Study Notes: Momentum (Further Mechanics 9231)
Welcome to the Momentum chapter! This is a core topic in Further Mechanics, focusing on what happens when objects collide. While A Level Mechanics introduced you to momentum conservation, Further Mechanics dives deeper, especially into how different types of collisions are modelled using the Coefficient of Restitution (e). Get ready to master collisions in one and two dimensions!
Note: This topic relies heavily on your prior knowledge of resolving forces and using kinematic equations (SUVAT) from A Level Mathematics Mechanics (9709, Paper 4).
1. The Fundamentals of Linear Momentum
Linear momentum is a measure of the mass and velocity of an object. It is a vector quantity, meaning direction matters crucially.
Definition and Conservation
The Linear Momentum (\(p\)) of a particle is defined as the product of its mass (\(m\)) and its velocity (\(v\)):
$$p = mv$$
The standard units for momentum are \(\text{kg m s}^{-1}\) (or sometimes \(\text{N s}\)).
Conservation of Linear Momentum (CLM)
In a closed system where no external forces act (e.g., gravity or friction is ignored during the brief moment of impact), the total linear momentum of the system remains constant.
- For two particles, mass \(m_1\) and \(m_2\), with initial velocities \(u_1\) and \(u_2\), and final velocities \(v_1\) and \(v_2\):
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
Important Tip for CLM:
When solving problems, always define a positive direction. If a velocity goes in the opposite direction, it must be represented by a negative value in the equation.
If a ball of mass 2 kg moving right at \(5 \text{ m s}^{-1}\) hits a wall and rebounds left at \(3 \text{ m s}^{-1}\).
- Initial velocity (\(u\)): \(+5\)
- Final velocity (\(v\)): \(-3\) (since we chose right as positive)
Getting the signs wrong is one of the most common mistakes in momentum problems!
2. Newton's Experimental Law (NEL) and Restitution
Conservation of Momentum gives us one relationship between the initial and final velocities. To solve for two unknowns (\(v_1\) and \(v_2\)), we need a second equation, which comes from Newton's Experimental Law (NEL), also known as the Law of Restitution.
Definition of Coefficient of Restitution (\(e\))
The coefficient of restitution (\(e\)) quantifies how "bouncy" a collision is. It is defined based on the relative speeds of the particles before and after the collision.
$$e = \frac{\text{Speed of separation}}{\text{Speed of approach}}$$
Mathematically, for two particles colliding, NEL is:
$$v_2 - v_1 = e (u_1 - u_2)$$
Where \(u_1, u_2, v_1, v_2\) are the velocities measured along the line of impact, respecting the established positive direction.
The Range of e and Collision Types
The coefficient of restitution is always between 0 and 1, inclusive: \(0 \le e \le 1\).
- 1. Perfectly Elastic Collision (\(e = 1\)):
If \(e=1\), then \(v_2 - v_1 = u_1 - u_2\).
In this ideal case, Kinetic Energy (KE) is conserved. This is the maximum possible bounce.
Did you know? No real-world collision is perfectly elastic, but collisions between billiard balls are often modelled as such because they are close approximations.
- 2. Inelastic (or Perfectly Plastic) Collision (\(e = 0\)):
If \(e=0\), then \(v_2 - v_1 = 0\), meaning \(v_1 = v_2\).
The particles move together as a single mass after the collision. This results in the maximum loss of KE possible while still conserving momentum.
Analogy: This is like a lump of clay hitting a stationary object and sticking to it.
- 3. General Case (\(0 < e < 1\)):
This covers most real-world collisions. Momentum is conserved, but KE is lost (converted into heat, sound, or deformation energy).
NEL must be applied carefully. Remember: \(v_2 - v_1\) is the separation velocity, and \(u_1 - u_2\) is the approach velocity. Ensure the particles are labelled consistently (e.g., Particle 1 is always the one approaching Particle 2).
3. Direct Impact (1D Collisions)
A direct impact (or head-on collision) occurs when the velocities of the particles are aligned along the line connecting their centres (the line of impact).
3.1 Direct Impact Between Two Smooth Spheres
To find the final velocities (\(v_1\) and \(v_2\)), you must solve two simultaneous equations:
Equation 1: Conservation of Momentum (CLM)
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \quad \text{(Always true)}$$
Equation 2: Newton's Experimental Law (NEL)
$$v_2 - v_1 = e (u_1 - u_2) \quad \text{(Always true)}$$
Encouragement: Don't worry if the algebra looks messy! Remember to substitute the expression for \(v_2\) or \(v_1\) from the NEL equation into the CLM equation to solve efficiently.
3.2 Direct Impact with a Fixed Surface
When a smooth sphere hits a fixed surface (like a wall or the ground), the fixed surface is modelled as having infinite mass. This means:
- CLM is not useful: Since the mass of the wall is infinite, we cannot apply the CLM equation effectively.
- NEL is the only principle needed: We only consider the motion perpendicular to the wall (the line of impact).
If a sphere hits a fixed wall perpendicularly with speed \(u\) and rebounds with speed \(v\):
$$v = eu$$
The final speed is simply the initial speed multiplied by \(e\).
4. Oblique Impact (2D Collisions)
Oblique impacts are collisions where the initial velocities are not parallel to the line connecting the centres of the spheres at the point of impact. This requires resolving velocities into components.
The key to solving 2D collisions is understanding which direction the momentum and NEL rules apply.
Identifying the Axes
The collision process only affects motion along the line of impact (LOI). Motion perpendicular to the line of impact (PDOI) is unaffected.
- Line of Impact (LOI): The line joining the centres of the two spheres at the moment of collision. This is where momentum changes and NEL applies.
- Perpendicular to the Line of Impact (PDOI): The common tangent plane at the point of impact. Velocities in this direction are conserved for each particle individually.
Step-by-Step Process for Oblique Impact of Two Spheres
Step 1: Resolve Initial Velocities
- For particle 1 (\(m_1\)), resolve \(u_1\) into:
- \(u_{1L}\) (Component along LOI)
- \(u_{1P}\) (Component along PDOI)
- Do the same for particle 2 (\(m_2\)): \(u_{2L}\) and \(u_{2P}\).
Step 2: Apply Conservation in the PDOI Direction
The velocity components perpendicular to the line of impact remain unchanged.
- Final PDOI velocity for particle 1: \(v_{1P} = u_{1P}\)
- Final PDOI velocity for particle 2: \(v_{2P} = u_{2P}\)
Step 3: Apply CLM and NEL in the LOI Direction
Treat this as a 1D direct collision using only the LOI components. You now have two unknowns, \(v_{1L}\) and \(v_{2L}\):
CLM (LOI):
$$m_1 u_{1L} + m_2 u_{2L} = m_1 v_{1L} + m_2 v_{2L}$$
NEL (LOI):
$$v_{2L} - v_{1L} = e (u_{1L} - u_{2L})$$
Solve these simultaneous equations to find \(v_{1L}\) and \(v_{2L}\).
Step 4: Recombine Final Velocities
The final velocity (\(v_1\)) of particle 1 is the vector sum of its two final components (\(v_{1L}\) and \(v_{1P}\)).
The magnitude is found using Pythagoras:
$$|v_1| = \sqrt{v_{1L}^2 + v_{1P}^2}$$
The direction (angle) is found using trigonometry (usually the angle with the LOI).
Oblique Impact with a Fixed Surface
If a sphere hits a fixed surface obliquely, we use the same principle: resolve into components perpendicular and parallel to the surface.
Let \(\alpha\) be the angle of incidence (the angle between the initial velocity vector \(u\) and the normal to the surface, which is the LOI).
- PDOI (Parallel to Surface / Tangent): Since the surface is smooth, there is no tangential friction or impulse. The velocity component parallel to the surface is conserved:
$$v_{\text{parallel}} = u_{\text{parallel}} = u \sin \alpha$$
- LOI (Perpendicular to Surface / Normal): This is where the collision occurs. We use the simple fixed-surface NEL:
$$v_{\text{normal}} = e u_{\text{normal}} = e (u \cos \alpha)$$
If the final velocity \(v\) makes an angle \(\beta\) with the normal, we can find \(\beta\) using the final components:
$$\tan \beta = \frac{v_{\text{parallel}}}{v_{\text{normal}}} = \frac{u \sin \alpha}{e u \cos \alpha} = \frac{1}{e} \tan \alpha$$
Always draw a clear diagram, mark the Line of Impact (LOI) and the Perpendicular Direction (PDOI). The components along the LOI change, governed by CLM and NEL. The components along the PDOI remain exactly the same.