Chapter 1.7: Linear Momentum and Impulse
Welcome to one of the most exciting and practical areas of mechanics! Have you ever wondered why catching a fast-moving cricket ball hurts your hands unless you pull them back, or how modern car crumple zones and airbags save lives in high-speed crashes? The answers lie in momentum and impulse.
In this chapter, we will break down Newton's laws in their most powerful form, explore how objects interact during collisions and explosions, and master the exact problem-solving techniques you need for your CCEA AS 1 exam.
Don't worry if mechanics has felt tricky before! We will build everything step-by-step with clear definitions, sign rules, and exam tips.
---1. Linear Momentum (\(p\))
What is Momentum?
In physics, linear momentum is a measure of how difficult it is to stop a moving object. It depends on both how massive the object is and how fast it is moving.
Formal Definition: Linear momentum (\(p\)) is defined as the product of an object's mass (\(m\)) and its linear velocity (\(v\)).
\(p = mv\)
Key Facts About Momentum:
1. Standard SI Units: \(\text{kg m s}^{-1}\) (kilogram metres per second). It can also be written equivalently as \(\text{N s}\) (Newton seconds).
2. Vector Quantity: Momentum has both magnitude and direction. The direction of an object's momentum is always identical to the direction of its velocity.
3. Real-World Comparison: A slow-moving supertanker has huge momentum because its mass \(m\) is enormous. A tiny bullet has huge momentum because its velocity \(v\) is immense. A stationary lorry has zero momentum (\(v = 0\)), making it easy to keep at rest!
Vector Sign Rule (Crucial for Calculations)
Because momentum is a vector, you must assign a positive direction before starting any calculation:
- Choose motion to the right (or forward) as positive (\(+\)).
- Motion to the left (or backward/rebound) must then be negative (\(-\)).
Forgetting this minus sign is the single most common mistake in AS physics exams!
Key Takeaway: Momentum is \(p = mv\). It is a vector quantity measured in \(\text{kg m s}^{-1}\) or \(\text{N s}\), and direction matters every single time.
---2. Newton's Second Law & Impulse of a Force
Newton's Second Law in Terms of Momentum
You may be familiar with \(F = ma\), but Isaac Newton originally formulated his Second Law in terms of momentum:
Statement: The rate of change of momentum of an object is directly proportional to the resultant force applied to it and occurs in the direction of the force.
\(F = \frac{\Delta p}{\Delta t} = \frac{m(v - u)}{t}\)
Where:
- \(F\) = Resultant force (\(\text{N}\))
- \(\Delta p = mv - mu\) = Change in linear momentum (\(\text{kg m s}^{-1}\) or \(\text{N s}\))
- \(\Delta t\) or \(t\) = Time duration over which the force acts (\(\text{s}\))
- \(u\) = Initial velocity, \(v\) = Final velocity
Note: When the mass \(m\) remains constant, \(\frac{m(v - u)}{t} = m\left(\frac{v - u}{t}\right) = ma\), which gives us the familiar formula \(F = ma\).
Impulse of a Force (\(J\) or \(\text{Impulse}\))
When a force acts on an object for a short time (like a bat hitting a ball), we calculate the impulse of that force.
Definition: The impulse of a force is defined as the product of the average resultant force acting on an object and the time duration for which it acts.
\(\text{Impulse} = F \Delta t\)
By rearranging Newton's Second Law (\(F \Delta t = \Delta p\)), we obtain the Impulse-Momentum Theorem:
\(\text{Impulse} = \Delta p = mv - mu\)
- SI Unit of Impulse: \(\text{N s}\) (or \(\text{kg m s}^{-1}\)).
- Vector Nature: Impulse is a vector quantity that acts in the same direction as the resultant force.
Force–Time (\(F\text{–}t\)) Graphs
Forces in real-life collisions are rarely constant; they spike rapidly upon contact and decrease as objects separate.
- On any Force–time graph, the area under the curve represents the Impulse (which equals the change in momentum, \(\Delta p\)).
- For a constant force: \(\text{Impulse} = \text{Area of rectangle} = F \Delta t\).
- For a variable force: \(\text{Impulse} = \text{Area under curve} = \int F \, dt\).
Examiner Warning: Never simply read the maximum peak value off the vertical axis when asked for impulse! You must calculate or estimate the total area under the graph.
Key Takeaway: \(\text{Impulse} = F \Delta t = \Delta p\). Impulse equals change in momentum and is represented by the area under a Force–time graph.
---3. Principle of Conservation of Linear Momentum
The Core Principle
One of the most fundamental laws of the universe governs what happens when objects collide or push apart.
Official Principle of Conservation of Linear Momentum:
For an isolated (or closed) system where no external resultant forces act, the total linear momentum before a collision or explosion is equal to the total linear momentum after the event.
Mathematical Statement for One Dimension:
\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\)
Where:
- \(m_1, m_2\) = Masses of objects 1 and 2
- \(u_1, u_2\) = Initial velocities before collision
- \(v_1, v_2\) = Final velocities after collision
Exam Condition Checklist:
Whenever you write down the definition in an exam, you must state the condition: "provided no external resultant force acts" or "in a closed/isolated system". Omitting this phrase will lose you the definition mark!
Step-by-Step Problem-Solving Method:
1. Draw a quick sketch showing the objects before and after the interaction.
2. Assign a positive direction (e.g. to the right \(\rightarrow\) is \(+\)).
3. Assign signs to every velocity (if an object moves left, write its velocity with a minus sign!).
4. Write down the equation: \(\text{Total momentum before} = \text{Total momentum after}\).
5. Substitute values carefully and solve for the unknown.
Key Takeaway: Momentum is always conserved in closed systems (\(\sum p_{\text{initial}} = \sum p_{\text{final}}\)). Always define your positive direction first!
---4. Classifying Collisions and Explosions
While total momentum is conserved in all closed systems, the behaviour of kinetic energy (\(E_k = \frac{1}{2}mv^2\)) allows us to classify interactions into three distinct types:
1. Perfectly Elastic Collisions
- Linear Momentum: Conserved (\(p_{\text{before}} = p_{\text{after}}\)).
- Total Energy: Conserved.
- Total Kinetic Energy: Conserved (\(\sum \frac{1}{2}m u^2 = \sum \frac{1}{2}m v^2\)).
- Characteristics: No kinetic energy is converted into heat, sound, or permanent deformation.
2. Inelastic Collisions
- Linear Momentum: Conserved.
- Total Energy: Conserved (energy cannot be created or destroyed).
- Total Kinetic Energy: Not conserved (some \(E_k\) is converted into thermal energy, sound energy, or work done in deforming the objects).
- Completely (Perfectly) Inelastic Collisions: A special case where colliding bodies stick together (coalesce) and move off with a single common final velocity \(v\):
\(m_1 u_1 + m_2 u_2 = (m_1 + m_2) v\)
3. Explosions
- A single object breaks apart, or two objects push away from each other.
- If the system is initially stationary, the initial momentum is zero (\(p_{\text{initial}} = 0\)).
- By conservation of momentum, the final momentum must also sum to zero:
\(0 = m_1 v_1 + m_2 v_2 \implies m_1 v_1 = -m_2 v_2\)
- The two fragments must travel in opposite directions.
- Kinetic energy increases in an explosion because stored chemical or elastic potential energy is released into kinetic energy.
Quick Comparison Table
- Elastic Collision: Momentum Conserved? Yes | Total Energy Conserved? Yes | Kinetic Energy Conserved? Yes
- Inelastic Collision: Momentum Conserved? Yes | Total Energy Conserved? Yes | Kinetic Energy Conserved? No (Decreases)
- Explosion: Momentum Conserved? Yes | Total Energy Conserved? Yes | Kinetic Energy Conserved? No (Increases)
Key Takeaway: Momentum is conserved in every collision in a closed system. Kinetic energy is ONLY conserved in elastic collisions.
---5. Real-World Applications: Vehicle Safety & Impacts
How Safety Features Protect Us
Safety devices like crumple zones, airbags, seatbelts, and gymnasium crash mats all use the exact same physics principle to prevent serious injury.
The Standard Exam Explanation (Step-by-Step):
Whenever you are asked to explain how a safety feature works, structure your answer in three clear steps:
Step 1: Fixed Change in Momentum
During a crash, a passenger of mass \(m\) moving at initial speed \(u\) must come to a complete stop (\(v = 0\)). Therefore, the total change in momentum \(\Delta p = m(0 - u) = -mu\) is completely fixed and constant regardless of what safety features are present.
Step 2: Increasing Impact Time
The safety feature (e.g. crumple zone deforming, airbag compressing, or seatbelt stretching) significantly increases the time duration of impact (\(\Delta t\)) over which the passenger comes to rest.
Step 3: Reducing Resultant Force
According to Newton's Second Law:
\(F = \frac{\Delta p}{\Delta t}\)
Because \(\Delta p\) is constant, increasing \(\Delta t\) decreases the rate of change of momentum, which significantly reduces the average resultant impact force (\(F\)) acting on the body, thereby reducing injury.
Why Pulling Back Your Hands Helps Catch a Ball:
When you catch a fast cricket ball, bringing your hands back increases the time \(\Delta t\) taken to bring the ball to rest. Since \(\Delta p\) is fixed, the impact force \(F\) on your hands is greatly reduced!
Key Takeaway: Fixed \(\Delta p\) + Larger \(\Delta t\) = Smaller impact force \(F\).
---6. Examiner Pitfalls & Common Mistakes Checklist
Avoid these frequent traps identified in CCEA examiner reports:
1. The "Incomplete Condition" Trap:
Wrong: "Conservation of momentum means total momentum before equals total momentum after."
Right: "Total momentum before equals total momentum after in a closed system / provided no external resultant force acts."
2. Ignoring Opposite Vector Signs:
When two objects rebound or move towards each other, you cannot add their speeds as positive numbers. If an object bounces backward at \(3\text{ m s}^{-1}\), its velocity is \(-3\text{ m s}^{-1}\).
3. Assuming Kinetic Energy is Always Conserved:
Never assume a collision is elastic unless the question explicitly states it, or you calculate \(\sum \frac{1}{2}m u^2\) and \(\sum \frac{1}{2}m v^2\) and prove they are identical.
4. Vague Answers in Safety Questions:
Phrases like "it cushions the blow", "it absorbs the impact", or "it reduces the momentum" receive zero marks. Always state that \(\Delta p\) is constant, \(\Delta t\) increases, and therefore \(F\) decreases.
5. Peak Force vs. Area:
Remember that on a Force–time graph, impulse is the area under the graph, not the highest force reached.
Quick Chapter Summary
- Momentum: \(p = mv\) (Vector, unit: \(\text{kg m s}^{-1}\) or \(\text{N s}\)).
- Impulse: \(\text{Impulse} = F \Delta t = \Delta p = mv - mu\) (Area under an \(F\text{–}t\) graph).
- Newton's 2nd Law: \(F = \frac{\Delta p}{\Delta t}\) (Resultant force = rate of change of momentum).
- Conservation of Momentum: \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\) (True for all closed systems where no external resultant forces act).
- Elastic Collisions: Momentum and Kinetic Energy are both conserved.
- Inelastic Collisions: Momentum is conserved; Kinetic Energy is lost to heat/sound/deformation.
- Safety Features: Increase \(\Delta t\) to reduce \(F = \frac{\Delta p}{\Delta t}\) for a fixed \(\Delta p\).