Impulse & Collisions · 10–15% of the AP Physics 1 exam
p = mv
J = F·Δt
J = Δp
p_i = p_f
The basics
What's covered: Linear momentum, impulse, the impulse-momentum theorem, conservation of momentum, and elastic vs. inelastic collisions.
Exam weight: 10–15%
The big question: How do we analyze collisions and explosions, where forces are complicated but momentum stays simple?
The master idea: Momentum is conserved when no external force acts. This works even when energy isn't conserved.
📐 Key equations
p = mv
Linear momentum. Vector quantity. Same direction as velocity. SI unit: kg·m/s.
J = F_avg · Δt
Impulse. Force times time. A vector in the direction of the net force. Units: N·s (same as momentum).
J = Δp = p_f − p_i
Impulse-momentum theorem. The impulse on an object equals its change in momentum.
F_net = Δp/Δt
Newton's 2nd law (momentum form). Force is the rate of change of momentum. Reduces to F = ma when mass is constant.
p_i,total = p_f,total
Conservation of momentum. When net external force is zero, total system momentum is constant.
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
General 1D collision. Total momentum before equals total momentum after.
m₁v₁ + m₂v₂ = (m₁+m₂)v_f
Perfectly inelastic collision. Objects stick together with common final velocity.
v_cm = (m₁v₁ + m₂v₂)/(m₁+m₂)
Center-of-mass velocity. Total momentum divided by total mass. Constant if no external force.
Elastic: KE_i = KE_f
Elastic collision. Kinetic energy is conserved AND momentum is conserved.
Inelastic: KE_f < KE_i
Inelastic collision. KE decreases (heat/sound/deformation), but momentum is STILL conserved.
The 4 topics at a glance
4.1 Linear Momentum
p = mv. Vector, same direction as velocity. Used to model collisions and explosions. Total system momentum is the vector sum of all individual momenta.
4.2 Change in Momentum and Impulse
J = F·Δt = Δp. Impulse measures how much momentum a force transfers. The area under a F-t graph equals impulse. The slope of a p-t graph equals net force.
4.3 Conservation of Momentum
When net external force = 0, total momentum is constant. Internal forces cancel via Newton's 3rd law. This makes collisions solvable without knowing the messy internal forces.
4.4 Elastic and Inelastic Collisions
Elastic: KE conserved (both KE and p). Inelastic: KE decreases (only p conserved). Perfectly inelastic: objects stick together; max KE lost.
🧠 How to solve a collision problem (5 steps)
1. Pick your system. Usually all the colliding objects together. If you can include them all, internal forces cancel and external forces are negligible during the brief collision.
2. Identify before and after. Note velocities (with signs!) for each object, both pre- and post-collision.
3. Write conservation of momentum. m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (or with the stuck-together form for perfectly inelastic).
4. Solve for the unknown. Often a single final velocity. For 1D problems, just be careful with signs.
5. Check KE if asked. Compare ½mv² before and after to determine elastic vs. inelastic.
⚖️ The conservation comparison
Momentum (p = mv): Conserved in ALL collisions when net external force = 0. Always check this first.
Kinetic energy (½mv²): Conserved ONLY in elastic collisions. In inelastic ones, some KE becomes heat/sound/deformation.
Total energy: Always conserved (Unit 3). It just changes form.
The exam trick: Use momentum to find final velocity. Then check KE separately to classify the collision.
⚠️ Common exam traps
Momentum is a vector! Two carts moving in opposite directions don't add their momenta — they SUBTRACT (with sign convention). A 5 kg cart moving right at 2 m/s and a 5 kg cart moving left at 2 m/s have total momentum = 0.
Don't confuse momentum and KE. Both depend on m and v, but differently: p = mv (linear, vector), KE = ½mv² (squared, scalar). Doubling speed doubles momentum but quadruples KE.
Inelastic ≠ momentum lost. Inelastic means KINETIC ENERGY decreases, NOT momentum. Momentum is always conserved (when no external force).
Choosing the right system matters. If you only put one cart in your system, that cart's momentum changes (due to external force from the other cart). Include BOTH carts and momentum is conserved.
Impulse depends on Δt, not just F. A small force for a long time can give a bigger impulse than a large force for a short time.
Time interval matters for safety. Crumple zones, airbags, padded gloves — all reduce force by extending the time of impact. The Δp is the same; F just drops.
Center of mass is conserved. v_cm doesn't change in any closed system. Two skaters pushing apart on ice will have a center of mass that doesn't move.