Welcome to Momentum and Collisions!
Hi there! Welcome to one of the most exciting parts of Mechanics. Have you ever wondered why a heavy truck is much harder to stop than a small car, even if they are going the same speed? Or why a tennis ball bounces higher than a beanbag? That is exactly what we are going to explore here.
In this chapter, we’ll look at how objects move, how they hit each other, and how we can use math to predict what happens next. Whether you love physics or find it a bit daunting, we’ll break this down step-by-step.
1. The Basics: What is Momentum?
Before we can talk about collisions, we need to understand momentum. Think of momentum as "mass in motion." Every moving object has it.
The formula for momentum is very simple:
\( \text{Momentum} = \text{mass} \times \text{velocity} \)
\( p = mv \)
Important Point: Momentum is a vector. This means direction matters! If one ball is moving right at \( 5 \text{ m/s} \) and another is moving left at \( 5 \text{ m/s} \), they have different momenta because their directions are opposite.
Conservation of Momentum (MB1)
This is the "Golden Rule" of collisions: In any collision, the total momentum before the crash is equal to the total momentum after the crash (as long as no outside forces like friction are acting on them).
Imagine two snooker balls, A and B:
\( m_A u_A + m_B u_B = m_A v_A + m_B v_B \)
(Where \( u \) is the starting velocity and \( v \) is the final velocity.)
Memory Aid: "What goes in must come out!" The total "moving power" of the system stays the same.
Quick Review:
- Momentum = \( mv \)
- Conservation: Total momentum before = Total momentum after.
- Direction: Always pick a "positive" direction (usually right) and stick to it!
2. Impulse: Changing the Momentum (MB3 & MB4)
If momentum is what an object has, Impulse is what changes it. When you kick a football, you apply a force for a short amount of time. This "kick" is the impulse.
Constant Force (MB3)
If the force is constant, we use this formula:
\( \text{Impulse} = \text{Force} \times \text{Time} \)
\( I = Ft \)
Since impulse changes momentum, we can also say:
\( I = mv - mu \)
(Impulse = Change in Momentum)
Variable Force (MB4)
Sometimes a force isn't constant; it might start weak and get stronger. For AQA Further Maths, when force depends on time, we use integration:
\( I = \int_{t_1}^{t_2} F \, dt \)
Don't worry if this seems tricky! Just remember that if you see a force as an equation with \( t \) in it (like \( F = 3t^2 \)), you just need to integrate it over the time interval given.
Did you know? This is how car crumple zones work. By increasing the time (\( t \)) it takes for a car to stop, the force (\( F \)) on the passengers is reduced, even though the change in momentum is the same!
Key Takeaway:
Impulse is the link between force/time and velocity change. Use \( Ft = mv - mu \) for constant forces and integration for variable forces.
3. Newton’s Experimental Law & Restitution (MB2)
Why do some things bounce more than others? This is where the Coefficient of Restitution (denoted by the letter \( e \)) comes in.
\( e \) is a measure of "bounciness" between two surfaces. It is always a number between 0 and 1.
- If \( e = 1 \): The collision is "perfectly elastic" (it's super bouncy!).
- If \( e = 0 \): The objects stick together (like throwing wet clay at a wall). This is called "coalescing."
The Formula (Newton's Law of Restitution)
\( \text{Speed of separation} = e \times \text{speed of approach} \)
\( v_2 - v_1 = -e(u_2 - u_1) \)
Common Mistake: Be careful with minus signs! If the objects are moving towards each other, one of their velocities must be negative in your calculation.
Impacts with a Fixed Surface (like a wall)
If a ball hits a solid wall, the wall doesn't move. The formula becomes even simpler:
\( \text{Speed after} = e \times \text{Speed before} \)
\( v = eu \)
(Note: The ball will change direction, so technically \( v = -eu \) if we consider the vector.)
Key Takeaway:
The coefficient \( e \) tells us how much speed is kept after a bounce. Use it alongside Conservation of Momentum to solve "Before and After" collision problems.
4. Collisions in 2D: Using Vectors (MB1 & MB3)
Sometimes objects don't just move in a straight line; they move in 2D. In this course, you will see velocities given as vectors (like \( \begin{pmatrix} 3 \\ -2 \end{pmatrix} \)).
Good News: The syllabus says "resolving will not be required." This means you don't need to mess around with \( \sin(\theta) \) or \( \cos(\theta) \) for these problems!
You can treat the \( i \) and \( j \) (or top and bottom) components exactly the same way as 1D numbers. The rules for Momentum and Impulse still apply:
- Conservation: \( m_1 \mathbf{u_1} + m_2 \mathbf{u_2} = m_1 \mathbf{v_1} + m_2 \mathbf{v_2} \)
- Impulse: \( \mathbf{I} = m\mathbf{v} - m\mathbf{u} \)
Just do the math for the top row and the bottom row separately within the vector brackets.
Example: If a mass of \( 2\text{kg} \) has initial velocity \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \) and final velocity \( \begin{pmatrix} 5 \\ 1 \end{pmatrix} \):
\( \text{Impulse} = 2 \begin{pmatrix} 5 \\ 1 \end{pmatrix} - 2 \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 10-6 \\ 2-8 \end{pmatrix} = \begin{pmatrix} 4 \\ -6 \end{pmatrix} \text{ Ns} \)
Key Takeaway:
When working with vectors, don't panic! Just keep the \( x \) and \( y \) parts separate and follow the same momentum rules as before.
Summary: The Problem-Solver’s Toolkit
When you face a "Momentum and Collisions" question, follow these steps:
Step 1: Draw a "Before" and "After" diagram. Label masses and velocities clearly with arrows for direction.
Step 2: Use Conservation of Momentum to get your first equation.
Step 3: If things bounce, use Newton's Experimental Law (\( e \)) to get your second equation.
Step 4: Solve the equations (usually simultaneous equations) to find the missing velocities.
Step 5: If the question asks for Impulse, use \( I = mv - mu \) for one of the objects.
You've got this! Practice a few problems, and you'll see these patterns repeating every time.