Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Linear momentum is mass times velocity. In this course, “momentum” on its own means linear momentum.
- Momentum is a vector. It has the same direction as the velocity, because mass is never negative.
- In one dimension, state an axis and give p_x the sign of v_x. In two dimensions, work with components.
- The momentum of a system is the vector sum of the momenta of its parts. Parts moving in opposite directions can cancel.
- A collision model fits when the forces between the objects are much larger than the net external force during the interaction. Only the states just before and just after are analysed, so each object can be treated as a point.
- An explosion model fits when forces inside the system push its parts apart.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| p = mv | m in kg, v in m/s, p in kg·m/s | find the momentum of one object |
| p_x = m v_x, p_y = m v_y | components, signed | handle two-dimensional motion |
| p = √(p_x² + p_y²) | kg·m/s | find the size of a 2D momentum |
| p_sys = p₁ + p₂ + … | signed sum (1D) or vector sum | find the momentum of a system |
| K = ½mv² = p² ÷ (2m) | K in J | compare momentum with kinetic energy |
| slope of p_x against v_x | kg | equals the mass |
Assumptions behind the numbers
- Each object is modelled as a point with one mass and one velocity.
- Masses are constant. Systems that gain or lose mass are not analysed with numbers in this course.
- The reference frame is at rest relative to the ground (inertial) unless stated.
- For collisions and explosions, external forces such as friction and gravity are small compared with the internal forces during the brief interaction.
Mistakes to avoid
- No axis stated. A negative momentum means nothing without one.
- Adding sizes instead of signed values. Opposite momenta cancel.
- Assuming the heavier object has more momentum. Check the speeds too.
- Treating momentum like kinetic energy. Doubling speed doubles p but multiplies K by 4.
- Saying a bounce at the same speed leaves momentum unchanged. The direction has reversed, so p has changed.
- Thinking zero system momentum means nothing moves. The parts can move in opposite directions.
- Using the collision model when an external force dominates, such as a slow slide to rest on a rough surface.
Quick self-check
- Take +x to the right. A 0.050 kg toy car moves left at 20 m/s. What is its momentum? (−1.0 kg·m/s)
- An object’s mass and speed both double. By what factor do its momentum and its kinetic energy change? (Momentum ×4; kinetic energy ×8.)
- A 2.0 kg cart moves at +1.0 m/s and a 1.0 kg cart moves at −2.0 m/s. What is the total momentum? (Zero, although both carts move.)
Next: practice questions.