Elastic collisions in two dimensions

Welcome to the World of 2D Collisions!

Hey there! Ready to take your mechanics skills into the second dimension? In your previous studies, you’ve likely looked at collisions where everything moves in a straight line. But in the real world—like on a pool table or in a game of marbles—objects often hit each other at angles. These are called oblique collisions.

In this chapter, we are going to learn how to predict exactly where spheres go after they hit a wall or each other at an angle. Don't worry if this seems tricky at first; we’re basically just taking what you already know about 1D collisions and applying it to two different directions at once. Let’s dive in!

1. Prerequisite Check: The Basics

Before we start, let's quickly remind ourselves of two "golden rules" from one-dimensional mechanics:

1. Conservation of Linear Momentum (CLM): Total momentum before = Total momentum after (\( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \)).
2. Newton’s Law of Restitution (NLR): The speed of separation is \( e \) times the speed of approach (\( v_2 - v_1 = e(u_1 - u_2) \)).

In 2D, we apply these rules, but we have to be careful about which direction we are looking at.

2. Impact of a Sphere with a Smooth Fixed Surface

Imagine a smooth ball hitting a floor or a wall at an angle. This is the simplest version of a 2D collision.

How it works:

When a ball hits a smooth surface, the only force acting on it is the impulse from the wall, which acts perpendicular (at 90 degrees) to the surface. Because the surface is "smooth," there is no friction acting parallel to the surface.

This leads us to two vital rules for the components of velocity:

1. Parallel to the surface: The velocity stays exactly the same. Nothing is pushing or pulling it in this direction!
2. Perpendicular to the surface: We use Newton's Law of Restitution (\( e \)). The component of velocity "into" the wall is reversed and multiplied by \( e \).

The Math Breakdown:

If a sphere hits a wall with velocity \( u \) at an angle \( \alpha \) to the normal (the line perpendicular to the wall):
- Velocity component parallel to wall: \( u \sin \alpha \)
- Velocity component perpendicular to wall: \( u \cos \alpha \)

After the collision (velocity \( v \), angle \( \beta \)):
- New parallel component: \( v \sin \beta = u \sin \alpha \)
- New perpendicular component: \( v \cos \beta = e(u \cos \alpha) \)

Quick Analogy: Imagine running diagonally towards a wall while wearing ice skates. Your speed along the wall doesn't change because the ice is slippery (smooth), but your "bounce" away from the wall depends on how bouncy (elastic) the wall is.

Did you know? If the collision is perfectly elastic (\( e = 1 \)), the angle of reflection equals the angle of incidence, just like light hitting a mirror!

Key Takeaway: For a wall impact, the velocity component parallel to the wall is constant, and the component perpendicular is changed by \( e \).

3. Oblique Impact of Two Smooth Spheres

Now, let's look at what happens when two moving spheres (both with the same radius) hit each other. The secret to solving these is the Line of Centres.

Step-by-Step Process:

1. Identify the Line of Centres: This is the imaginary line connecting the centers of the two spheres at the moment they touch. The impulse of the collision acts only along this line.
2. Split velocities into components: Find the components of velocity along the line of centres and perpendicular to the line of centres.
3. Perpendicular Direction: The components of velocity perpendicular to the line of centres do not change for either sphere. (Because they are smooth, there’s no force to change them).
4. Along the Line of Centres: Treat this exactly like a 1D collision! Use CLM and NLR for the components in this direction.

Common Mistake to Avoid: Students often try to use CLM for the whole velocity vector at once. Remember: apply CLM and NLR only to the components along the line of centres.

Key Takeaway: Perpendicular to the impact line, velocities are preserved. Along the impact line, use 1D collision rules.

4. Working with Vectors

Sometimes, the exam will give you velocities in vector form (using i and j). This is actually helpful!

If a sphere with velocity \( (u_x \mathbf{i} + u_y \mathbf{j}) \) hits a wall that is parallel to the i unit vector:
- The i component (parallel) stays the same: \( v_x = u_x \).
- The j component (perpendicular) changes: \( v_y = -e u_y \).

Memory Aid: "Parallel is Permanent; Perpendicular is Product." (The perpendicular component is the product of the original and \( -e \)).

5. Loss of Kinetic Energy

In most collisions (\( e < 1 \)), some energy is "lost" to heat or sound. To find the Loss of Kinetic Energy, calculate the total energy before and subtract the total energy after.

\( \text{KE Loss} = \left( \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 \right) - \left( \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 \right) \)

Note: When calculating \( v^2 \), remember that \( v^2 = (v_{\text{parallel}})^2 + (v_{\text{perpendicular}})^2 \).

6. Successive Impacts

What if a ball hits a wall, then another wall? Or hits a sphere and then a wall?

Don't panic! Just treat them as separate events. Solve the first collision to find the new velocity, then use that as your "starting" velocity for the second collision. Just keep track of your angles and components at each stage.

Quick Review Box: - Smooth surface? No change in velocity parallel to the surface.
- Line of centres? The direction where the "bounce" happens.
- Elasticity (\( e \))? Always applies to the direction of the impact.
- Energy? Always use the total magnitude of the velocity vector to find KE.

Final Encouragement: You've got this! Most problems in this chapter are solved by drawing a clear diagram, resolving your velocities into components, and applying the two rules. Practice a few "wall bounce" questions first, then move on to "sphere-to-sphere" collisions. You'll be a mechanics pro in no time!