Welcome to the World of Collisions!
Ever wondered why a tennis ball bounces high while a lump of playdough just thuds onto the floor? Or how snooker players predict exactly where the balls will go after they hit each other? That is exactly what we are going to explore in this chapter! In Mechanics 2 (M2), we take the basic ideas of momentum you learned in M1 and "level them up" to handle more complex situations involving energy and elasticity.
1. Momentum and Impulse in Vector Form
In your M1 studies, you looked at objects moving in a straight line. In M2, we use vectors to describe motion in more detail. Don't worry if vectors feel a bit scary—just think of them as a way to give a "GPS coordinate" to a direction!
What is Momentum?
Momentum is a measure of how hard it is to stop a moving object. It depends on mass and velocity. In vector form, we write it as:
\( \mathbf{p} = m\mathbf{v} \)
Where \( m \) is mass (kg) and \( \mathbf{v} \) is the velocity vector (m/s).
The Impulse-Momentum Principle
An Impulse (\( \mathbf{I} \)) is what happens when a force acts on an object for a short time (like kicking a football). It causes a change in momentum.
\( \mathbf{I} = m\mathbf{v} - m\mathbf{u} \)
Example: If a ball of mass 0.5kg with initial velocity \( (2\mathbf{i} + 3\mathbf{j}) \) is hit and ends up with velocity \( (6\mathbf{i} - 1\mathbf{j}) \), the impulse is just the final momentum minus the initial momentum!
Conservation of Linear Momentum (CLM)
This is a "Golden Rule" in Mechanics: In any collision, the total momentum before the crash equals the total momentum after the crash (provided no external forces like friction are acting).
\( m_A\mathbf{u}_A + m_B\mathbf{u}_B = m_A\mathbf{v}_A + m_B\mathbf{v}_B \)
Quick Review:
• Momentum and Impulse are now vectors.
• Impulse = Change in Momentum.
• Total momentum stays the same before and after a collision.
2. Direct Impact and Newton's Law of Restitution
When two particles hit each other directly (head-on), they don't always stick together. They usually "spring" apart. How "springy" they are is measured by the Coefficient of Restitution, denoted by the letter \( e \).
What is \( e \)?
The value of \( e \) tells us how much "bounciness" there is. It is always between 0 and 1:
• \( e = 0 \): The objects are "perfectly inelastic" (like two pieces of wet clay sticking together).
• \( e = 1 \): The objects are "perfectly elastic" (like super-bouncy balls where no energy is lost).
• \( 0 < e < 1 \): Real-world objects (most collisions you will calculate).
The Restitution Formula
Newton’s Law of Restitution states:
\( e = \frac{\text{speed of separation}}{\text{speed of approach}} \)
To make this easy to calculate, if two particles A and B are moving along a line:
\( e = \frac{v_B - v_A}{u_A - u_B} \)
Note: Make sure your "after" velocities are in the same direction convention as your "before" velocities!
Analogy: The Fast Friend and the Slow Friend
Imagine you are running at 5 m/s (\( u_A \)) towards a friend running at 2 m/s (\( u_B \)) in the same direction. Your speed of approach is 3 m/s (you are catching up by 3 meters every second). After you bump into them, they fly forward at 4 m/s (\( v_B \)) and you slow down to 3 m/s (\( v_A \)). Your speed of separation is 1 m/s. The value of \( e \) would be \( 1/3 \).
Key Takeaway: \( e \) is the ratio of how fast they move apart compared to how fast they were coming together.
3. Loss of Kinetic Energy
In most collisions, the total Kinetic Energy (KE) decreases. This doesn't mean the energy vanished! It just turned into heat, sound, or the energy used to dent the objects.
Calculating the Loss
1. Find the KE before the collision: \( \sum \frac{1}{2}mu^2 \)
2. Find the KE after the collision: \( \sum \frac{1}{2}mv^2 \)
3. Energy Loss = Total KE Before - Total KE After
Did you know? If \( e = 1 \), there is zero loss of kinetic energy. This is the only time kinetic energy is conserved in a collision!
4. Successive Impacts
Sometimes, a question will involve more than one collision. For example, Particle A hits Particle B, and then Particle B goes on to hit a wall or a third Particle C.
Step-by-Step Approach for Multiple Impacts
Don't worry if this seems tricky at first! Just treat it like two separate, smaller problems:
1. Collision 1: Use CLM and the Law of Restitution to find the velocities of the first two particles after they hit.
2. The "Link": The "final" velocity of a particle from the first collision becomes its "initial" velocity for the second collision.
3. Collision 2: Apply the rules again for the next impact.
Impacts with a Smooth Plane Surface (a Wall)
When a particle hits a stationary wall, the wall doesn't move. The rules simplify greatly:
• Speed of approach = \( u \)
• Speed of separation = \( v \)
• Law of Restitution: \( v = eu \)
The particle simply bounces back with its speed multiplied by \( e \). If \( e = 0.5 \), it comes back at half the speed it went in.
Common Mistake to Avoid: When a particle hits a wall and reverses direction, its velocity changes sign (e.g., from positive to negative). When calculating Impulse, remember that \( I = m(v - u) \). If it hits at 10 m/s and bounces back at 5 m/s, the change is \( 5 - (-10) = 15 \), not \( 5 - 10 = -5 \)!
5. Final Summary Checklist
Before you tackle exam questions, check that you are comfortable with these points:
• Conservation of Momentum: \( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \)
• Newton's Law of Restitution: \( e = \frac{\text{sep}}{\text{app}} \)
• Energy Loss: Calculate KE before and after; subtract the smaller from the larger.
• Direction: Always draw a clear diagram with arrows showing which way you’ve decided is "positive." If a ball moves left, its velocity must be negative in your equations!
Top Tip: In most M2 exam questions, you will end up with two equations (one from CLM and one from Restitution) and two unknowns (\( v_A \) and \( v_B \)). Solving these simultaneously is the "bread and butter" of this chapter!