Welcome to FM1.3: Collisions in One Dimension!
Hello! This chapter takes the foundational mechanics you know (like forces and motion) and applies them to one of the most exciting events in Physics: collisions. Whether it's a snooker ball hitting another, or a lump of putty sticking to a moving trolley, understanding these impacts requires powerful tools: Momentum and the Coefficient of Restitution.
Don't worry if mechanics sometimes feels abstract. Collisions are highly practical, and we will break down the laws governing these sudden, high-force interactions step-by-step. Let’s get started!
1. Momentum – The Measure of Motion
Before we analyze a crash, we need to know what quantity is conserved (stays the same). That quantity is momentum.
What is Momentum?
Momentum is, quite simply, the measure of an object’s motion, calculated from its mass and its velocity. It tells you how difficult it is to stop a moving object.
- Definition: Momentum (\(p\)) is the product of mass (\(m\)) and velocity (\(v\)).
- Formula: \(p = mv\)
- Units: \(\text{kg m s}^{-1}\) or, equivalently, \(\text{N s}\) (Newton-seconds).
Crucial Point: Momentum is a vector quantity. This means direction is just as important as magnitude. If an object is moving to the right, its momentum is positive; if it moves to the left, its momentum is negative. You MUST define a positive direction at the start of every problem!
Key Takeaway:
Momentum is mass multiplied by velocity, and since velocity is a vector, momentum is too!
2. Impulse – The Force of Impact
A collision is a very short event where a massive force acts on an object, dramatically changing its velocity. The concept used to measure this change is Impulse.
Impulse as Change in Momentum
Impulse (\(I\)) is defined as the total effect of a force acting over a time interval, and it is numerically equal to the change in momentum of the body.
- Formula (Change in Momentum):
\(I = mv - mu\)
Where \(m\) is mass, \(u\) is initial velocity, and \(v\) is final velocity.
Analogy: Imagine catching a tennis ball. Impulse is the total "hit" the ball delivers to your hand. If you catch a heavy ball or a very fast ball, the impulse is greater.
Impulse as Force times Time
Newton’s Second Law (\(F = ma\)) helps us connect impulse to the force and the time duration of the collision.
- Formula (Constant Force):
\(I = Ft\)
Did you know? This link explains why car safety features (like airbags and crumple zones) work. They increase the time (\(t\)) over which the collision happens, which reduces the average force (\(F\)) needed to achieve the same necessary impulse (\(I\)), thereby minimizing injury.
Impulse for Variable Forces (Advanced Note)
Sometimes, the force during a collision is not constant. In this case, we calculate the impulse using integration:
- Formula (Variable Force):
\(I = \int F dt\)
Don't worry if this seems tricky at first. In many FM1 problems, you will be able to assume the force is constant or simply use the change in momentum formula.
Key Takeaway:
Impulse is the ‘push’ that causes a change in momentum (\(I = \Delta p\)), and its units are \(\text{N s}\).
3. The Golden Rule: Conservation of Momentum (CoM)
The most important concept in collision mechanics is that momentum is always conserved in a collision between two bodies, provided no external forces (like friction) are acting in that direction.
The Principle
In any collision (or explosion) involving objects \(M_1\) and \(M_2\):
Total momentum BEFORE collision = Total momentum AFTER collision
Applying the Formula
Let \(m_1\) and \(m_2\) be the masses, \(u_1\) and \(u_2\) be the initial velocities, and \(v_1\) and \(v_2\) be the final velocities (all in the same dimension).
Conservation of Momentum Equation:
\[m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\]
Step-by-step for using CoM:
- Choose a Direction: Define one direction (e.g., right) as positive.
- Input Initial Velocities: If \(u_1\) is positive (moving right) and \(u_2\) is negative (moving left), you must include the negative sign in the equation.
- Input Final Velocities: Typically, \(v_1\) and \(v_2\) are unknown. Assume they are both positive. If your final calculated value for a velocity is negative, it simply means that object moved in the opposite direction (left) after the collision.
Common Mistake to Avoid:
Forgetting that stationary objects still have mass! If a body is initially at rest, its velocity \(u\) is zero, so its initial momentum is zero. Do not just delete the term; write \(m_2(0)\) to show you have considered it.
Quick Review: CoM
CoM gives you one equation relating the four velocities. Since you usually have two unknowns (\(v_1, v_2\)), you need a second equation!
4. The Bounciness Factor: Coefficient of Restitution (e)
The Conservation of Momentum equation alone cannot solve for two unknown velocities. We need to know how "bouncy" the collision is, which is measured by the Coefficient of Restitution (\(e\)).
Newton's Experimental Law
This law relates the relative speed of the two bodies *before* the collision to their relative speed *after* the collision.
\[v_1 - v_2 = -e(u_1 - u_2)\]
This equation can be understood simply as:
\[\text{Speed of Separation} = e \times \text{Speed of Approach}\]
Memory Aid: Remember the acronym COR (Coefficient of Restitution) gives you the second equation needed for collisions.
Interpreting the Value of \(e\)
The coefficient of restitution \(e\) is a number between 0 and 1, inclusive: \(0 \leq e \leq 1\).
- \(e = 1\): Perfectly Elastic Collision
- Kinetic energy is conserved (no energy lost as heat or sound).
- Example: Idealised collision between very hard, bouncy spheres.
- \(e = 0\): Perfectly Inelastic Collision (The Sticky Case)
- Maximum loss of kinetic energy.
- The objects stick together after impact, meaning they have the same final velocity: \(v_1 = v_2\).
- Example: Two lumps of soft clay colliding.
- \(0 < e < 1\): Inelastic Collision
- This is the most common case for real-world collisions.
- Some kinetic energy is lost (usually converted to heat or sound).
- The closer \(e\) is to 1, the bouncier the collision.
The Single Object Bounce
When an object hits a fixed surface (like a wall or the floor), we treat the surface as the second object, \(m_2\), with an initial and final velocity of zero (\(u_2 = 0\), \(v_2 = 0\)).
If a ball approaches the floor with speed \(u\) and separates with speed \(v\), the COR equation simplifies to:
\[(v) - (0) = -e((-u) - (0)) \implies v = eu\]
(Note: You must ensure \(u\) and \(v\) are defined consistently based on your chosen positive direction. Often, we just use the magnitudes here: speed after bounce = \(e \times\) speed before bounce.)
Key Takeaway:
The Coefficient of Restitution, \(e\), provides the second simultaneous equation needed to solve collision problems.
5. Strategy for Solving Collision Problems (The Checklist)
Collisions in one dimension almost always require you to solve two simultaneous equations derived from the two fundamental laws.
Step-by-Step Procedure
- Set up the Diagram and Directions:
- Draw a simple diagram showing the initial masses and velocities (\(m_1, u_1, m_2, u_2\)).
- CRITICAL: Clearly state your chosen positive direction (e.g., "Positive direction is to the right").
- Apply Conservation of Momentum (CoM):
- Write down the CoM equation, ensuring all known velocities moving against the positive direction are written as negative numbers.
- (Equation 1: \(\sum mu_{initial} = \sum mv_{final}\))
- Apply the Coefficient of Restitution (COR):
- Write down Newton’s Experimental Law: \(v_1 - v_2 = -e(u_1 - u_2)\).
- Substitute the known values for \(e, u_1,\) and \(u_2\).
- (Equation 2: Relating \(v_1\) and \(v_2\))
- Solve Simultaneously:
- Use substitution or elimination to solve Equation 1 and Equation 2 for the unknown final velocities \(v_1\) and \(v_2\).
- Interpret the Results:
- If a calculated velocity is positive, the object moves in the defined positive direction.
- If a calculated velocity is negative, the object moves opposite to the defined positive direction.
Working with Impulse in Problems
Sometimes, you might be asked to find the impulse exerted on one of the colliding bodies.
If you want to find the impulse exerted on body 1, use the formula:
\[I_1 = m_1 v_1 - m_1 u_1\]
Remember, by Newton’s Third Law (action and reaction), the impulse on body 2 will be equal in magnitude but opposite in direction: \(I_2 = -I_1\).
Keep practising these steps. Once you master defining directions and handling the two simultaneous equations, you will find collisions manageable!