Mechanics 2: Study Notes on Collisions

Hello future physicist! Welcome to the exciting chapter on Collisions. This is where we apply your knowledge of momentum, energy, and impulse to understand what happens when objects crash into each other—from tiny particles to colliding billiard balls. Don't worry if this seems tricky at first; we will break down the complexity into two simple, powerful laws. By the end of this chapter, you’ll be able to predict the velocities of objects after they hit!

Why Study Collisions?

Understanding collisions is crucial in engineering, physics, and even safety design (think car crash testing!). In M2, collisions are modeled using perfect mathematical rules, allowing us to accurately solve problems involving direct (1D) and oblique (2D) impacts.

Section 1: The Foundation - Impulse and Momentum Review

A collision is a very short-lived event where large forces are exerted between two bodies. The primary concepts governing this interaction are Impulse and Momentum.

Momentum (\(p\))

Momentum is the 'quantity of motion' an object possesses.

\(p = m v\)
Where m is mass (kg) and v is velocity (\(m s^{-1}\)). Momentum is a vector quantity, meaning its direction matters!

Impulse (\(I\))

When a collision occurs, the force of impact results in an impulse. Impulse is defined as the change in momentum.

\(I = F t = m v - m u\)

Key Takeaway: Momentum is always conserved in a collision, but the impulse dictates how the velocity changes for individual particles.

Section 2: The Governing Law - Conservation of Momentum

The Principle of Conservation of Momentum (PCM) is the most fundamental rule in collision dynamics.

What does PCM state?

When two or more bodies collide, provided there are no external forces (like air resistance or friction) acting on the system, the total momentum before the collision is equal to the total momentum after the collision.

The PCM Equation (Direct Collision)

Consider two particles, Particle 1 (mass \(m_1\)) and Particle 2 (mass \(m_2\)), moving along the same straight line.

  • Initial velocities (before collision): \(u_1\) and \(u_2\)
  • Final velocities (after collision): \(v_1\) and \(v_2\)

Momentum Before = Momentum After
\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\)

Crucial Convention: Direction!

Since momentum is a vector, you must be consistent with direction.

  • Step 1: Define a positive direction (e.g., Right).
  • Step 2: Any velocity moving in the opposite direction must be entered as a negative value in the equation.

Example: If \(m_2\) is moving left (\(u_2\)), and you defined right as positive, you must use \(-u_2\) in the equation.

Quick Review: PCM

Always draw a clear diagram first, indicating initial and final directions. You will usually have two unknown variables in a collision problem, so you need a second equation (coming up next!) to solve it.

Section 3: Direct Collisions (1D) and the Coefficient of Restitution

In most collision problems, applying the PCM alone leaves us with one equation and two unknowns (\(v_1\) and \(v_2\)). We need a second law that describes how "bouncy" the collision is. This is Newton's Experimental Law (NEL).

Newton's Experimental Law (NEL)

NEL introduces the Coefficient of Restitution (\(e\)), which measures the ratio of the relative speed of separation to the relative speed of approach. Think of \(e\) as the "bounciness factor."

The Coefficient of Restitution (\(e\))

The value of \(e\) must always be between 0 and 1:

\(0 \le e \le 1\)

The definition equation is:

\(e = \frac{\text{Speed of Separation}}{\text{Speed of Approach}}\)

Assuming Particle 1 is approaching Particle 2 (i.e., \(u_1 > u_2\)):

\(v_2 - v_1 = e (u_1 - u_2)\)

Analogy: Imagine dropping a tennis ball. If it bounces back up with exactly the same speed it hit the floor, \(e=1\). If it hits the floor and stops dead (like a lump of clay), \(e=0\).

Solving 1D Collision Problems (The 4-Step Process)

Every standard 1D collision problem can be solved by combining PCM and NEL.

  1. Diagram and Convention: Draw the 'Before' and 'After' diagrams. Choose a positive direction (e.g., right) and clearly mark all known masses and velocities (using negative signs for opposing velocities).
  2. Apply PCM: Set up the equation \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\). This gives Equation (1).
  3. Apply NEL: Set up the equation \(v_2 - v_1 = e (u_1 - u_2)\). This gives Equation (2).
  4. Solve Simultaneously: Use substitution or elimination to find the unknown velocities (\(v_1\) and \(v_2\)). Remember to check the sign of your final answers to confirm the direction of movement.
Common Mistake Alert!

When applying NEL, ensure the order of subtraction is consistent. If you use \(v_2 - v_1\) on the left, you must use \(u_1 - u_2\) on the right. This maintains the "Relative Velocity of Separation (2 relative to 1)" = \(e \times\) "Relative Velocity of Approach (1 relative to 2)."

Section 4: Types of Collisions and Energy Loss

The value of the coefficient of restitution, \(e\), tells us exactly what kind of collision we are dealing with.

Case 1: Perfectly Elastic Collision (\(e = 1\))

An elastic collision is a collision where Kinetic Energy (KE) is conserved.

  • The bodies separate with the same relative speed they approached with.
  • KE Before = KE After.
Case 2: Perfectly Inelastic Collision (\(e = 0\))

An inelastic collision is a collision where the bodies coalesce (stick together) after impact.

  • They move off with the same final velocity (\(v_1 = v_2\)).
  • This results in the maximum possible loss of Kinetic Energy.
Loss of Kinetic Energy (LKE)

Unless \(e=1\), kinetic energy is always lost in a collision, usually converted into heat and sound. You must be able to calculate this loss.

LKE is defined as:

LKE = Initial Total KE - Final Total KE

Remember the Kinetic Energy formula:

\(KE = \frac{1}{2} m v^2\)

Important: KE is a scalar quantity, so it is always positive, regardless of the velocity direction. You do not use negative signs for velocity when calculating KE!

Did you know?

In physics, there is no such thing as a "perfectly elastic" collision (\(e=1\)) because some energy is always lost to sound and heat. However, billiard ball collisions come very close!

Section 5: Oblique Collisions (2D)

When a particle strikes another object at an angle (not head-on), this is an oblique collision. Since this happens in two dimensions, we must use vector resolution.

The key to solving 2D collisions is recognizing the direction of the impulsive force:

  • The impulsive force always acts along the line of centres (or line of impact).
  • No impulse acts perpendicular to the line of impact.
The M2 Rule for Oblique Collisions

We analyze the motion in two perpendicular directions:

A. Along the Line of Impact (Parallel Component)

This is the direction the collision forces act. All the rules from 1D collisions apply here.

  1. Velocity components: Resolve all initial velocities (\(u\)) along the line of impact.
  2. Apply PCM: Conservation of momentum holds for this line.
    \(m_1 u_{1, \text{parallel}} + m_2 u_{2, \text{parallel}} = m_1 v_{1, \text{parallel}} + m_2 v_{2, \text{parallel}}\)
  3. Apply NEL: Newton's law applies ONLY along this line.
    \(v_{2, \text{parallel}} - v_{1, \text{parallel}} = e (u_{1, \text{parallel}} - u_{2, \text{parallel}})\)

B. Perpendicular to the Line of Impact (Tangential Component)

Since there is no impulsive force in this direction, the momentum (and therefore the velocity) of each individual particle is conserved in this direction.

For Particle 1:

\(v_{1, \text{perpendicular}} = u_{1, \text{perpendicular}}\)

For Particle 2:

\(v_{2, \text{perpendicular}} = u_{2, \text{perpendicular}}\)

Putting it Back Together

Once you have found the final parallel and perpendicular components (\(v_{parallel}\) and \(v_{perpendicular}\)) for each particle, you must combine them using Pythagoras' Theorem and trigonometry to find the overall final velocity magnitude and direction.

\(v = \sqrt{(v_{parallel})^2 + (v_{perpendicular})^2}\)

Key Takeaway for 2D Collisions

When solving 2D collisions, remember the core principles:

  • Parallel: Use PCM and NEL (the equations involve both particles).
  • Perpendicular: Velocity is unchanged (the equations only involve one particle at a time).

Section 6: Further Consideration - Collisions with a Fixed Surface

A common special case is when a particle collides with a large, fixed object, like a wall or the ground.

If a particle hits a fixed surface, the mass of the surface is effectively infinite. We don't use PCM because the surface can absorb or transfer infinite momentum (it's an external body). We only use NEL.

If a ball hits a wall, the wall's velocity before and after the collision is \(u_{wall} = 0\) and \(v_{wall} = 0\).

Applying NEL to a Fixed Surface

Consider a particle velocity \(u\) hitting a fixed surface and rebounding with velocity \(v\). The relative speed of approach is \(u\), and the relative speed of separation is \(v\).

\(v = e u\)

Note: If the collision is oblique, the NEL rule only applies to the velocity component perpendicular to the wall. The velocity component parallel to the wall remains unchanged (assuming the surface is smooth and frictionless).

You've made it through the trickiest chapter in M2! Collisions rely heavily on your algebraic skills for solving simultaneous equations and your ability to manage vector components. Practice makes perfect—keep applying those two powerful laws (PCM and NEL), and you'll succeed!

Good luck with your studies!