Conservation of linear momentum
During a direct impact, the two bodies exchange momentum. Each body can gain or lose momentum, but the total signed momentum of the chosen system remains constant under the collision model.
Choose the system and the short event
Definition
What is meant by conservation of linear momentum?
Model answer: During the very short modelled impact, external forces are ignored. The total signed momentum of the complete chosen system is then the same before and after the impact.
- The system is the complete set of bodies whose momenta you add.
- The event is the brief impact between the before and after snapshots.
- External forces such as friction are ignored during that brief event in the particle model.
- The contact forces between the two bodies are an equal and opposite pair inside the system. They change each body's momentum, but their effects cancel when the two momenta are added.
Common mistake
Write one signed before-and-after equation
Swipe left or right to see the whole diagram →
- Draw the bodies before and after the event.
- Choose one positive direction and label signed velocities.
- Write every momentum term as mass × signed velocity.
- Set the total before equal to the total after.
- Solve first; translate the sign of the answer last.
Worked example
A direct impact with one body initially at rest
A 4 kg particle moving right at 6 m s⁻¹ hits a stationary 2 kg particle. After the impact, the first particle moves right at 1.5 m s⁻¹. Find the second particle's velocity.
The positive result means 9 m s⁻¹ to the right.
Examiner note
Count equations and unknowns before solving
In one-dimensional motion, momentum conservation supplies one number equation. It can determine one unknown directly. If two quantities are unknown, the question must provide another relationship, such as one body being stationary, both bodies having equal speed, or one velocity being a multiple of another.
Worked example
Use the extra relationship
A 3 kg particle moving right at 5 m s⁻¹ collides with a stationary 2 kg particle. Afterwards the first moves left with speed and the second moves right with speed .
Two particles A and B of masses 0.4 kg and 0.6 kg move directly towards one another. A moves right at 5 m s⁻¹ and B moves left at 2 m s⁻¹. After impact A moves left at 1 m s⁻¹. Find the velocity of B after impact and state its speed and direction.
Show worked answer
Take right as positive.
B moves at 2 m s⁻¹ to the right.