Short on time? This page is the recap. For diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A system’s total momentum is the vector sum of the momenta of its parts. In one dimension, add signed values.
- A system can be treated as one object moving with its centre-of-mass velocity, v_cm = p_sys / M_total.
- Forces between parts of the system are internal. By Newton’s third law their impulses are equal and opposite, so they cannot change the total.
- Only an external impulse changes a system’s momentum. Momentum then moves between the system and its surroundings; none is created or destroyed.
- The choice of system decides whether p_sys is constant. Put both interacting objects inside it when you can.
- In a collision or explosion, internal forces are much larger than external ones for a short time, so momentum just before = momentum just after.
- In two dimensions, conserve x and y components separately. Set up the equations and reason with them; you will not solve simultaneous equations.
Key relationships
| Relationship | Units | Use it to |
|---|---|---|
| p_sys = m₁v₁ + m₂v₂ + … | kg·m/s | find total momentum (signed) |
| v_cm = p_sys / (m₁ + m₂ + …) | m/s | find the velocity of the system as a whole |
| Δp_A = −Δp_B | kg·m/s | link two objects that interact only with each other |
| Δp_sys = J_ext = F_net,ext Δt | kg·m/s = N·s | find a change caused by an external force |
| Σp_x and Σp_y each constant | kg·m/s | set up a two-dimensional collision |
Assumptions behind the numbers
- Objects are treated as particles; only states just before and just after a collision are analysed.
- External forces (friction, air resistance) are small compared with collision forces and act only briefly.
- Motion in one dimension unless the question says otherwise.
Mistakes to avoid
- No system stated. “Momentum is conserved” means nothing until you say for which objects.
- Adding speeds, not velocities. Opposite directions need opposite signs.
- Plain mean as v_cm. Weight each velocity by its mass.
- “Momentum is lost to friction.” It is transferred to Earth; your system’s momentum changes.
- Equal forces, equal velocity changes. The momentum changes are equal in size; the lighter object’s velocity changes more.
- Using conservation over a long time when an external force acts throughout.
Quick self-check
- With +x to the right, a 2.0 kg cart moves at +1.5 m/s and a 1.0 kg cart at −3.0 m/s. What is v_cm? (0 m/s)
- A trolley at rest on a smooth floor fires a 0.20 kg beanbag it was carrying backwards at 6.0 m/s. The empty trolley has a mass of 4.0 kg. How fast does it move? (0.30 m/s forwards)
- A 3.0 N friction force acts on a box for 4.0 s while it is still sliding. By how much does the box’s momentum change? (12 kg·m/s, opposite to its motion)
Next: practice questions.