3.5MechanicsStretch

Momentum, Impulse and Collisions

Two cars of identical speed collide head-on — one pair crumples together and stops dead, another pair bounces cleanly apart. Both collisions obey exactly the same conservation law, and the difference between them is precisely what separates an "elastic" collision from an "inelastic" one.

45 min Video by The Organic Chemistry Tutor Module 3.5 Newton’s laws of motion and momentum

Impulse and Momentum Conservation — Inelastic & Elastic CollisionsWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Define linear momentum and use $p = m v$
State the principle of conservation of momentum
Define impulse and relate it to the change in momentum and to a force–time graph
Distinguish elastic from inelastic collisions
Solve collision problems using conservation of momentum

Momentum

$Linear momentum$ is the product of an object’s mass and velocity — a vector quantity, in the same direction as the velocity, measured in kg m s⁻¹.

p = m v

Linear momentum.

1Use

p = m v

p = 1400 \times 18

Answer

p = 25200

kg m s⁻¹

Conservation of momentum

The principle of conservation of momentum states that, for a system with no external resultant force acting on it (such as during a collision or explosion), the total momentum before an event equals the total momentum after. This follows directly from Newton’s third law: the internal forces the colliding objects exert on each other are always equal and opposite, so they cause equal and opposite changes in momentum, leaving the total unchanged.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Conservation of momentum for a two-object collision.

1Total momentum before:

(1200 \times 20) + (800 \times 0) = 24000

kg m s⁻¹.

2After the collision, both move together with combined mass

1200 + 800 = 2000

kg.

3Conservation of momentum:

2000 \times v = 24000

, so

v = \frac{24 000}{2000}

Answer

v = 12

m s⁻¹

Tip — Always define a positive direction before starting a collision problem — velocities in the opposite direction must be given a negative sign, or the momentum totals will come out wrong.

Impulse

$Impulse$ is the product of a force and the time for which it acts, and is exactly equal to the change in momentum it produces. On a force–time graph, impulse is the area under the graph, which is particularly useful when a force varies during a collision (as almost all real collision forces do).

Impulse = F Δ t = Δ p = m Δ v

Impulse equals the change in momentum.

1Taking the initial direction as positive: initial momentum

= 0.15 \times 8.0 = 1.2

kg m s⁻¹; final momentum

= 0.15 \times (- 12) = - 1.8

kg m s⁻¹.

2Change in momentum:

Δ p = - 1.8 - 1.2 = - 3.0

kg m s⁻¹.

3Average force:

F = \frac{Δ p}{Δ t} = \frac{- 3.0}{0.010}

AnswerF = −300 N (300 N, in the direction of the rebound).

Tip — A longer contact time reduces the average force needed to produce the same change in momentum — exactly why airbags, crumple zones and bending your knees on landing all reduce injury by extending the collision time.

Elastic and inelastic collisions

In a perfectly $elastic collision$ , both momentum AND total kinetic energy are conserved — the objects separate afterward with no permanent deformation or energy loss. In an $inelastic collision$ , momentum is still conserved, but total kinetic energy is NOT — some is transferred to other forms, such as heat, sound, or the energy of permanent deformation. A collision where the objects stick together afterward (like the car example above) is a special case called a perfectly inelastic collision, where the maximum possible kinetic energy is lost consistent with conserving momentum.

1KE before:

\frac{1}{2} \times 1200 \times 2 0^{2} + 0 = 240000

2KE after:

\frac{1}{2} \times 2000 \times 1 2^{2} = 144000

3KE lost

= 240000 - 144000

Answer96,000 J (96 kJ) of kinetic energy is lost — transferred to heat, sound and deformation.

Tip — Momentum is ALWAYS conserved in a collision with no external force, whether elastic or inelastic — it is only kinetic energy conservation that distinguishes the two types.

Equation recap

p = m v

Linear momentum.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Conservation of momentum.

Impulse = F Δ t = Δ p

Impulse-momentum relationship.

Common mistakes to avoid

Assuming kinetic energy is always conserved in a collision.

Only MOMENTUM is always conserved (with no external force) — kinetic energy is conserved only in a perfectly elastic collision.

Forgetting to assign a negative sign to velocities in the opposite direction during a collision calculation.

Momentum is a vector — define a positive direction at the start and consistently apply negative signs to anything moving the opposite way.

Believing a longer collision time produces a LARGER force for the same change in momentum.

For a fixed change in momentum, a LONGER contact time gives a SMALLER average force (F=Δp/Δt) — this is the entire principle behind airbags and crumple zones.

Key takeaways

Momentum p = mv, a vector quantity.
Conservation of momentum: total momentum before = total momentum after, provided no external resultant force acts.
Impulse = FΔt = change in momentum; a longer contact time reduces the average force for a given momentum change.
Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum.

Test yourself

Ready to lock in Momentum, Impulse and Collisions? Pick a mode and earn XP & Dobloons.

Momentum, Impulse and Collisions

45 min Video by The Organic Chemistry Tutor Module 3.5 Newton’s laws of motion and momentum