Collisions in one dimension

Welcome to Collisions in One Dimension!

Ever wondered why car bumpers are designed to crumple during a crash, or why a tennis ball bounces higher than a beanbag? It all comes down to the physics of collisions. In this chapter, we are going to explore how objects move, interact, and exchange energy when they hit each other in a straight line.

Don't worry if Mechanics feels a bit "heavy" at first. We are going to break it down into simple rules that work every single time. By the end of these notes, you’ll be able to predict exactly what happens after two objects collide!

1. Momentum: The "Power" of a Moving Object

Before we talk about crashes, we need to understand Momentum. Think of momentum as how difficult it is to stop a moving object.

A massive truck moving slowly has a lot of momentum. A tiny bullet moving very fast also has a lot of momentum.

The formula is simple:
\( \text{Momentum} = \text{mass} \times \text{velocity} \)
\( p = mv \)

Important Point: Momentum is a vector. This means direction matters! If moving to the right is positive (+), then moving to the left must be negative (-).

Key Takeaway:

To have momentum, an object must be moving. If \( v = 0 \), then \( p = 0 \).

2. Impulse: Changing the Momentum

If you want to change an object's momentum (make it speed up, slow down, or change direction), you need to apply a force over a period of time. This "hit" or "push" is called Impulse (\( I \)).

The Three Ways to Calculate Impulse:

1. As a change in momentum:
If you know the start and end speeds:
\( I = mv - mu \)
(Where \( v \) is final velocity and \( u \) is initial velocity).

2. As Force multiplied by Time (Constant Force):
If the force is steady:
\( I = Ft \)

3. Using Calculus (Variable Force):
Sometimes a force isn't steady (like a golf club hitting a ball—the force starts small, peaks, and then drops). In this case, we integrate:
\( I = \int F \, dt \)

Analogy: Think of catching an egg. If you pull your hands back (increasing the time \( t \)), the force \( F \) decreases, and the egg doesn't break! The total Impulse needed to stop the egg is the same, but you've spread it out.

Quick Review: Impulse is measured in Newton-seconds (\( Ns \)) or \( kg \cdot m/s \). They are the same thing!

3. Conservation of Momentum

This is the "Golden Rule" of mechanics. In any collision where no external forces (like friction) are acting, the total momentum before the collision equals the total momentum after.

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

Step-by-Step for Problems:
1. Draw a diagram: Draw two circles for your objects.
2. Pick a direction: Label which way is positive (usually to the right).
3. Label everything: Write the mass and velocity for "Before" and "After".
4. Watch your signs: If an object is moving left, its velocity is negative!

Key Takeaway:

Momentum is never lost; it just gets handed over from one object to another like a baton in a relay race.

4. Newton’s Experimental Law (The "Bounciness" Factor)

While momentum is always conserved, speed isn't always kept. Some collisions are "bouncy," and some are "sticky." To measure this, we use the Coefficient of Restitution (\( e \)).

The value of \( e \) is always between 0 and 1:
- If \( e = 1 \): The collision is perfectly elastic (perfectly bouncy).
- If \( e = 0 \): The collision is inelastic (the objects stick together).

The Law:

Newton discovered that the speed at which objects move apart is proportional to the speed at which they came together.

Simple Version:
\( \text{Speed of separation} = e \times \text{Speed of approach} \)

The Math Version:
\( v_2 - v_1 = e(u_1 - u_2) \)
Note: The syllabus also expresses this as \( v_1 - v_2 = -e(u_1 - u_2) \). Both work! Just stay consistent with your directions.

Did you know? A "superball" has an \( e \) value close to 0.9, while a lump of wet clay hitting a floor has an \( e \) value of 0.

5. Common Mistakes to Avoid

Don't worry if this seems tricky at first; most students lose marks on simple things. Watch out for these:
- Mixing up units: Ensure mass is in \( kg \) and speed is in \( m/s \).
- Forgetting signs: This is the #1 mistake! If an object bounces back, its final velocity must have a different sign than its starting velocity.
- Misidentifying \( u \) and \( v \): Remember, \( u \) is initial (start) and \( v \) is final (end).

Summary Checklist

Before moving to the next chapter, make sure you can:
- Calculate momentum using \( p = mv \).
- Use \( I = Ft \) or \( I = mv - mu \) to find Impulse.
- Set up a Conservation of Momentum equation for two colliding particles.
- Apply \( e = \frac{\text{separation speed}}{\text{approach speed}} \) to find unknown velocities.
- Use integration to find Impulse if the force is given as a function of time.

Pro-Tip: Most exam questions will require you to use both the Conservation of Momentum and Newton’s Experimental Law. This usually creates two equations with two unknowns—just use simultaneous equations to solve for \( v_1 \) and \( v_2 \)!