Is momentum always conserved?

In a closed system with no external resultant force, yes. If friction or another external force acts, the total momentum of the objects changes.

Why do crumple zones make cars safer?

They increase the time over which the car stops. The change in momentum is fixed, so a longer time means a smaller average force on the occupants.

What are the units of impulse?

N s, which is equivalent to the units of momentum, kg,m,s^-1.

Why is F = p/ t more general than F = ma?

It also handles situations where mass changes or where a stream of matter is involved, such as rockets or jets of water.

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21–22, 97–98MechanicsCore

Momentum and Impulse

$\textbf{Momentum}$ is mass times velocity, and it is conserved in every interaction between objects. Newton actually stated his second law in terms of momentum — force is the rate at which momentum changes — and that form explains why airbags, crumple zones and bending your knees on landing all work.

35 min Topic 6 · Further Mechanics

What you'll be able to do

Calculate momentum as a vector quantity
State Newton’s second law as $F = \Delta p/\Delta t$
Define impulse and find it from a force–time graph
Apply conservation of momentum in one dimension

Momentum

Momentum is $p = mv$ , measured in $\mathrm{kg\,m\,s^{-1}}$ (equivalently N s). It is a $\textbf{vector}$ , so direction matters: an object moving left has momentum of opposite sign to one moving right.

Choosing a positive direction and applying it consistently is essential in every momentum problem. A ball bouncing off a wall reverses its momentum, so its change is $\textit{twice}$ its magnitude, not zero.

p = mv

Momentum — a vector.

Example 1A 0.15 kg ball hits a wall at

20\ \mathrm{m\,s^{-1}}

and rebounds at

15\ \mathrm{m\,s^{-1}}

. Find its change in momentum.

1Take towards the wall as positive: initial

p = 0.15 \times 20 = +3.0

2Final

p = 0.15 \times (-15) = -2.25

\Delta p = -2.25 - 3.0

Answer

-5.25\ \mathrm{kg\,m\,s^{-1}}

(5.25 away from the wall).

Tip — A rebound change in momentum is the sum of the two magnitudes. Getting 0.75 instead of 5.25 means the sign of the rebound velocity was ignored.

Force as rate of change of momentum

Newton’s second law in its general form is $F = \dfrac{\Delta p}{\Delta t}$ : the resultant force equals the rate of change of momentum.

For constant mass this reduces to $F = ma$ , since $\Delta p/\Delta t = m\Delta v/\Delta t$ . But the momentum form also covers changing mass, such as a rocket expelling fuel or water hitting a surface.

F = \dfrac{\Delta p}{\Delta t}

General form of Newton’s second law.

F = ma

Special case for constant mass.

Example 1Water leaves a hose at

12\ \mathrm{m\,s^{-1}}

at a rate of

0.50\ \mathrm{kg\,s^{-1}}

and stops dead on a wall. Find the force on the wall.

1Each second, 0.50 kg loses

12\ \mathrm{m\,s^{-1}}

F = \dfrac{\Delta p}{\Delta t} = 0.50 \times 12

Answer

6.0\ \mathrm{N}

Impulse

$\textbf{Impulse}$ is force multiplied by the time it acts: $F\Delta t$ . Rearranging the second law shows impulse equals change in momentum.

When the force varies, impulse is the $\textbf{area under a force–time graph}$ . That is how sports scientists analyse a kick or a jump from force-plate data.

The same change in momentum can come from a large force over a short time or a small force over a long time. Safety features — airbags, crumple zones, crash mats, bending knees on landing — all increase $\Delta t$ to reduce the force for the same $\Delta p$ .

F\Delta t = \Delta p

Impulse equals change in momentum.

\text{impulse} = \text{area under } F\text{–}t

For a varying force.

Example 1A 70 kg passenger travelling at

15\ \mathrm{m\,s^{-1}}

is stopped by an airbag in 0.15 s. Without it, a dashboard would stop them in 0.010 s. Compare the average forces.

\Delta p = 70 \times 15 = 1050\ \mathrm{kg\,m\,s^{-1}}

in both cases.

2Airbag:

F = 1050/0.15 = 7000

3Dashboard:

F = 1050/0.010 = 105\,000

AnswerThe airbag reduces the force fifteenfold, from about 105 kN to 7 kN.

Tip — In safety explanations, say that $\Delta p$ is fixed by the situation and the device increases $\Delta t$ , so $F = \Delta p/\Delta t$ decreases.

Conservation of momentum

In a closed system — no external resultant force — the total momentum before an interaction equals the total momentum after.

It follows from Newton’s third law: during a collision each object exerts an equal and opposite force on the other for the same time, so their momentum changes are equal and opposite and cancel in the total.

It applies to explosions too. A stationary system that breaks apart has zero total momentum, so the fragments move off with equal and opposite momenta — which is why a gun recoils.

\Sigma p_{\text{before}} = \Sigma p_{\text{after}}

Closed system.

Example 1A 2.0 kg trolley at

3.0\ \mathrm{m\,s^{-1}}

collides with a stationary 1.0 kg trolley and they stick together. Find their common velocity.

1Before:

2.0 \times 3.0 + 0 = 6.0

2After:

(2.0 + 1.0)v = 3.0v

v = 6.0/3.0

Answer

2.0\ \mathrm{m\,s^{-1}}

Example 2A 4.0 kg rifle fires a 0.010 kg bullet at

400\ \mathrm{m\,s^{-1}}

. Find the recoil velocity.

1Total momentum before is zero.

0 = 0.010 \times 400 + 4.0v

v = -4.0/4.0

Answer

-1.0\ \mathrm{m\,s^{-1}}

— backwards.

Equation recap

p = mv

Momentum.

F = \Delta p/\Delta t

Second law, general form.

F\Delta t = \Delta p

Impulse.

\Sigma p_{\text{before}} = \Sigma p_{\text{after}}

Conservation.

Common mistakes to avoid

Ignoring direction when calculating a rebound change in momentum.

Treat momentum as a vector; a rebound change adds the magnitudes.

Saying airbags reduce the change in momentum.

The change is the same; the airbag increases the time, reducing the force.

Applying conservation when an external force acts.

Conservation needs a closed system.

Assuming kinetic energy is conserved whenever momentum is.

Momentum is always conserved in a closed system; kinetic energy only in elastic collisions.

Key takeaways

Momentum $p = mv$ is a vector; signs matter.
Resultant force is the rate of change of momentum, $F = \Delta p/\Delta t$ .
Impulse $F\Delta t$ equals change in momentum and is the area under $F$ – $t$ .
Safety devices increase impact time to reduce force.
Total momentum is conserved in a closed system, including collisions and explosions.

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