Momentum & Collisions

Canada Grade 11 Physics Revision Notes

Momentum is one of the most important concepts in Grade 11 Physics because it allows us to understand what happens when objects interact, collide, or push against one another. From hockey pucks and basketballs to vehicle safety systems and spacecraft, momentum provides a powerful way to analyse motion.

In this chapter, we will study linear momentum, conservation of momentum, elastic and inelastic collisions, impulse, and the impulse-momentum theorem.

These revision notes are designed for Canadian Grade 11 Physics students, with a focus on the mathematical problem-solving approach commonly used in Ontario Grade 11 Physics.

1. What Is Momentum?

Momentum describes the quantity of motion an object has. It depends on both the object's mass and its velocity.

Linear momentum is represented by the symbol \(p\).

\[ \vec{p}=m\vec{v} \]

where:

  • p = momentum
  • m = mass in kilograms (kg)
  • v = velocity in metres per second (m/s)

Because velocity is a vector, momentum is also a vector quantity. This means momentum has both magnitude and direction.

The SI unit of momentum is:

\[ \text{kg}\cdot\text{m/s} \]

A larger mass or a larger velocity produces greater momentum.

Rahul Sir's Key Idea: Momentum depends on mass and velocity. Because velocity includes direction, you must consider direction when solving momentum problems.

2. Understanding the Direction of Momentum

Momentum has the same direction as velocity.

Suppose east is positive. A 5 kg object moving east at 4 m/s has:

\[ p=mv \]
\[ p=(5)(+4) \]
\[ \boxed{p=+20\,\text{kg}\cdot\text{m/s}} \]

Now suppose the same object moves west at 4 m/s:

\[ p=(5)(-4) \]
\[ \boxed{p=-20\,\text{kg}\cdot\text{m/s}} \]

The negative sign indicates that the momentum is directed west, according to our coordinate system.

Exam Tip: Always choose a positive direction before solving a collision or momentum problem.

3. Momentum and Newton's Second Law

Momentum is closely connected to force.

Newton's Second Law can be written in terms of momentum as:

\[ \vec{F}_{\text{net}} = \frac{\Delta\vec{p}}{\Delta t} \]

This means that a net force causes momentum to change.

If an object's momentum changes rapidly, a large force may be involved. If the same momentum change occurs over a longer time, the average force can be smaller.

This idea is extremely important when understanding airbags, helmets, protective padding, and other safety systems.

4. Conservation of Momentum

The law of conservation of momentum states that the total momentum of an isolated system remains constant when the net external impulse on the system is zero.

For a collision involving two objects:

\[ \vec{p}_{\text{total, initial}} = \vec{p}_{\text{total, final}} \]

For two objects, this becomes:

\[ m_1v_{1i}+m_2v_{2i} = m_1v_{1f}+m_2v_{2f} \]

where:

  • \(m_1,m_2\) = masses of the objects
  • \(v_{1i},v_{2i}\) = initial velocities
  • \(v_{1f},v_{2f}\) = final velocities

Remember that velocity includes direction, so signs are important.

Momentum before the interaction = momentum after the interaction

5. What Is an Isolated System?

Momentum conservation is most straightforward when the system is isolated.

An isolated system is one in which the net external force or external impulse is negligible during the interaction being analysed.

For example, during a very short collision between two hockey pucks, friction from the ice may be small compared with the forces between the pucks. The two pucks can therefore be treated approximately as an isolated system for the collision.

In real situations, external forces may exist. The key question is whether their effect is significant during the time interval being studied.

6. Momentum Conservation Example

A 2 kg cart moving at 6 m/s to the right collides with a 4 kg cart that is initially at rest. Suppose the carts stick together. Find their final velocity.

Step 1: Choose a positive direction

Let right be positive.

Step 2: Identify the values

\[ m_1=2\,kg \]
\[ v_{1i}=+6\,m/s \]
\[ m_2=4\,kg \]
\[ v_{2i}=0 \]

Step 3: Apply conservation of momentum

\[ m_1v_{1i}+m_2v_{2i} = (m_1+m_2)v_f \]
\[ (2)(6)+(4)(0) = (2+4)v_f \]
\[ 12=6v_f \]
\[ \boxed{v_f=2\,m/s} \]

The two carts move together at 2 m/s to the right.

7. Types of Collisions

Collisions can be classified according to what happens to kinetic energy during the interaction.

The two major categories you should understand are:

  • Elastic collisions
  • Inelastic collisions

In both types, momentum is conserved for an isolated system. The important difference is what happens to kinetic energy.

8. Elastic Collisions

An elastic collision is a collision in which both momentum and total kinetic energy are conserved.

Momentum conservation gives:

\[ m_1v_{1i}+m_2v_{2i} = m_1v_{1f}+m_2v_{2f} \]

Kinetic energy conservation gives:

\[ \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \]

Perfectly elastic collisions are an idealized model. Some interactions in physics can be approximated as elastic, especially when relatively little kinetic energy is transformed into other forms.

Important

Elastic does not mean that the objects cannot deform temporarily. What matters is that total kinetic energy is conserved before and after the collision.

9. Inelastic Collisions

In an inelastic collision, momentum is conserved for an isolated system, but kinetic energy is not conserved.

Some kinetic energy is transformed into other forms of energy, such as:

  • Thermal energy
  • Sound
  • Deformation
  • Other internal energy

The total energy of the system is still conserved, but kinetic energy alone is not.

Momentum can be conserved even when kinetic energy is not.

10. Perfectly Inelastic Collisions

A special type of inelastic collision occurs when the objects stick together after the collision.

For two objects:

\[ m_1v_{1i}+m_2v_{2i} = (m_1+m_2)v_f \]

Therefore:

\[ v_f= \frac{m_1v_{1i}+m_2v_{2i}} {m_1+m_2} \]

This equation is particularly useful for collision problems where two objects join together after impact.

11. Comparing Elastic and Inelastic Collisions

Feature Elastic Collision Inelastic Collision
Momentum conserved? Yes, for an isolated system Yes, for an isolated system
Kinetic energy conserved? Yes No
Other energy transformations? Ideally negligible Often significant
Objects stick together? No They may
Exam Tip: Momentum conservation is common to both elastic and inelastic collisions. The key difference is kinetic energy.

12. Impulse

Impulse describes the effect of a force acting over a period of time.

For a constant force:

\[ \vec{J}=\vec{F}\Delta t \]

where:

  • J = impulse
  • F = force
  • Δt = time interval

Impulse is a vector quantity because force is a vector.

The SI unit of impulse is:

\[ \text{N}\cdot\text{s} \]

This is equivalent to the unit of momentum:

\[ 1\,\text{N}\cdot\text{s} = 1\,\text{kg}\cdot\text{m/s} \]

13. The Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum.

\[ \vec{J}=\Delta\vec{p} \]

Therefore:

\[ \vec{F}_{\text{avg}}\Delta t = \Delta\vec{p} \]

Since:

\[ \Delta\vec{p} = m\vec{v}_f-m\vec{v}_i \]

for an object with constant mass:

\[ \vec{F}_{\text{avg}}\Delta t = m(\vec{v}_f-\vec{v}_i) \]

This equation connects force, time and momentum change.

14. Why Increasing Collision Time Reduces Force

One of the most important applications of impulse is understanding safety equipment.

Suppose an object must undergo the same change in momentum:

\[ \Delta p=\text{constant} \]

From the impulse-momentum relationship:

\[ F_{\text{avg}}\Delta t=\Delta p \]

Therefore:

\[ F_{\text{avg}} = \frac{\Delta p}{\Delta t} \]

If the collision time increases, the average force decreases for the same change in momentum.

This principle helps explain the physics behind:

  • Vehicle airbags
  • Seat belts
  • Helmets
  • Protective sports equipment
  • Vehicle crumple zones
  • Safety padding
Increasing the time over which momentum changes can reduce the average force experienced by an object.

15. Force-Time Graphs and Impulse

Impulse can also be found from a force-time graph.

The area under a force-time graph represents impulse:

\[ J=\text{Area under the }F\text{-}t\text{ graph} \]

For a constant force, the graph is a rectangle:

\[ J=F\Delta t \]

For a changing force, calculate the appropriate area or use the relevant mathematical method provided in the problem.

16. Worked Example: Impulse

A hockey stick applies an average force of 120 N to a puck for 0.05 s. Find the impulse delivered to the puck.

Step 1: Identify the values

\[ F=120\,N \]
\[ \Delta t=0.05\,s \]

Step 2: Use the impulse equation

\[ J=F\Delta t \]

Step 3: Substitute

\[ J=(120)(0.05) \]
\[ \boxed{J=6\,N\cdot s} \]

Therefore, the puck experiences an impulse of 6 N·s in the direction of the applied force.

17. Worked Example: Change in Momentum

A 0.20 kg hockey puck initially travels east at 10 m/s. After being struck, it travels west at 15 m/s. Find the change in momentum.

Step 1: Choose a positive direction

Let east be positive.

Therefore:

\[ v_i=+10\,m/s \]
\[ v_f=-15\,m/s \]

Step 2: Calculate initial momentum

\[ p_i=mv_i \]
\[ p_i=(0.20)(10) \]
\[ p_i=2\,kg\cdot m/s \]

Step 3: Calculate final momentum

\[ p_f=mv_f \]
\[ p_f=(0.20)(-15) \]
\[ p_f=-3\,kg\cdot m/s \]

Step 4: Find the change in momentum

\[ \Delta p=p_f-p_i \]
\[ \Delta p=-3-2 \]
\[ \boxed{\Delta p=-5\,kg\cdot m/s} \]

The negative sign indicates that the change in momentum is directed west.

18. Momentum in Explosions and Recoil

Conservation of momentum is not limited to collisions. It can also be used to analyse explosions, recoil and objects separating from one another.

Suppose an initially stationary object separates into two pieces. Before separation:

\[ p_{\text{initial}}=0 \]

Therefore, after separation:

\[ p_{\text{final}}=0 \]

For two pieces:

\[ m_1v_1+m_2v_2=0 \]

The two pieces must therefore have momenta that are equal in magnitude and opposite in direction.

19. Momentum and Energy: Do Not Confuse Them

Momentum and kinetic energy are related to motion, but they are different physical quantities.

Quantity Formula Type SI Unit
Momentum \(p=mv\) Vector kg·m/s
Kinetic Energy \(E_k=\frac{1}{2}mv^2\) Scalar J
Impulse \(J=F\Delta t\) Vector N·s

A collision can conserve momentum while kinetic energy decreases. This is why momentum conservation and energy conservation must not be treated as identical ideas.

20. Common Momentum and Collision Mistakes

Mistake 1: Forgetting Direction

Momentum is a vector. A collision problem involving objects moving in opposite directions requires positive and negative signs.

Mistake 2: Conserving Kinetic Energy in Every Collision

Momentum is conserved in an isolated collision. Kinetic energy is only conserved in an elastic collision.

Mistake 3: Forgetting That Sticking Together Means One Final Velocity

If two objects stick together, they share the same final velocity.

Mistake 4: Confusing Impulse with Force

Force and impulse are different quantities:

\[ J=F\Delta t \]

Impulse depends on both force and the time over which the force acts.

Mistake 5: Ignoring External Forces

Momentum conservation should be applied to an appropriate system where the external impulse is negligible or otherwise accounted for.

Mistake 6: Mixing Up Initial and Final Values

Clearly label quantities with \(i\) for initial and \(f\) for final.

21. A Reliable Momentum Problem-Solving Method

  1. Identify the system. Decide which objects are included in your momentum analysis.
  2. Choose a positive direction. For example, right or east can be positive.
  3. List the masses and velocities. Separate initial and final values.
  4. Determine the type of interaction. Ask whether the objects bounce apart, stick together, or separate.
  5. Apply conservation of momentum when appropriate.
    6. \[ p_i=p_f \]
  6. Use kinetic energy conservation only when the collision is elastic.
  7. For impulse problems, use:
    8. \[ J=\Delta p \]
  8. Check the direction, units and magnitude of your answer.

22. Grade 11 Momentum Formula Sheet

Linear Momentum

\[ \vec{p}=m\vec{v} \]

Total Momentum

\[ \vec{p}_{\text{total}} = \sum m\vec{v} \]

Conservation of Momentum

\[ \vec{p}_{i}=\vec{p}_{f} \]

Two-Object Collision

\[ m_1v_{1i}+m_2v_{2i} = m_1v_{1f}+m_2v_{2f} \]

Perfectly Inelastic Collision

\[ v_f= \frac{ m_1v_{1i}+m_2v_{2i} } { m_1+m_2 } \]

Impulse

\[ \vec{J}=\vec{F}\Delta t \]

Impulse-Momentum Theorem

\[ \vec{J}=\Delta\vec{p} \]

Average Force

\[ \vec{F}_{\text{avg}} = \frac{\Delta\vec{p}}{\Delta t} \]

Kinetic Energy

\[ E_k=\frac{1}{2}mv^2 \]

Elastic Collision

\[ p_i=p_f \]
\[ E_{k,i}=E_{k,f} \]

23. Quick Revision Checklist

Before your Grade 11 Physics assessment, make sure you can:

  • Define linear momentum.
  • Calculate momentum using \(p=mv\).
  • Identify momentum as a vector quantity.
  • Choose a positive direction for collision problems.
  • Explain conservation of momentum.
  • Identify an appropriate isolated system.
  • Solve one-dimensional collision problems.
  • Distinguish elastic and inelastic collisions.
  • Recognize perfectly inelastic collisions.
  • Calculate final velocity when objects stick together.
  • Explain the difference between momentum and kinetic energy.
  • Define impulse.
  • Calculate impulse using \(J=F\Delta t\).
  • Apply the impulse-momentum theorem.
  • Interpret force-time graphs.
  • Explain how increasing collision time can reduce average force.
  • Apply momentum conservation to recoil and explosions.
  • Use correct SI units.
  • Include direction in vector calculations.

24. Rahul Sir's Final Revision Tip

Momentum problems become much easier when you organize the information before writing an equation.

Start by asking:

What is my system, and is the external impulse small enough that I can use conservation of momentum?

Then choose a positive direction and write the initial and final momentum clearly.

\[ p_i=p_f \]

For a collision where two objects stick together, remember:

\[ m_1v_{1i}+m_2v_{2i} = (m_1+m_2)v_f \]

For an elastic collision, momentum and kinetic energy are both conserved. For an inelastic collision, momentum is conserved but some kinetic energy is transformed into other forms.

When you see the words force and time, think about impulse:

\[ J=F\Delta t=\Delta p \]

This relationship is especially important for understanding safety equipment. Increasing the time over which an object changes momentum can reduce the average force involved.

The key sequence to remember is:

Momentum = mass × velocity

Collision → conserve momentum

Elastic collision → conserve momentum + kinetic energy

Inelastic collision → conserve momentum, but kinetic energy changes

Impulse → change in momentum

Master these relationships and you will have a strong foundation for Grade 11 Physics problems involving collisions, recoil, impulse, momentum and real-world safety applications.

Don't just memorize \(p=mv\). Understand what momentum is telling you about the motion and interaction of objects.

— Rahul Sir | ODTutor Canada