Momentum and Impulse

Overview

Many interactions happen over a short time:

• a bat striking a ball
• a car braking suddenly
• a hammer driving a nail
• a tennis racket returning a serve

In such situations, analysing forces directly with:

$$\sum \vec{F}=m\vec{a}$$

may be inconvenient because the force changes rapidly with time.

Instead, use the ideas of momentum and impulse.

These concepts are essential for collisions, safety design, recoil, and force-time graph questions.

Related hub: Dynamics

Why It Matters

Momentum methods are often simpler than force-based methods when interactions are brief and forces vary strongly with time.

Definition

Linear momentum measures motion using mass and velocity, while impulse measures the effect of a force acting over time.

Key Representations

$$\vec{p}=m\vec{v}$$ $$\vec{F}=\frac{d\vec{p}}{dt}$$ $$J=\Delta p$$ $$J=F\Delta t$$

Linear Momentum

Definition

Linear momentum is defined as:

$$\vec{p}=m\vec{v}$$

Where:

• $m$ = mass
• $\vec{v}$ = velocity

Hence momentum is a vector quantity.

Its direction is the same as velocity.

SI Unit

$${\text{kg m s}}^{-1}$$

Equivalent to:

$$\text{N s}$$

Meaning of Momentum

Momentum measures how difficult it is to stop or change the motion of an object.

Large momentum may arise from:

• large mass
• high speed
• both

Examples:

• truck moving slowly
• bullet moving rapidly

Both may have large momentum.

Direction and Sign Convention

Momentum depends on chosen positive direction.

Example:

Take right as positive.

• object moving right:

$$p=+mv$$

• object moving left:

$$p=-mv$$

Always state sign convention clearly.

Newton’s Second Law in Momentum Form

A resultant force causes momentum to change:

$$\vec{F}=\frac{d\vec{p}}{dt}$$

This is the most general form of Newton’s Second Law.

For constant mass:

$$\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}$$

Impulse

Definition

Impulse is the effect of a force acting over a time interval.

For constant force:

$$J=F\Delta t$$

For variable force:

$$J=\int Fdt$$

Impulse is a vector quantity and has direction of resultant force.

SI Unit

$$\text{N s}$$

Equivalent to:

$${\text{kg m s}}^{-1}$$

Impulse-Momentum Theorem

Impulse equals change in momentum:

$$J=\Delta p$$

or in vector form:

$$\vec{J}=\Delta \vec{p}$$

Thus:

$$J=mv-mu$$

where:

• $u$ = initial velocity
• $v$ = final velocity

Force-Time Graphs

Core Idea

Impulse equals area under the force-time graph.

$$J=\int Fdt$$

This is commonly tested.

Constant Force Rectangle

If force constant:

• height = $F$
• width = $\Delta t$

Area:

$$J=F\Delta t$$

Triangular Graph

If force rises then falls linearly:

$$J=\frac{1}{2}\times \text{base}\times \text{height}$$

Irregular Graph

Split into shapes:

• rectangles
• triangles
• trapezia

Then add signed areas.

Average Force

If momentum changes by $\Delta p$ in time $\Delta t$:

$$F_{\text{avg}}=\frac{\Delta p}{\Delta t}$$

Useful in collisions and braking questions.

Why Increasing Contact Time Reduces Force

From:

$$F_{\text{avg}}=\frac{\Delta p}{\Delta t}$$

For same momentum change:

• larger $\Delta t$
• smaller average force

Applications:

• airbags
• crumple zones
• helmets
• bending knees on landing
• catching ball by moving hands backward

Worked Examples

Example 1: Momentum of Moving Trolley

A 4.0 kg trolley moves right at 3.0 m s${}^{-1}$.

$$p=mv=4(3)=12$$

Answer:

$$12{\text{kg m s}}^{-1}$$

to the right.

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Example 2: Impulse from Constant Force

A force of 20 N acts for 0.50 s.

$$J=F\Delta t=20(0.50)=10$$

Answer:

$$10\text{N s}$$

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Example 3: Change in Momentum

A 2.0 kg ball changes velocity from +5.0 m s${}^{-1}$ to -3.0 m s${}^{-1}$.

$$\Delta p=mv-mu$$ $$=2(-3)-2(5)$$ $$=-6-10=-16$$

Answer:

$$16{\text{kg m s}}^{-1}$$

towards the left.

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Example 4: Average Force During Collision

A 0.20 kg ball moving at 10 m s${}^{-1}$ is stopped in 0.040 s.

Initial momentum:

$$2.0$$

Final momentum:

$$0$$

Change:

$$\Delta p=-2.0$$

Average force:

$$F=\frac{-2.0}{0.040}=-50$$

Answer:

$$50\text{ N}$$

opposite to motion.

Reversal of Direction

If an object rebounds, the momentum change is larger than if it simply stops.

Example:

• moving right initially
• rebounds left finally

Then initial and final momenta have opposite signs, so:

$$|\Delta p|$$

is large.

Hence rebound collisions often involve large forces.

Momentum vs Kinetic Energy

Momentum:

$$p=mv$$

Kinetic energy:

$$KE=\frac{1}{2}m{v}^2$$

Do not confuse them.

Momentum is vector.
Kinetic energy is scalar.

Common Exam Pitfalls

1. Ignoring Direction

Momentum and impulse require sign convention.

2. Forgetting Final Minus Initial

Use:

$$\Delta p=p_f-p_i$$

not the reverse.

3. Missing Units

Use:

• kg m s${}^{-1}$
• N s

4. Using Distance Instead of Time

Impulse depends on time, not displacement.

5. Wrong Graph Area

Use area under graph, not perimeter or peak force.

6. Confusing Force with Impulse

Force is instantaneous.
Impulse includes duration.

Summary Table

Quantity	Formula	Type
Momentum	$\vec{p}=m\vec{v}$	Vector
Impulse	$J=F\Delta t$	Vector
General impulse	$J=\int Fdt$	Vector
Change in momentum	$J=\Delta p$	Vector
Avg force	$F=\frac{\Delta p}{\Delta t}$	Vector

Formula Summary

$$\vec{p}=m\vec{v}$$ $$\vec{F}=\frac{d\vec{p}}{dt}$$ $$J=F\Delta t$$ $$J=\int Fdt$$ $$J=\Delta p$$ $$F_{\text{avg}}=\frac{\Delta p}{\Delta t}$$

Summary

Momentum and impulse provide a clean way to analyse short, rapid interactions when force is changing and awkward to model directly.

Links

Related Links

• Dynamics
• Free-Body Diagrams and Force Analysis
• Momentum Conservation and Collisions
• Forces
• Kinematics

The graph must show resultant force for the area to represent change in momentum of the body.

Impulse-Momentum Theorem

The impulse of the resultant force equals the change in momentum. This explains why increasing the collision time reduces the average force for the same momentum change.

Common Exam Points

• Momentum and impulse are vectors.
• Only resultant force changes the momentum of a body.
• Area under a force-time graph gives impulse, not work.
• A longer impact time can reduce average force even if the momentum change is the same.

Links

• Related: dynamics
• Related: vectors
• Related: newtonian dynamics applications
• Related: momentum conservation and collisions
• Related: dynamics methods and non constant forces
• Misconception: force time graph area
• Misconception: momentum conservation system