wall, acts. A ball hitting a wall changes its own momentum because the wall (attached to the Earth) takes the difference.

• Using weight instead of mass.

Safety: If you try this with real carts or toy cars, keep fingers out from between them, and do not use anything heavy or sharp. A collision that is gentle on a track can pinch.

Where you will use it next

Rockets, recoil and explosions use the same law run in reverse: total momentum starts at zero, so the pieces fly off in opposite directions. The energy bookkeeping continues in work, energy and power. Collisions are also a good thing to try in a simulation, and DuckyHelper can run simulations for you while you ask questions out loud.

Frequently asked questions

Why does a small bullet do so much damage if its mass is tiny?

Its momentum is small compared with a truck, but it stops over a very short time and distance, so the force on the target is large. Impact force is the change in momentum divided by the time it takes.

Is momentum conserved if friction is acting?

Not exactly. Friction is an outside force. But during the instant of a collision it is usually small compared with the collision forces, so the law is still a good approximation across the impact.