Applications to Kinematics | DP IB Maths: AA HL Revision Notes 2021

Kinematics using Vectors

How are vectors related to kinematics?

Vectors are often used in questions in the context of forces, acceleration or velocity
If an object is moving in one dimension then its velocity, displacement and time are related using the formula s = vt

where s is displacement, v is velocity and t is the time taken

If an object is moving in more than one dimension then vectors are needed to represent its velocity and displacement

Whilst time is a scalar quantity, displacement and velocity are both vector quantities

For an object moving at a constant speed in a straight line its velocity, displacement and time can be related using the vector equation of a line

r = a + $λ$ b
Letting

r be the position of the object at the time, t
a be the position vector, r₀ at the start (t = 0)
$λ$ represent the time, t
b be the velocity vector, v

Then the position of the object at the time, t can be given by

r = r₀+ t v

The speed of the object will be the magnitude of the velocity |v |

Examiner Tip

Kinematics questions can have a lot of information in, read them carefully and pick out the parts that are essential to the question
Look out for where variables used are the same and/or different within vector equations, you will need to use different techniques to find these

Worked example

A car, moving at constant speed, takes 2 minutes to drive in a straight line from point A (-4, 3) to point B (6, -5).

At time t, in minutes, the position vector (p) of the car relative to the origin can be given in the form $p = a + t b$ .

Find the vectors a and b.

3-10-2-ib-aa-hl-kinematics-vectors-we-solution

Applications to Kinematics (DP IB Maths: AA HL)

Revision Note

How are vectors related to kinematics?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs

2.9.1 Modulus Functions