Physics › 🚀 Mechanics

Momentum & Collisions

Why a slow truck and a fast baseball can hit with the same punch — and how one conservation law predicts every collision.

High schoolAP Physics 1 level
💡
The big idea: Momentum is the "quantity of motion" an object carries — mass times velocity — and it never disappears without a fight. Whenever objects push on each other, whatever momentum one loses, the other gains. This one conservation law lets you predict the outcome of car crashes, rocket launches, and pool-ball collisions without ever needing to untangle the messy forces happening inside the collision itself.
🎯 By the end, you'll be able to
  • Calculate the momentum of an object and reason about how mass and velocity trade off to produce it.
  • Use the impulse-momentum theorem (FΔt = Δp) to find the force, time, or velocity change involved in an impact.
  • Apply conservation of momentum to solve for unknown velocities before or after a collision.
  • Tell elastic collisions apart from inelastic ones by checking whether kinetic energy survives the crash.
📎 You should already know
  • Newton's Laws of Motion
  • Vectors and vector addition
  • Velocity, acceleration, and force basics

The Quantity of Motion

Picture two things barreling toward you: a bowling ball rolling at walking speed, and a golf ball fired from a cannon. Which one do you dodge? Speed alone doesn't tell the whole story — what matters is a combination of how much stuff is moving and how fast it's moving. Physicists bottle that combination into a single number called momentum, and it turns out to be one of the most reliable bookkeeping tools in all of physics.

Momentum matters because it doesn't just describe motion — it's conserved. When objects collide, push off each other, or fly apart, the total momentum right before the event exactly equals the total momentum right after. No exceptions, no fine print.

🔑 The core idea

Momentum is mass times velocity: \(p = mv\). It's a vector, so direction counts — a truck moving north and one moving south don't just add their speeds, they partly cancel. And in any interaction where no outside force interferes, total momentum before equals total momentum after. That one sentence lets you solve collisions you could never untangle by tracking the internal forces alone.

\[ p = mv \]
Momentum is the product of mass (kg) and velocity (m/s); its units are kg·m/s, and its direction matches the velocity's direction.
\[ \Delta p = F \Delta t \]
The impulse-momentum theorem: a force applied over a time interval changes momentum by exactly that amount — this is why stretching out the time of impact softens the force felt.
\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]
Conservation of momentum for a two-object collision: the sum of momenta before the crash equals the sum after.
🎮 Interactive: Collision Lab LIVE
Set the mass and speed of two carts, choose elastic or inelastic, and watch momentum stay constant while kinetic energy does — or doesn't — survive the crash.

Two Very Different Kinds of Crash

Momentum is conserved in every collision — that part never changes. What changes is what happens to kinetic energy.

  • Elastic collisions conserve kinetic energy too. Objects bounce off each other and separate cleanly — think billiard balls or two magnets repelling.
  • Inelastic collisions lose some kinetic energy to heat, sound, or bent metal, even though momentum is still conserved. A perfectly inelastic collision is the extreme case: the objects stick together and move as one afterward.

Real-world crashes — cars, football tackles, a dropped phone — are almost always inelastic. The energy doesn't vanish; it just stops being the tidy, back-and-forth kinetic kind.

📝 Worked example: A 0.15 kg baseball arrives at a catcher's mitt traveling at 20 m/s and is brought to rest in 0.01 s. What average force does the mitt exert on the ball?
  1. Find the change in momentum: \(\Delta p = m(v_f - v_i) = 0.15\,\text{kg} \times (0 - 20\,\text{m/s}) = -3\,\text{kg·m/s}\).
  2. Apply the impulse-momentum theorem: \(F = \dfrac{\Delta p}{\Delta t}\).
  3. Divide: \(F = \dfrac{-3\,\text{kg·m/s}}{0.01\,\text{s}} = -300\,\text{N}\).
✓ About 300 N, directed opposite to the ball's motion. Notice that if the catcher 'gives' with the mitt and stretches the stopping time to 0.1 s instead, the force drops to just 30 N — the same principle behind airbags and bent knees when you land a jump.
📝 Worked example: A 1000 kg car moving at 15 m/s rear-ends a stationary 1500 kg car, and the two lock bumpers and move together. What is their common velocity right after the collision?
  1. Write momentum conservation: \(m_1 v_{1i} + m_2 v_{2i} = (m_1+m_2)v_f\).
  2. Plug in values: \(1000(15) + 1500(0) = (1000+1500)v_f\).
  3. Solve: \(15000 = 2500\,v_f \Rightarrow v_f = 6\,\text{m/s}\).
✓ 6 m/s, in the direction the first car was already moving.
⚠️ Common trap: don't confuse momentum with energy

It's tempting to assume that if momentum is conserved, kinetic energy must be too — it isn't, except in the special elastic case. Also remember momentum is a vector: two objects moving toward each other have momenta that partly (or fully) cancel, not simply add as plain numbers. Finally, conservation of momentum only holds for an isolated system — if an outside force (friction from the ground, a wall, gravity pulling sideways) acts during the collision, some momentum leaks out of your two-object system into the outside world.

Check your understanding

1. Which equation correctly defines momentum?
Momentum is mass times velocity (a vector). ½mv² is kinetic energy, Fd is work, and mgh is gravitational potential energy — different quantities entirely.
2. A 0.5 kg ball moving at 4 m/s hits a wall and bounces straight back at 4 m/s. If the contact time is 0.02 s, what average force does the wall exert on the ball?
Δp = m(v_f − v_i) = 0.5 kg × (−4 − 4) m/s = −4 kg·m/s. Dividing by the 0.02 s contact time gives F = 200 N. Bouncing back doubles the momentum change compared to just stopping in place.
3. Two objects collide and stick together, moving as one afterward. What is this type of collision, and what quantity is NOT conserved?
Objects sticking together after impact defines a perfectly inelastic collision. Momentum is always conserved in an isolated system, but kinetic energy converts into heat, sound, and deformation, so it is not conserved here.
4. A 3000 kg truck moving at 10 m/s collides head-on with a stationary 1000 kg car, and the two lock together. What is the speed of the wreckage right after the collision?
Total momentum before = 3000 kg × 10 m/s + 1000 kg × 0 = 30,000 kg·m/s. After the collision the combined mass is 4000 kg, so v = 30,000 / 4000 = 7.5 m/s.
✅ Key takeaways
  • Momentum p = mv is a vector; it depends on both how much mass is moving and how fast.
  • Impulse (FΔt) equals the change in momentum — this is why airbags, bent knees, and longer stopping times make impacts safer.
  • Total momentum of an isolated system is always conserved, even when kinetic energy is not.
  • Elastic collisions conserve kinetic energy; inelastic collisions — including 'stick together' ones — don't, though momentum still balances out.