Welcome to the World of Bumping Into Things!
In this chapter, we are going to explore Elastic Collisions in One Dimension. This is a core part of your Further Mechanics 1 (Paper 3C) studies. Have you ever wondered why a pool ball stops dead when it hits another one, or why a superball bounces much higher than a lump of playdough? That is exactly what we are going to calculate!
We will be looking at objects moving in a straight line (one dimension) and hitting each other. To keep things simple, we model these objects as particles (tiny dots with mass).
1. Prerequisite Knowledge: The Basics
Before we dive into the new stuff, let’s quickly refresh two concepts you’ve likely seen before:
- Momentum: Calculated as mass multiplied by velocity, \(p = mv\).
- Principle of Conservation of Momentum (PCLM): In a closed system, the total momentum before a collision equals the total momentum after. \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\).
2. Newton’s Law of Restitution
This is the "star" of this chapter. While momentum tells us about the total "oomph" in a system, Newton’s Law of Restitution tells us how "bouncy" the collision is.
We use a value called the coefficient of restitution, denoted by the letter \(e\). It is a ratio of speeds:
\(e = \frac{\text{speed of separation}}{\text{speed of approach}}\)
What does \(e\) actually mean?
The value of \(e\) is always between 0 and 1 (\(0 \le e \le 1\)):
- If \(e = 0\): The collision is perfectly inelastic. The objects stick together after they hit (like a piece of gum hitting a wall).
- If \(e = 1\): The collision is perfectly elastic. No kinetic energy is lost, and the objects bounce apart with maximum efficiency.
- If \(0 < e < 1\): This is the real world! Some energy is lost, and the objects bounce apart but slower than they approached.
Quick Review Box:
Remember: \(e\) is just a number. It has no units because it is a ratio of speeds! If you calculate \(e = 1.5\), go back and check your work—it’s impossible!
An Everyday Analogy
Imagine you are running toward a friend to give them a high-five. If you both stop dead and hug, that’s \(e = 0\). If you hit hands and fly backward at the same speed you ran in, that’s \(e = 1\). Most high-fives are somewhere in the middle!
3. Solving Collision Problems Step-by-Step
When two spheres (let's call them \(A\) and \(B\)) collide directly, you will usually need to find their new velocities (\(v_A\) and \(v_B\)). Don't worry if this seems tricky at first; we always use the same two "secret weapons" together.
Step 1: Draw a clear diagram.
Draw the spheres "Before" and "After". Use arrows to show the direction of travel. Pro Tip: Always pick one direction (usually to the right) to be positive. If a ball moves left, its velocity is negative!
Step 2: Apply Conservation of Momentum.
Write out the equation: \(m_Au_A + m_Bu_B = m_Av_A + m_Bv_B\).
Step 3: Apply Newton's Law of Restitution.
Use the formula: \(v_B - v_A = e(u_A - u_B)\).
Note: Be very careful with your signs here!
Step 4: Solve the simultaneous equations.
You now have two equations with two unknowns (\(v_A\) and \(v_B\)). Solve them just like you did in GCSE Maths!
Key Takeaway: You almost always need both the Momentum equation and the Restitution equation to solve these problems.
4. Loss of Kinetic Energy
In most collisions (\(e < 1\)), some Kinetic Energy (KE) is "lost." It doesn't disappear from the universe; it just turns into other forms like heat or sound (the "clack" of pool balls).
The formula for Kinetic Energy is \(KE = \frac{1}{2}mv^2\).
To find the Loss of KE:
\(\text{Loss} = (\text{Total KE Before}) - (\text{Total KE After})\)
Common Mistake to Avoid: When calculating KE, do not use the momentum! You must calculate \(\frac{1}{2}mv^2\) for each object separately and then add them up. Also, because velocity is squared, it doesn't matter if the object is moving left or right—KE is always positive!
5. Impact with a Fixed Surface (The Wall)
What happens if a sphere hits a solid, immovable wall? Since the wall doesn't move, its velocity is always 0.
The Law of Restitution simplifies to:
\(\text{Speed of separation} = e \times \text{Speed of approach}\)
\(v = eu\)
Wait! Why is momentum not conserved here?
Good catch! Momentum is only conserved when no external forces act. The wall is attached to the Earth, which is an external "force" relative to the ball. So, for the ball alone, momentum changes, but the Law of Restitution still works perfectly.
Did you know?
This is why squash players warm up the ball. Heating the air inside the ball increases its "bounciness," effectively raising its coefficient of restitution (\(e\))!
6. Successive Impacts
Sometimes, the fun doesn't stop after one hit. You might have three spheres in a line (\(A\), \(B\), and \(C\)).
- First, \(A\) hits \(B\). Solve this collision to find the new velocities of \(A\) and \(B\).
- Then, the "new" velocity of \(B\) is used as the "starting" velocity for its collision with \(C\).
- Solve the collision between \(B\) and \(C\).
Memory Aid: Think of this like a relay race. The "After" velocity of one collision becomes the "Before" velocity of the next one.
7. Summary of Key Points
- Coefficient of Restitution (\(e\)): Measures bounciness. \(e = \frac{\text{sep speed}}{\text{app speed}}\).
- Direct Impact: Use PCLM and Newton's Law of Restitution together.
- Energy: KE is lost unless \(e = 1\). \(\text{Loss} = KE_{\text{initial}} - KE_{\text{final}}\).
- Walls: Separation speed is just \(e \times\) approach speed.
- Signs: Always, always, always check your direction arrows! A negative velocity means the object is moving backward.
Final Encouragement: Mechanics is all about the setup. If you spend an extra minute drawing a clear diagram with labels for mass, initial velocity (\(u\)), and final velocity (\(v\)), the math becomes much easier. You've got this!