Module title & Course code
Vectors and Mechanics
MAS112
Lecturer
Prof Elizabeth Winstanley
The module begins with the algebra of vectors, essential for the study of many branches of applied
mathematics. The theory is illustrated by many examples, with emphasis on geometry including lines
and planes. Vectors are then used to define the velocity and acceleration of a moving particle, thus
leading to an introduction to Newtonian particle mechanics. Newton's laws are applied to particle
models in areas such as sport, rides at theme parks and oscillation theory.
Course description
Corequisites: MAS100 (Mathematics with Maple); MAS103 (Differential and Difference Equations)
University course content (semester/week)
Edexcel A’level module reference
Autumn
Semester Week 1
Introduction. Definition of a vector, magnitude of a vector, unit vector. Position and displacement
vectors. Basic operations on vectors: addition, subtraction, multiplication by a scalar. Cartesian
components.
Parametric vector equation of a line: r = a + λ c
Parametric vector equation of a plane through the origin: r = λ a + µ b
C4, M1, (GCSE Maths Higher)
Scalar product of two vectors. Geometric definition and computation using Cartesian components.
Use of the scalar product in finding the angle between two vectors.
Application to the vector equation of a plane: r . n = d. Cartesian equation of a plane.
Algebraic properties of the vector product (commutativity, distributivity over addition)
Vector product of two vectors. Geometric definition and computation using Cartesian components
Use of the vector product in finding a vector perpendicular to two given vectors. Triple scalar
product. Geometric interpretation. Coplanar vectors.
Algebraic properties of the vector product ( anti-commutativity, distributivity over addition)
Product of three vectors
Vector equation of a line in the form r x c = d .Perpendicular distance between two lines.
Distance of a point from a line.
Vector triple product. Non-associativity of the vector triple product.
Line of intersection of two planes.
Kinematics: path of a moving particle. Velocity and acceleration. Velocity and acceleration in
components. Geometric interpretation of velocity.
General differentiation of vectors.
Units and dimension. Relative motion: relative displacement.
Relative displacement and relative velocity.
Rules of differentiation, differentiation of scalar and vector products. Differentiation of unit
vectors.
C4
Autumn
Semester Week 2
Autumn
Semester Week 3
Autumn
Semester Week 4
Autumn
Semester Week 5
Autumn
Semester Week 6
Autumn
Semester Week 8
C4
FP3
FP3
Not covered.
FP3
Not covered
FP3
C4/FP3
Not covered
FP3
M2
M2
M1 – only briefly mentioned
M4
Not covered
Autumn
Semester Week 9
Autumn
Semester Week
10
Autumn
Semester Week
11
Autumn
Semester Week
12
Spring Semester
Week 1
Spring Semester
Week 2
Spring Semester
Week 3
Spring Semester
Week 4
Spring Semester
Week 5
Motion with constant acceleration in a straight line. Derivation of standard formulae in this case.
Velocity-time graphs. Motion near the Earth’s surface under gravity in a straight line.
Projectile: basic model and integration of equation of motion (without forces). Path of a projectile.
Maximum height of a projectile above the ground, time of flight on level ground, range on level
ground. Minimum speed of projection to pass through a fixed point.
Region of accessibility.
Forces. Force as a vector. Newton’s Laws of Motion. Momentum. Units of physical quantities.
Weight. Normal contact force. Resolving forces.
M1
Block on an inclined plane. Friction. Impulse. Collision: conservation of momentum. Modelling.
M1
Gravity.
Newton’s Laws of Gravitation. Direction of the gravitational force.
Body close to the Earth’s surface – approximation of constant acceleration due to gravity.
Planetary orbits.
Resisted motion in a straight line under gravity. Resistive force proportional to either v or v2.
Derivation of the equation of motion. Integration of equation of motion. Terminal speed.
M1
M3
Not covered
Circular Motion. Kinematics of motion in a circle (not necessarily with constant angular speed).
Derivation of velocity and acceleration. Motion in a horizontal circle with constant angular speed.
The conical pendulum. Derivation of equation of motion. Features of the motion depending on the
angular speed.
Penny on a flat rotating turntable, maximum angular speed.
M3
Penny on a banked rotating turntable. Angle of friction. Comparison of model with observations.
Not covered (M3 handles motion
on a banked surface)
M2 (not covered)
Spring Semester
Week 6
Work done by a force. Constant force in 3-D. (Non-constant force in a straight line.) Kinetic
energy. Gravitational potential energy.
The work-energy equation. Conservation of mechanical energy. Power. Collisions. Newton’s Law of
restitution. Elastic and non-elastic collisions
Spring Semester
Week 7
Motion in a vertical circle. Kinematics and derivation of equation of motion. Conservation of
energy. Application to hot wheels.
Spring Semester
Week 8
Elasticity. Hooke’s Law. Natural length, tension, stiffness, modulus of elasticity. Elastic potential
energy. Conservation of mechanical energy in problems involving elasticity. Oscillations. Free
oscillations. Horizontal spring. Derivation of equation of motion. Solution of equation of motion.
M2
M2
Not covered
M1
Not covered
M3
M2
M3 (deals with motion in avertical
circle and motion of particles that
can leave the circular path)
M3
Amplitude and frequency of oscillations.
Spring Semester
Week 9
Vertical Spring. Equilibrium and oscillations.
Damped oscillations. Natural frequency. Critical damping.
M3
M4
Spring Semester
Week 10
Forced Oscillations. Forcing frequency, forcing amplitude. Resonance
M4
Key: C4 – “Core Maths module 4”, FP – “Further Pure”, M – “Mechanics” modules. M1 is occasionally part of a an A2 in Single Mathematics or part of the AS
in Further Mathematics, M2 is part of an A2 in Further Mathematics. Occasionally some students self-study M3/M4 in addition to an A2in Further
Mathematics.
Specification as summarised on the departmental website:
Vector geometry
Vectors as displacements of space.(M1) Addition, subtraction, multiplication by a scalar.(GCSE/C4) Position vector.(C4) Cartesian basis, co-ordinates.(C4)
Scalar product.(C4) Vector product.(FP3) Triple products.(FP3) Applications throughout geometry especially lines and planes.(FP3)
Kinematics
The path of a particle given its position vector as a function of time. (M2) Differentiation of vectors with respect to a scalar; velocity, acceleration. (M2)
Motion in a circle with constant speed. (M3)Relative motion. (M4)
Motion with constant acceleration
Motion in a straight line. (M1) Motion near the Earth's surface under gravity. (M1) Projectiles (no air resistance). (M2) Examples from sport.
Newton's laws
Force, momentum. Newton's laws of motion.(M1) Newton's law of gravitation, gravitational acceleration. (M3) Planetary orbits, Kepler's third law. (NC)
Impulse, conservation of momentum. (M1)Types of force. (M1) Resistance proportional to speed : one-dimensional case. (NC)
Circular motion
Kinematics of circular motion. (M3) Conical pendulum. (M3) Normal contact force.(M1) Friction. (M1) Penny on turntable. (M3) Penny on turntable with
banking. (M3) Vertical circular motion. (M3)
Work and energy
Kinetic energy. (M2) Work. (M2) Work-energy equation.(M2) Gravitational potential energy. (M2)Conservation of mechanical energy (KE+PE). (M2) Power.
(M2) Hooke's law. (M3) Elastic potential energy (EPE). KE + PE + EPE = constant. (M3)
Oscillations
Oscillations : horizontal spring, simple harmonic motion, amplitude, frequency. (M3) Vertical spring. (M3) Damped oscillations: damping factor, weak,
strong and critical damping. Forced oscillations, resonance. (M4)