Classical Mechanics · Chapter 5

Momentum & Collisions

Momentum is the most conserved quantity in mechanics — it's conserved even when energy is not, even across relativity.

Prerequisites

Newton's laws \cdot Vectors \cdot Energy concepts (helpful

Learning Goals

Use momentum and impulse to connect force, time, and changes in motion.
Recognize when a system is isolated enough to conserve total momentum.
Classify collisions as elastic, perfectly inelastic, or partially inelastic.
Use coefficient of restitution with momentum conservation to solve 1D collisions.
Separate collision stages from later energy-conversion stages in ballistic pendulum problems.

5.1 Linear Momentum

Newton's second law is most generally stated as F = dp/dt, not F = ma. When mass is constant these are equivalent, but the momentum form extends naturally to variable-mass systems (rockets) and special relativity.

Definition 5.1 — Linear Momentum and Impulse

The linear momentum of an object is:

p = mv \qquad [units: kg\cdotm/s

The impulse J is the change in momentum, equal to the integral of force over time:

J = \Deltap = \intF dt = F_{avg} \cdot \Deltat

This is why airbags save lives: same \Deltap (car decelerates to rest either way), but larger \Delta

t \to smaller F_{avg} \to survivable force

Theorem 5.1 — Conservation of Momentum

For an isolated system (no net external forces), total linear momentum is conserved:

p_{total} = \summ_{i}v_{i} = const. \qquad (when F_{external} = 0

This follows directly from Newton's third law: internal forces come in equal-and-opposite pairs and cancel in the total. Conservation of momentum holds even when energy is not conserved.

5.2 Collisions and Coefficient of Restitution

All collisions conserve momentum. They differ in what happens to kinetic energy:

Elastic (e = 1): KE is conserved. Microscopic collisions (billiard balls at low speed, atomic collisions) are approximately elastic.
Perfectly inelastic (e = 0): Objects stick together. Maximum KE loss consistent with momentum conservation.
Partially inelastic (0 < e < 1): Most real macroscopic collisions. KE is lost to heat, sound, deformation.

The coefficient of restitution e is defined as the ratio of relative speed after to relative speed before:

e = \frac{v_2' - v_1'}{v_1 - v_2} \qquad 0 \le e \le 1

(5.1)

Combined with conservation of momentum m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′, solving the system:

v_1' = \frac{(m_1 - em_2)v_1 + (1+e)m_2v_2}{m_1+m_2}

(5.2)

v_2' = \frac{(m_2 - em_1)v_2 + (1+e)m_1v_1}{m_1+m_2}

(5.3)

Example 5.1 — Newton's Cradle — Equal Mass Elastic

$Ball 1 (m = 1 kg) moving at 3 m/s strikes stationary Ball 2 (m = 1 kg) elastically (e = 1$ .

Apply (5.2):

v_{1}′ = [(1-1\times1)(3) + (1+1)(1)(0)] / 2 = 0/2 =

0 m/s

Apply (5.3):

v_{2}′ = [(1-1)(0) + (1+1)(1)(3)] / 2 = 6/2 =

3 m/s

Result:Ball 1 stops completely; Ball 2 moves at 3 m/s. This is the Newton's Cradle effect — for equal masses, elastic collisions transfer velocity completely.

Example 5.2 — Ballistic Pendulum

A 10 g bullet embeds in a 2 kg pendulum bob at rest. The pendulum rises 8 cm. Find the bullet's initial speed.

Phase 2 (energy):½(

m+M)v_{f}^{2} = (m+M)gh \to v_{f} = \sqrt(2gh) = \sqrt(2\times9.81\times0.08) = 1.253 m/s

Phase 1 (momentum):

mv_bullet = (m+M)v_{f} \to v_{bullet} = (m+M)v_{f}/m = (2.01)(1.253)/0.01 =

251.9 m/s

Note:Momentum conserved in the collision (Phase 1); energy conserved in the swing (Phase 2). You can't use energy conservation across the collision itself — the impact is inelastic.

Block 1 (blue)

Mass3 kg

Initial velocity4 m/s

Block 2 (orange)

Mass2 kg

Initial velocity-1 m/s

Coefficient of restitution (e)Elastic

Predicted outcome

Before: p10.00 kg·m/s

Before: KE25.00 J

v₁ after0.00 m/s

v₂ after5.00 m/s

After: p10.00 kg·m/s ✓

After: KE25.00 J

KE lost0.00 J (0%)

Figure 5.1. 1D collis

ion simulation. The coefficient of restitution slider morphs from perfectly inelastic (e =

ts stick) to elastic (e = 1, billiard ball behavior). The stats panel shows momentum and K

E before and after.

Definition 5.2 — Common Traps

Momentum is vectorial: signs or directions matter even in one dimension.
Momentum conservation needs a system: external impulse breaks conservation for the chosen objects.
Kinetic energy is not always conserved: only elastic collisions conserve KE.
Objects sticking together is not enough information for energy: use momentum for the collision, then energy for later motion if appropriate.
Impulse depends on time: $increasing stopping time reduces average force for the same \Deltap$

Exercises — 5.1–5.2 Momentum

A 145 g baseball moving at 40 m/s is hit straight back at 40 m/s. Contact time is 2 ms. Find (a) the impulse and (b) the average force during contact.

Straightforward

3 kg block moving at 4 m/s collides elastically with a 2 kg block moving at -1 m/s. Find t

he velocities after and verify KE conservation.

m/s

Straightforward

3.A 1200 kg car traveling 20 m/s rear-ends a stationary 2000 kg truck. They stick together. Find the final speed and the kinetic energy lost.

Intermediate

4.Derive a general formula

for the fraction of kinetic energy lost in a perfectly inelastic collision between mass m_{1}

ary). Evaluate for equal masses and for m_{1} ≪ m_{2}

Challenging

Key Takeaways

$Momentum p = mv is always conserved in an isolated system — even when energy is not$ .
$Impulse J = \Deltap = F_{avg} \Deltat; increasing collision time reduces peak force (airbags, crumple$ zones).
$Elastic (e=1): KE conserved. Perfectly inelastic (e=0): objects stick. Most real collision$ s are in between.
For equal-mass elastic collisions, velocities are exchanged — the basis of Newton's cradle.
The ballistic pendulum is a classic example of combining inelastic collision (momentum) with energy conservation separately.