Elastic collisions in one dimension

Welcome to Elastic Collisions!

In this chapter of Further Mechanics 1, we are going to explore what happens when two objects—like snooker balls or bumper cars—hit each other in a straight line. This is called a direct impact.

By the end of these notes, you’ll understand how to predict where objects go after they collide and how much energy they lose in the process. Don't worry if this seems tricky at first! Mechanics is all about drawing a good picture and following a few reliable "rules of the road."

1. The Fundamentals: Momentum Revisited

Before we dive into collisions, we need to remember the golden rule of mechanics: Conservation of Linear Momentum (CoM).

In any collision where no external forces act, the total momentum before the crash must equal the total momentum after the crash.

The Formula:
\( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \)

Where:
- \( m \) is mass.
- \( u \) represents velocities before the collision.
- \( v \) represents velocities after the collision.

Quick Review Box:
Always pick a direction to be positive (usually to the right). If an object is moving to the left, its velocity must be written as a negative number. This is the most common mistake students make!

Key Takeaway: Momentum is always conserved in these problems. It gives us our first equation to solve for unknown speeds.

2. Newton’s Law of Restitution (NLR)

While momentum tells us part of the story, it doesn't tell us how "bouncy" the objects are. For that, we use Newton’s Law of Restitution.

This law relates the speed of approach (how fast they are coming together) to the speed of separation (how fast they are moving apart).

The Coefficient of Restitution (\( e \))

The "bounciness" is represented by the letter \( e \).
The value of \( e \) is always between 0 and 1 (\( 0 \le e \le 1 \)).

• If \( e = 0 \): The objects are "perfectly inelastic." They stick together like wet clay.
• If \( e = 1 \): The objects are "perfectly elastic." No energy is lost, and they bounce apart perfectly.
• Real life: Most things fall somewhere in between, like a tennis ball (\( e \approx 0.7 \)).

The Formula:

\( e = \frac{\text{speed of separation}}{\text{speed of approach}} \)

To make this easier for calculations, we usually write it as:
\( v_2 - v_1 = e(u_1 - u_2) \)

Memory Aid: "Sep over App"
Just remember that separation is on top and approach is on the bottom. You can also think of \( e \) as the "ratio of bounciness."

Key Takeaway: Newton's Law of Restitution gives us our second equation. Between CoM and NLR, you have two equations to solve for two unknown final velocities (\( v_1 \) and \( v_2 \)).

3. Loss of Kinetic Energy

Did you know? In most real-world collisions, some energy is "lost." It isn't actually gone; it just turns into heat, sound, or the energy used to dent the objects.

To find the Loss of Kinetic Energy, you simply calculate the energy at the start and subtract the energy at the end.

The Process:
1. Calculate Total KE Before: \( \frac{1}{2}m_1(u_1)^2 + \frac{1}{2}m_2(u_2)^2 \)
2. Calculate Total KE After: \( \frac{1}{2}m_1(v_1)^2 + \frac{1}{2}m_2(v_2)^2 \)
3. Loss = KE Before - KE After

Common Mistake to Avoid:
Kinetic energy uses velocity squared (\( v^2 \)). This means even if a velocity is negative (moving left), the energy will be positive because a negative number squared is always positive!

Key Takeaway: Unless \( e = 1 \), you will always lose some kinetic energy in a collision.

4. Collisions with a Fixed Surface (Walls)

Sometimes a sphere hits a solid, smooth, stationary wall. This is actually easier because the wall doesn't move!

For a wall, the "speed of approach" is just the speed the ball hits the wall with (\( u \)), and the "speed of separation" is the speed it bounces back with (\( v \)).

The Formula for Walls:
\( v = eu \)

Example: If a ball hits a wall at \( 10 \text{ m/s} \) and \( e = 0.5 \), it will bounce back at \( 5 \text{ m/s} \).

Key Takeaway: When hitting a wall, the object simply reverses direction and its speed is multiplied by \( e \).

5. Successive Impacts

In some exam questions, you might have successive impacts. This is just a fancy way of saying "one thing happens after another."

Analogy: Imagine Sphere A hits Sphere B, and then Sphere B carries on to hit a wall.
- Step 1: Solve the collision between A and B to find their new velocities.
- Step 2: Use the new velocity of B as the "before" speed for its impact with the wall.

Successive Impact Tip:
Always draw a fresh diagram for the second impact. Use the results from your first calculation to label your new "Before" speeds. This keeps your work organized and prevents "information overload."

Key Takeaway: Treat each impact as a separate, mini-problem. The "After" velocity of the first collision becomes the "Before" velocity of the next.

Final Summary Checklist

When tackling an Elastic Collision problem, ask yourself these four questions:

1. Have I drawn a diagram? (Label masses, speeds, and directions clearly.)
2. Have I picked a positive direction? (Stick to it for both CoM and NLR equations!)
3. Have I set up my two equations? (Conservation of Momentum and Newton’s Law of Restitution.)
4. Does my answer make sense? (If \( e = 0.5 \), the balls should be moving apart slower than they came together.)

You've got this! Practice a few problems starting with these steps, and you'll find the patterns start to repeat themselves.