Work, Energy, Power Revision Notes for Leaving Cert Applied Maths

Conservation of Momentum

Understanding the principle of conservation of momentum

The principle of conservation of momentum is one of the most fundamental concepts in physics. It states that when no external forces act on a system, the total momentum of that system remains constant. This principle comes directly from Newton's Second Law, which tells us that the rate of change of momentum is proportional to the applied force.

infoNote

The conservation of momentum is a direct consequence of Newton's laws of motion. When Newton formulated his second law as "the rate of change of momentum equals the applied force," he laid the foundation for understanding how momentum behaves in isolated systems.

Momentum is defined as the product of an object's mass and velocity:

$p = mv$

Where:

$p$ = momentum
$m$ = mass (kg)
$v$ = velocity (m/s)
Units: kg m/s

When external forces are absent, momentum is conserved. This means the total momentum before an event (like a collision) equals the total momentum after the event.

What is impulse?

Impulse represents the change in momentum of an object. When a force acts on an object for a certain time, it changes the object's momentum. This change is called impulse.

The impulse-momentum relationship can be expressed as:

$I = \Delta p = p\_f - p\_i = mv\_f - mv\_i$

Where:

$I$ = impulse
$\Delta p$ = change in momentum
$p\_f$ = final momentum
$p\_i$ = initial momentum
Units: newton-seconds (Ns)

Understanding impulse helps us analyse collisions and impacts. In any collision, the impulses experienced by the colliding objects are equal and opposite, following Newton's Third Law.

chatImportant

Newton's Third Law and Impulse: During any collision, the impulse experienced by object A is equal in magnitude but opposite in direction to the impulse experienced by object B. This is why momentum is conserved in isolated systems.

Applying conservation of momentum in one dimension

Let's examine how conservation of momentum works in a simple collision scenario. When a hammer strikes a stake, we can apply the conservation principle to find the final velocities after impact.

The key steps are:

Calculate the momentum of each object before collision
Apply conservation: total momentum before = total momentum after
Solve for unknown final velocities
Calculate impulse for each object

For the hammer and stake example, the hammer initially moves downward at 2 m/s while the stake is initially at rest. After collision, momentum is conserved between both objects, and the impulse on each object is equal in magnitude but opposite in direction.

lightbulbExample

Worked Example: Hammer and Stake Collision

Given:

Hammer: mass = 2 kg, initial velocity = 2 m/s downward
Stake: mass = 0.5 kg, initial velocity = 0 m/s

Step 1: Calculate initial momentum $p\_i = m\_{hammer} \times v\_{hammer} + m\_{stake} \times v\_{stake}$ $p\_i = (2 \text{ kg})(2 \text{ m/s}) + (0.5 \text{ kg})(0 \text{ m/s}) = 4 \text{ kg⋅m/s}$

Step 2: Apply conservation of momentum If they stick together after collision: $p\_f = (m\_{hammer} + m\_{stake}) \times v\_f$ $4 = (2 + 0.5) \times v\_f$ $v\_f = 1.6 \text{ m/s downward}$

Two-dimensional collisions

Conservation of momentum also applies when objects collide at angles. In these cases, we must consider momentum components in both horizontal and vertical directions separately.

Consider a van and car colliding at right angles:

Van: 800 kg moving at 30 m/s horizontally
Car: 700 kg moving at 10 m/s vertically
After collision, they become entangled (perfectly inelastic collision)

The solution involves setting up momentum conservation equations for both x and y directions, using the fact that after collision, both objects move together as one mass, then finding the resultant velocity using vector addition.

lightbulbExample

Worked Example: Two-Dimensional Collision

Step 1: Calculate initial momentum in x-direction $p\_{x,i} = 800 \text{ kg} \times 30 \text{ m/s} = 24,000 \text{ kg⋅m/s}$

Step 2: Calculate initial momentum in y-direction $p\_{y,i} = 700 \text{ kg} \times 10 \text{ m/s} = 7,000 \text{ kg⋅m/s}$

Step 3: Apply conservation (total mass = 1,500 kg) $v\_{x,f} = \frac{24,000}{1,500} = 16 \text{ m/s}$ $v\_{y,f} = \frac{7,000}{1,500} = 4.67 \text{ m/s}$

Step 4: Find resultant velocity $v\_f = \sqrt{v\_{x,f}^2 + v\_{y,f}^2} = \sqrt{16^2 + 4.67^2} = 16.67 \text{ m/s}$

Conservation of momentum with connected systems

When objects are connected by strings or ropes (like in pulley systems), conservation of momentum principles still apply, but we must also consider the constraint that connected objects move together.

In pulley systems, the string creates a connection between masses. When one mass picks up additional mass, momentum is conserved at the instant of change. However, the system's behaviour changes as the total mass changes.

infoNote

Connected Systems Complexity: In connected systems, momentum is conserved at the instant when masses change, but immediately after, the system's motion is governed by the new force balance. This requires analysing both the moment of change (using momentum conservation) and the subsequent motion (using force analysis).

These problems typically involve:

Analysing the system before and after mass changes
Applying conservation of momentum at the moment of change
Using force analysis (Newton's Second Law) to find accelerations
Calculating distances travelled using kinematic equations

Exam tips

When tackling momentum problems in exams, follow these proven strategies:

Always draw clear diagrams showing the situation before and after
Define your coordinate system clearly at the start
Remember that momentum is a vector quantity - direction matters
In two-dimensional problems, treat x and y components separately
Check your units throughout calculations
For connected systems, identify when momentum conservation applies versus when force analysis is needed

chatImportant

Common Mistake to Avoid: Don't confuse the moment when momentum is conserved (instantaneous change) with the ongoing motion afterward. In connected systems, momentum conservation applies at the instant of change, but forces determine the subsequent motion.

Remember!

bookmarkSummary

Key Points to Remember:

Momentum: $p = mv$ and is measured in kg⋅m/s
Conservation principle: Total momentum before = Total momentum after (when no external forces act)
Impulse: $I = \Delta p = F \times \Delta t$ and equals the change in momentum
In two-dimensional collisions, apply conservation separately to x and y components
Connected systems require careful analysis of when momentum is conserved versus when forces dominate
Always consider the direction of momentum - it's a vector quantity

Work, Energy, Power (Leaving Cert Applied Maths): Revision Notes

Conservation of Momentum

Understanding the principle of conservation of momentum

What is impulse?

Applying conservation of momentum in one dimension

Two-dimensional collisions

Conservation of momentum with connected systems

Exam tips

Remember!

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